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https://mathoverflow.net/questions/140840 | 9 | Are equivalent (in ZFC) the following two statements, for any infinite cardinal $\mu$?
(i) For every infinite cardinal $\kappa$, $|\{ \lambda \in \kappa : \lambda \textrm{ is a singular cardinal and} \operatorname{pp} (\lambda) \geq \kappa \}| \leq \mu$.
(ii) For every infinite cardinal $\kappa$, $|\{ \lambda \in \... | https://mathoverflow.net/users/39086 | Some variants of the Shelah's Weak Hypothesis | They are equivalent, though it took me a while to see it. This is perhaps my fifth attempt at getting a proof, so caveat lector.
Clearly (i) implies (ii), so assume by way of contradiction that (i) fails while (ii) holds.
Choose $\kappa$ least such that
$$|\{\lambda<\kappa:\rm{pp}(\lambda)\geq\kappa\}|>\mu.$$
We as... | 7 | https://mathoverflow.net/users/18128 | 140936 | 76,947 |
https://mathoverflow.net/questions/139795 | 25 | I am thinking about reading a course on motivic integration. I have already read certain introductions to the subject; yet I am not sure that they mention all the significant parts of the theory. So, my questions are:
1. Which papers and results are the most important for the theory of motivic integrations?
2. What a... | https://mathoverflow.net/users/2191 | Most significant results in motivic integration theory? | As for the first two questions (papers, results, and applications): For motivation, I'd recommend understanding the content of Batyrev's paper ["Birational Calabi-Yau n-folds have equal Betti numbers"](http://arxiv.org/abs/alg-geom/9710020) which proves the claim in its title. Using motivic integration techniques analo... | 21 | https://mathoverflow.net/users/6950 | 140943 | 76,949 |
https://mathoverflow.net/questions/140828 | 3 | For $1\leq k \leq n+1$, consider the set $S\_k$ of functions $f:\{1,\ldots,n\} \rightarrow \{1,\ldots,n+1\}$, with the property that $|f^{-1}\{1,\ldots,k\}| < k$. Note that $|S\_1|=n^n$, and $|S\_{n+1}| = (n+1)^n$. Is it true that $|S\_k|$ grows with $k$?
It's not hard to come up with the formula
$$ |S\_k| = \sum\_{... | https://mathoverflow.net/users/39359 | Counting discrete functions | OK, once I went into "preaching", I feel I am obliged to post a solution too.
Let's consider $n$ i.i.d. Bernoulli variables $X\_j$ such that $P(X\_j=1)=\frac k{n+1}$ and $P(X\_j=0)=1-\frac k{n+1}$. We want to show that the probability of the event $\sum\_j X\_j<k$ is increasing in $k$. Note that we can model it as $n... | 5 | https://mathoverflow.net/users/1131 | 140948 | 76,952 |
https://mathoverflow.net/questions/140906 | 3 | Let C be a compact convex subset of 3-dimensional Euclidean space E(3) whose interior is non-empty and whose diameter is d. What is the largest volume that C can have if every subset of C that is a straight line segment of length d is a subset of the boundary of C (or-equivalently-if no chord of C that contains an inte... | https://mathoverflow.net/users/4423 | A question about maximizing the volume of a particular kind of convex set | I would suggest yet another larger set. Take two points $A$ and $B$ in the plane with $|AB|=d$. Let $S$ be the set of all points $X$ in this plane lying in one halfplane defined by $AB$ and satisfying $|XA|,|XB|\leq d$. Then rotate $S$ around the perpendicular bisector of $AB$; you get the convex body containing your c... | 2 | https://mathoverflow.net/users/17581 | 140959 | 76,956 |
https://mathoverflow.net/questions/140942 | 4 | Let $M$ be a manifold, $TM$ its tangent bundle, and $N:TM\to TM$ a vector bundle morphism. It is possible to find a torsionless linear connection $\nabla$ on $TM$ such that $\nabla N=0$?
| https://mathoverflow.net/users/39404 | Existence of connections making a bundle endomorphism parallel | My answer is almost contained in the answers/comments of Mariano Suárez-Alvarez, Ben McKay and Robert Bryant; I summarize the answers and give a reference.
There exists a torsion-free connection such that a given endomorphism $A$ is parallel if and only if the following two conditions are fulfilled:
(1) The Jordan t... | 6 | https://mathoverflow.net/users/14515 | 140961 | 76,957 |
https://mathoverflow.net/questions/140954 | 23 | Not so long ago I took a class called "Discrete analysis". I remember that I couldn't find any "novice" level material on Mobius functions in combinatorics. So then I went to the roots and read Rota's original paper "On the foundations of combinatorial theory I" and it really impressed me. So I wonder is there other ma... | https://mathoverflow.net/users/39110 | Papers better than books? | Very recently I and Misha Sodin had a strong incentive to learn the Ito-Nisio lemma (which, roughly speaking, says that weak convergence in probability of a series of symmetric independent random variables with values in a separable Banach space implies almost sure norm convergence to the same limit). The textbooks we ... | 45 | https://mathoverflow.net/users/1131 | 140966 | 76,960 |
https://mathoverflow.net/questions/140965 | 1 | To avoid trival cases, we assume that $f$ is neither a constant nor a finite Blaschke product.
Two celebrated theorems of Frostman say that $f\_a(z)$ is actually a Blaschke product for every $|a|<1$ with the possible exception of a set of logarithmic capacity zero and if $w=f(z)$ is not reducing to a finite Blaschke ... | https://mathoverflow.net/users/30754 | Some questions about inner functions | I think that the answer to all three questions is no. The simplest is 3.
Take any inner function $f\neq 1$ with arbitrary non-zero singular measure, and
infinitely many zeros.
Then $f$ is an not a Blaschke product,
thus $0\in E\_2(f)\backslash E\_1(f)$, and the singular measure of $f=f\_0$
is arbitrary.
Negative an... | 3 | https://mathoverflow.net/users/25510 | 140974 | 76,963 |
https://mathoverflow.net/questions/140746 | 8 | Let $G$ be the group $\mathbb{Z}^2\rtimes\_{\sigma} \mathbb{Z}=\langle y,z\rangle\rtimes\_{\sigma}\langle x\rangle$, where $\sigma(x)=\begin{pmatrix}a, b\\c,d\end{pmatrix}\in SL\_2(\mathbb{Z})$, which means that we have relations $xyx^{-1}=y^az^c, xzx^{-1}=y^bz^d$. Then we can form the group ring $R=\mathbb{Z}G$, note ... | https://mathoverflow.net/users/9305 | For $G=\mathbb{Z}^2\rtimes \mathbb{Z}$, $Spec(\mathbb{Z}G)$=? | The natural map $\mathbb{Z} \to \mathbb{Z}G$ has central image and therefore induces a map between prime spectra $Spec(\mathbb{Z}G) \to Spec(\mathbb{Z})$. The preimage of the ideal generated by $(p)$ under this map is in a natural bijection with $Spec( kG )$ where $k = \mathbb{F}\_p$ if $p$ is a prime number and $k = \... | 11 | https://mathoverflow.net/users/6827 | 140978 | 76,967 |
https://mathoverflow.net/questions/140937 | 0 | Suppose that $M\subseteq \mathbb{Z}^n$ is a module such that $\mathbb{Z}^n/M$ is free and $S\subseteq \mathbb{R}^n$ is a bounded, symmetric (around $0$) convex set. Let $M'$ be the module generated by $S\cap M$.
Question: Is $\mathbb{Z}^n/M'$ free?
I think it is free if the following is true: for any $x\in M\setmi... | https://mathoverflow.net/users/nan | sublattice generated by lattice points intersecting a convex set | This is true if $M$ has rank at most $2$ but not beyond that.
For a counterexample in rank $r \geq 3$, choose coordinates
so that $M$ is the body-centered cubic lattice, that is, the
subgroup of ${\bf Z}^r$ consisting of all-even and all-odd vectors;
and let $S$ be the $l^1$ ball of radius $2$, that is,
$$
S = \{ (x\_1... | 2 | https://mathoverflow.net/users/14830 | 140986 | 76,972 |
https://mathoverflow.net/questions/140991 | -1 | I am trying currently why we (the mathematicians) imply that whenever the derivative of a real function $f$ is involved, always the domain $D\_f$ of $f$ should be open interval or at least semi-open (then we talk about the right or left derivative). $Thanks$ $in$ $advance$.
| https://mathoverflow.net/users/39414 | derivative of a real function without open domain | Hassler Whitney proposed a definition for $C^1$ function (and more generally $C^k$ and $C^\infty$) for a function defined on a subset of $\mathbb R^n$ that is not necessarily open.
Whitney, Hassler,
Trans. Amer. Math. Soc. 36 (1934), no. 1, 63–89.
[LINK](http://www.ams.org/journals/tran/1934-036-01/S0002-9947-19... | 4 | https://mathoverflow.net/users/454 | 140994 | 76,975 |
https://mathoverflow.net/questions/140969 | 18 | Let $T$ be the theory consisting of Zermelo's original set theoretic axioms (extensionality, empty set, pairing, union, powerset, infinity, separation, choice) together with foundation. Put more succinctly, $T$ consists of ${\rm ZFC}$ axioms without the replacement axiom scheme. The theory $T$ is too weak for most set ... | https://mathoverflow.net/users/5984 | Does the existence of the von Neumann hierarchy in models of Zermelo set theory with foundation imply that every set has ordinal rank? | Take the Zermelo ordinals to be defined by $Z(0) = 0$, $Z(\alpha+1) = \{Z(\alpha)\}$, and $Z(\lambda) = \{Z(\alpha): \alpha<\lambda\}$ (where $\alpha, \lambda$ are von Neumann ordinals). Then if we add $Z(\omega+ \omega)$ to $V\_{\omega +\omega}$ and close under pairing, union, subsets, and powersets, we get a model of... | 12 | https://mathoverflow.net/users/17968 | 141000 | 76,980 |
https://mathoverflow.net/questions/141004 | 5 | Consider a smooth tame Deligne-Mumford stack $[Y/G]$, a point $[p]$ on it with stabilizer group $H$. Is it true that every representation of $H$ can be extended to a locally free sheaf on $[Y/G]$?
Alternatively, consider a smooth scheme (or algebraic space) $Y$ with a $G$ action. Let $p$ be a point on $Y$ with a fini... | https://mathoverflow.net/users/10332 | Extend a representation of a stabilizer group on a smooth DM stack to a locally free sheaf? | Unfortunately that is not always possible. For instance, let $Y$ be $\mathbb{A}^n$, and let $\rho:G\times \mathbb{A}^n\to \mathbb{A}^n$ be a faithful, linear representation. Then the locally free sheaves on $[Y/G]$ are the same as $G$-representations. If there exists a point $p$ of $Y$ whose stabilizer is $H$, then you... | 9 | https://mathoverflow.net/users/13265 | 141005 | 76,981 |
https://mathoverflow.net/questions/140693 | 3 | Tarski proved that if a group $G$ is exponentially bounded, then for $a$, $b$ and $c$ in the associated (equidecomposability) type semigroup, we have $a+c=b+2c \Rightarrow a=b+c$.
**Question:** Can this Tarski condition on the type semigroup be proved if one replaces "exponentially bounded" with "supramenable"? (I a... | https://mathoverflow.net/users/26809 | Does supramenability imply that $a+c=b+2c \Rightarrow a=b+c$ on the type semigroup? | The Tarski condition $a+c=b+2c\Rightarrow a=b+c$ is equivalent to supramenability, given AC.
Proof:
First, note that the Tarski condition on the type space $S=S(G)/G$ is easily equivalent to the condition that $a+b=2a\Rightarrow a=b$. (This is called strong separativity, I am told.) This is equivalent to the condit... | 1 | https://mathoverflow.net/users/26809 | 141014 | 76,985 |
https://mathoverflow.net/questions/141008 | 2 | I wonder what is the most explicit characterization that can be given for the solution to the ($N$-dimensional) problem of maximizing the criterion
$$
-\textrm{trace}[AS^{-1}] - b^\top Sb
$$
over positive semidefinite (symmetric) $S$ is, where $A$ is also positive semidefinite, and nonzero $b \in [0,\infty)^N$.
... | https://mathoverflow.net/users/39430 | optimization over positive semidefinite matrices | $S$ must be positive-definite, not just positive-semidefinite, else
$AS^{-1}$ does not exist. Suppose $A$ is positive-definite,
and let $A^{1/2}$ be its positive-definite square root.
Then the supremum over positive-definite $S$ of
$-{\rm tr}(AS^{-1}) - b^\top S b$ is $-2 |A^{1/2}b|$,
as in the scalar case. But once $N... | 2 | https://mathoverflow.net/users/14830 | 141018 | 76,986 |
https://mathoverflow.net/questions/140997 | 10 | Let $t\_n$ be a sequence of real numbers and $C,r>1.$ Suppose that for every $n\geq 1$ we have $\frac{1}{C}r^n\leq t\_n \leq Cr^n.$ Does there exist a real number $\xi$ and an $\varepsilon>0$ such that $|| \xi t\_n ||\geq \varepsilon$ for every $n\geq1$?
Here $|| x ||$ denotes the distance between $x$ and the nearest... | https://mathoverflow.net/users/34640 | Distribution mod 1 of exponential growth sequences | I think the answer is yes. For any natural number $n$, let $P(n)$ denote the assertion that there exists an interval $I\_n$ of length $\sqrt{\varepsilon}/t\_n$ such that $\| \xi t\_m \| \geq \varepsilon$ for all $\xi \in I\_n$ and $m \leq n$. If $\varepsilon$ is small enough, it appears that $P(n)$ implies $P(n+A)$ for... | 10 | https://mathoverflow.net/users/766 | 141025 | 76,988 |
https://mathoverflow.net/questions/141034 | 1 | Let $p\_i\in (c,1-c)$ for some fixed $c\in(0,1)$ . Consider a sum $X=\varepsilon\_{1}+\cdots+\varepsilon\_{n}$ where $\varepsilon\_{i}$ are independent Bernoulli random variables with parameters $p\_{i}$. Let $Z$ be a normal random variable with the same mean and variance as $X$. I would like to approximate probabiliti... | https://mathoverflow.net/users/24494 | Local limit theorem for Bernoulli sums | There are a lot of results along these lines in
>
> V. V. Petrov, Sums of independent random variables, Springer-Verlag, 1975
>
>
>
| 2 | https://mathoverflow.net/users/nan | 141043 | 76,990 |
https://mathoverflow.net/questions/141044 | 1 | Let $A$ be a matrix. If $A$ is "almost" equal to $A^\*$, it follows from an argument of continuity that the eigenvalues of $A$ are "almost" real. Same argument can be made for $A$ "almost" $-A^\*$, in which case the eigenvalues are "almost" purely imaginary.
Question: Is there a way of getting a quantitative estimate o... | https://mathoverflow.net/users/39451 | Estimate on the real and imaginary parts of eigenvalues | Suppose $A=B+C$ where $B$ is self adjoint. Then you can diagonalize $B$, and then by applying Gersgorin's criterion, you get the following: if $\{\lambda\_i\}$ are the eigenvalues of $B$ and $\hat \lambda$ is an eigenvalue of $A$, then
there is an $i$ so that $|\hat \lambda-\lambda\_i|\leq \|C\|\_2$.
(Here, $\|C\|\_2$ ... | 1 | https://mathoverflow.net/users/35520 | 141047 | 76,991 |
https://mathoverflow.net/questions/141036 | 0 | I hope this question is well suited for this site; please excuse me if not.
I recently read that the value of $\delta(x^2)$ is an open question [1], with $\delta(x)$ the Dirac delta. Now I'm trying to get my head around what $\delta(|x|)$ might be, where $|x|$ is the absolute value of x. I know from [2] that
$\delta(... | https://mathoverflow.net/users/39447 | Dirac delta composed with absolute value | Let $\kappa:\mathbb R\rightarrow\mathbb R$ be a diffeomorphism with $\kappa(0)=0$. Mimicking the change of variable formula, we would like to have
$$
\int \delta(\kappa(x))\vert \kappa'(x)\vert \phi(\kappa(x))dx=\int \delta(y)\phi(y) dy,
$$
where the integrals should be replaced by brackets of duality.
We shall in fact... | 2 | https://mathoverflow.net/users/21907 | 141070 | 77,000 |
https://mathoverflow.net/questions/141067 | 2 | Let $SL(n, \mathbb{R})$ be the group of $n \times n$ invertible matrices of determinant $1$ in real numbers. Let $G:=SL(n, \mathbb{R}\_{\geq 0})$ be its subgroup $\{M \in SL(n,\mathbb{R}) \mid M, M^{-1} \text{ both have non-negative entries}\}$. Is there any known results on this group? I am particular interest in how ... | https://mathoverflow.net/users/29730 | Subgroup of $SL(n,\mathbb{R})$ with positive entries | This is answered as Pietro says in <http://archive.numdam.org/ARCHIVE/CM/CM_1969__21_4/CM_1969__21_4_376_0/CM_1969__21_4_376_0.pdf>
In general semigroup theorists have heavily studied maximal subgroups of semigroups of nonnegative matrices.
Again all maximal subgroups are isomorphic to the group of monomial matric... | 7 | https://mathoverflow.net/users/15934 | 141073 | 77,001 |
https://mathoverflow.net/questions/141065 | 8 | According to Deligne's "yoga of weights", the cohomology of an algebraic variety should have a weight filtration. For concreteness we can consider the rational cohomology of complex varieties, with their mixed Hodge structure.
It seems to me that in the yoga of weights there is a kind of duality between singularitie... | https://mathoverflow.net/users/1310 | Duality between singularities and non-compactness in the yoga of weights | This may be already clear to you, but from my perspective, the clearest manifestation of this duality is in the setting of mixed Hodge modules (or some other version of ``mixed sheaves'').
Let $f: X \to Y$ be a morphism of complex algebraic varieties, and let $D\_m(X)$ and $D\_m(Y)$ refer to the derived categories o... | 6 | https://mathoverflow.net/users/7762 | 141074 | 77,002 |
https://mathoverflow.net/questions/141080 | 4 | I really couldn't figure out the answer to the following question: Let $X$ be a scheme of finite type over a field $k$ and let $K$ be an extension field of $k$. Let $X\_K := K \times\_k X$ be the base extension and let $p: X\_K \rightarrow X$ be the projection. Is it true that $p$ maps closed points to closed points?
... | https://mathoverflow.net/users/8070 | Closed points of field extension of k-scheme under projection | No, it is not ! For instance, if $k=\bar{\mathbb{Q}}$, $K=\mathbb{C}$ and $X$ has dimension $\geq 1$, there is exactly one closed point of $X\_K$ above each closed point of $X$. But there is only countably many closed points of $X$, whereas $X\_K$ has uncountably many closed points.
In general, a point of $X$ will be... | 9 | https://mathoverflow.net/users/2868 | 141083 | 77,006 |
https://mathoverflow.net/questions/141081 | 2 | Let $F:\mathbb{C}\to \mathbb{C}$ be a homogeneous map of degree $k$ (i.e., $F(tx)=t^kF(x)$, $t>0$). It is true that $F$ has topological degree less than or equal to k?
This is true if F is polynomial!
| https://mathoverflow.net/users/39472 | Topological degree of homogeneous function of degree k | No. Consider the map $F(r\,\cos(\theta),r\,\sin(\theta))=(r\cos(n\theta),r\,\sin(n\theta))$. This is homogeneous of degree $1$ but has topological degree $n$.
| 5 | https://mathoverflow.net/users/10366 | 141084 | 77,007 |
https://mathoverflow.net/questions/140914 | 5 | I have been reading Levitt's paper *Automorphisms of Hyperbolic groups and Graphs of Groups*. I am having some trouble trying to fit all the bits together, and would appreciate some help with this last step.
In the paper, Levitt considers *minimal* graphs of groups, and gives results regarding (a specific subgroup of... | https://mathoverflow.net/users/35478 | Automorphisms of Hyperbolic groups and Graphs of Groups | This has been dealt with in comments, but since MO works better if answers are given, I'll elaborate a little bit here.
A graph of groups satisfying your definition of 'minimal' is usually called 'reduced'. You're absolutely correct that Bowditch's JSJ is often not reduced (though note that, in your example, the grap... | 6 | https://mathoverflow.net/users/1463 | 141085 | 77,008 |
https://mathoverflow.net/questions/37278 | 40 | There are several questions in the Euler-Goldbach correspondence that
I am unable to answer. Sometimes it does not take very much: in his
letter to Goldbach dated June 9th, 1750, Euler conjectured
that every odd number can be written as a sum of four squares
in such a way that $n = a^2 + b^2 + c^2 + d^2$ and $a+b+c+d... | https://mathoverflow.net/users/3503 | Euler and the Four-Squares Theorem | We address the problem of Euler, showing an asymptotic lower bound for the
number of ways of writing $n\equiv 2 \pmod 4$ as a sum of four squares $a^2+b^2+c^2+d^2$
where neither $a^2+b^2$ nor $c^2+d^2$ is divisible by any prime $\equiv 3 \pmod 4$. For simplicity
we shall only count the solutions where $a^2+b^2\equiv... | 31 | https://mathoverflow.net/users/38624 | 141096 | 77,009 |
https://mathoverflow.net/questions/141095 | 3 | For the past few weeks I've been trying to get myself acquainted with the language and basic theory of linear algebraic group schemes. In an attempt to see whether I have learned enough to read a paper that I once told myself that I would read once I know enough about linear algebraic groups, I have been stumped by the... | https://mathoverflow.net/users/5309 | Why does the expression "the largest quotient of a linear algebraic group that is multiplicative type" make sense? | This is a subtle question, since the formation of that quotient need *not* commute with extension of the ground field (and it is not said if the ground field is algebraically closed, or separably closed, or what).
Let's take up the story at the start; I assume you are working over a field. Let $k$ be a field, and let... | 6 | https://mathoverflow.net/users/36938 | 141097 | 77,010 |
https://mathoverflow.net/questions/90603 | 9 | *Are there (known) examples of non-isometric Riemannian metrics on the projective plane that have the same length spectrum?*
This question is related to MO questions [Length spectrum and Zoll surfaces of revolution](https://mathoverflow.net/questions/90530) and [Length spectrum of spheres](https://mathoverflow.net/qu... | https://mathoverflow.net/users/21123 | Length spectrum for Riemannian metrics in the projective plane | The answer is positive; in fact any smooth manifold has two nonisometric metrics with conjugate geodesic flows. A construction is in C. Croke, B. Kleiner, Conjugacy and rigidity for manifolds with a parallel
vector field. J. Differential Geom. 39(1994), 659–680.
The idea is quite simple: consider the followins two b... | 5 | https://mathoverflow.net/users/14515 | 141110 | 77,012 |
https://mathoverflow.net/questions/141103 | 1 | By the hypercube I mean the lattice formed by all n-bit strings ordered by pointwise inequality. For example, $000 \leq 110$, $010 \leq 110$, $110$ and $001$ are not comparable. Further we have the meet and join operations $\wedge$ and $\vee$ that take the pointwise max and min. For example $010 \wedge 110 = 010$ and $... | https://mathoverflow.net/users/11541 | Submodular measures on the hypercube | These submodular measures on the hypercube are the same as
non-negative [submodular set
functions](http://en.wikipedia.org/wiki/Submodular_function). A minimal set of inequalities is $$ f(x+e\_j)+f(x+e\_j)\geq
f(x)+f(x+e\_i+e\_j) $$ over all $x\in\{0,1\}^n$ and $1\leq i < j\leq n$
such that $x\_i=x\_j=0$. I'm using $e\... | 3 | https://mathoverflow.net/users/35453 | 141111 | 77,013 |
https://mathoverflow.net/questions/141108 | 5 | Can the number of minimal vertex covers of a graph be super-polynomial (like exponential)? I suspect it can, but can't think of any examples.
**Vertex cover** $C$ of a graph $G$ is a subset of its vertices that any edge has an incident vertex in that set. That is:
$$
C\subseteq V(G)
\hspace{1cm}
\text{s.t.}
\hspace{1... | https://mathoverflow.net/users/39492 | Bound on the number of minimal vertex covers of a graph | The union of $k$ triangles has $3^k$ minimum vertex covers. You can easily find connected examples.
| 11 | https://mathoverflow.net/users/9025 | 141116 | 77,015 |
https://mathoverflow.net/questions/141119 | 14 | Ordinal numbers are generalizations of natural numbers. In this sense the "proper class" of all ordinals ($Ord$) is very similar to "infinite" set of all natural numbers ($\omega$). In the other direction we know that many large cardinal axioms are generalizations of the properties of $\omega$ and without assumption of... | https://mathoverflow.net/users/nan | How strong are large cardinal properties of Ord? | $\newcommand{\Ord}{\text{Ord}}
\newcommand{\ZFC}{\text{ZFC}}$
Here is one way to formalize your concept a little more tightly,
which provides the answers to your questions. For any large
cardinal property $P$, let's take the phrase "*$\Ord$ is $P$*" to
be the theory asserting $\sigma$, for any sentence $\sigma$ that
... | 18 | https://mathoverflow.net/users/1946 | 141135 | 77,023 |
https://mathoverflow.net/questions/141130 | 3 | If we look at reverse mathematics statements as mass problems, considering the class of solutions of an instance, it is known that Weak König's lemma has a maximal instance in the sense that there is an infinite computable binary tree $T$ such that for every path $P$ in the tree and every infinite computable binary tre... | https://mathoverflow.net/users/8833 | Reverse mathematics, Ramsey theorem and mass problem | This is due to Joe Mileti and can be found in [his thesis](http://www.math.grinnell.edu/~miletijo/research/thesis.pdf), Corollary 5.4.7.
| 5 | https://mathoverflow.net/users/2000 | 141136 | 77,024 |
https://mathoverflow.net/questions/141102 | 1 | Is there a nice/unique way to express the pseudofunctor laws (<http://ncatlab.org/nlab/show/pseudofunctor>) when you don't have horizontal composition, only whiskering? I don't want to define horizontal composition because it's only unique up to homotopy, as per the argument in [chapter 9 of the HoTT book](http://books... | https://mathoverflow.net/users/30462 | Expressing the pseudofunctor laws uniquely with whiskering rather than horizontal composition? | I did not realize that all the horizontal compositions listed on the [nlab page](http://ncatlab.org/nlab/show/pseudofunctor) had the identity on one side or the other. Since whiskering is equivalent to horizontal composition with the identity, it's easy to rephrase all the horizontal compositions as whiskerings.
| 1 | https://mathoverflow.net/users/30462 | 141148 | 77,029 |
https://mathoverflow.net/questions/141063 | 3 | Let $A$ and $B$ be finitely generated $\mathbf{Z}$-algebra. Suppose that there exists two coprime integers $m$ and $n$ and an isomorphism of $\mathbf{Z}$-algebra
$\phi:A\otimes\_{\mathbf{Z}}\mathbf{Z}[1/n]\simeq B\otimes\_{\mathbf{Z}}\mathbf{Z}[1/m] $. Then we can glue $A$ and $B$ along $\phi$ so that we obtain
a schem... | https://mathoverflow.net/users/11765 | When is the gluing of two finite type affine Z-schemes affine? | By Zariski glueing on affine schemes,
$$C=\{(a,b)\in A×B\mid\phi(a\otimes1)=b\otimes1\}.$$
Of course the beautiful thing in this story is the answer to Q1: if $f:Y\to X$ is an affine morphism and $X$ is an affine scheme, then $Y$ is affine. Once you know this, the only possible answer to Q2 is the above.
| 6 | https://mathoverflow.net/users/7666 | 141154 | 77,032 |
https://mathoverflow.net/questions/141125 | 1 | Let $k$ be a finite field of char $p \geq 3$. Given an absolutely irreducible, continuous, odd representation $\overline{\rho}: G\_\mathbb{Q} \longrightarrow GL\_2(k)$ and a deformation condition $D$ for $\overline{\rho}$, let $S(D)$ be the collection of all newforms with associated $p$-adic representation in $D$. If $... | https://mathoverflow.net/users/5310 | Bounding the level for eigenforms satisfying a deformation condition | Let's bound the level of such an $f$ in two stages. Firstly, let's look at a prime $\ell \ne p$. Here there is a theorem of Livne and (independently) Carayol which says that if $\rho$ is a lifting of $\bar\rho$, the exponent of $\ell$ dividing the Artin conductor of $\rho$ is bounded (it's at most 2 more than the $\ell... | 0 | https://mathoverflow.net/users/2481 | 141161 | 77,035 |
https://mathoverflow.net/questions/141150 | 1 | Let $p\equiv 5 [8]$ be a prime number, and consider $K=\mathbb{Q}(\sqrt{-p})$.
I would like to check that the $2$-Sylow subgroup of the class group $C\_K$ has order $2$ (I'm pretty sure it's true).
Apparently, this can be done using genus theory, but I don't know anything about it or class field theory , really.
... | https://mathoverflow.net/users/36683 | 2-class group of a quadratic imaginary extension | Here's a simple argument using Hilbert's theorem 90. No doubt it's the same as Will's (deleted) argument couched in quadratic form language, and I suppose it's the same as the Frohlich -Taylor argument as well.
First one shows that the 2-torsion in the class group has order 2 and is generated by the prime P of norm 2... | 3 | https://mathoverflow.net/users/6214 | 141166 | 77,037 |
https://mathoverflow.net/questions/141167 | 6 | In *Embedding theorems for groups*, (J. London Math. Soc. 34 1959 465–479.) Neumann and Neumann (NB: this is not the Higman-Neumann-Neumann paper of the same name) make the following definition.
**Definition:** A subgroup $H$ of a group $G$ is an *E-subgroup* of $G$ if for every normal subgroup $N\triangleleft H$, th... | https://mathoverflow.net/users/29437 | Is there an agreed-upon name for this type of subgroup? | I think the magic acronym is CEP. See <http://en.wikipedia.org/wiki/CEP_subgroup>
**Update** I've just noticed that the question is also tagged reference-request, so [here's](http://arxiv.org/abs/math/0208237) one (Google reveals many, I'm not sure if there is a canonical one): *Non-amenable finitely presented torsio... | 6 | https://mathoverflow.net/users/18263 | 141170 | 77,038 |
https://mathoverflow.net/questions/141157 | 36 | After having read Gunnar Carlsson's [*Topology and Data*](http://www.ams.org/journals/bull/2009-46-02/S0273-0979-09-01249-X/S0273-0979-09-01249-X.pdf) I feel enthusiastic to use some topological data analysis (TDA) methods in my current research, mostly in social sciences. We often handle huge databases and I think it ... | https://mathoverflow.net/users/39229 | Inference using Topological Data Analysis: Is it worth it for a regular statistician to learn TDA? | Let me answer the broad question first: depending on what you actually want to do, the barcode-type invariants extracted by topological data analysis could be quite useful in your work. And it doesn't take too much prior knowledge to *use* the TDA tools. For instance, if all you want to do is show that two datasets are... | 26 | https://mathoverflow.net/users/18263 | 141174 | 77,039 |
https://mathoverflow.net/questions/141173 | 24 | I am an undergraduate student. I am not sure if it's OK to ask this question here.
I want to learn Hodge theory. But I do not know how to start it, and how much mathematics I should need before I read Deligne's paper.
Is there an elementary book or note on Hodge theory for undergraduate students? Is it worthy to r... | https://mathoverflow.net/users/39470 | The prerequisites for Deligne's Théorie de Hodge I, II, III | I would recommend Voisin's "Hodge Theory and Complex Algebraic Geometry" as an introduction to Hodge theory--Volume I should suffice for your purposes. The book also does a bit of the Hodge theory of non-compact varieties, in Section 8.4. The relevant sections in Griffiths-Harris aren't bad either. Deligne's papers als... | 32 | https://mathoverflow.net/users/6950 | 141186 | 77,041 |
https://mathoverflow.net/questions/141179 | 0 | Let $P$ be a homogenous polynomial with real coefficients in several variable(at least three variable)
Is the following statement true:
>
> For every $\epsilon$ there is a $\delta$ such that for every x with $|P(x)|< \delta$ we have $d(x,Z)<\epsilon$.
>
>
>
Here $Z=P^{-1}(\{0\})$ is the set of roots of $P$, an... | https://mathoverflow.net/users/36688 | Ulam stability of homogeneous polynomials | EDIT. My previous answer was incorrect. So I replace it.
The answer is no. A counterexample is
$$y^{2m}+(z^{m-1}y-x^m)^2.$$
This is of degree $2m$ but $\delta$ is like $\epsilon^{2m^2}$ near the
point $(0,0,1)$.
I found this example in the paper of Kollar and Shiffman, TAMS 329 (1992), on the very first page. They cr... | 1 | https://mathoverflow.net/users/25510 | 141187 | 77,042 |
https://mathoverflow.net/questions/141171 | 4 | Looking for literature / known results on the following class of problems:
Consider the domain bounded, open $\Omega\in \mathbb R^2$ with smooth boundary, divergence free drift $u=u(x,t)$, scalar field $T=T(x,t)$
with no slip and steady Dirichlet conditions:
$u(\partial\Omega,t)=0\:\forall t,\: T(\partial\Omega,t)=T\... | https://mathoverflow.net/users/30684 | Boundary flux maximizing drift (velocity) vector fields for 2D heat equation | this may not be exactly what you are asking but may have some related materials..
<http://www.math.cmu.edu/~gautam/research/papers/200911-bad-mixing-2d.pdf>
| 2 | https://mathoverflow.net/users/29444 | 141193 | 77,045 |
https://mathoverflow.net/questions/141172 | 12 | The category of abelian groups $\mathsf{Ab}$ is the $\mathcal{Ind}$-completion of the full subcategory of finitely presentable abelian groups $\mathsf{Ab}\_{fp}$. This is not so special, since the analogous statement holds for any finitary variety e.g. groups, rings, boolean algebras etc.
However one nice property of... | https://mathoverflow.net/users/5152 | Looking for concrete description of a category derived from abelian groups | Very interesting problem! I'd not seen it before, but from what I can make out, it looks as though this category can be described concretely as having for its objects triples $(A, T, i: A \otimes \mathbb{Q}/\mathbb{Z} \to T)$ where $A$ is an abelian group, $T$ is a torsion abelian group, and $i$ is an injective homomor... | 7 | https://mathoverflow.net/users/2926 | 141195 | 77,046 |
https://mathoverflow.net/questions/141198 | 4 | While working on a research problem (algebraic cycles), I bumped into a question that I want to prove, though I couldn't yet prove. After several days of attempts, I realized that if the following statement on projective geometry holds, then my original question is most likely answered affirmatively.
>
> **Statemen... | https://mathoverflow.net/users/3168 | A question on infinitely many closed points on a smooth projective variety and their behavior under embeddings | Let me explain why this statement cannot be true in general. I will give a counterexample where $X$ is a complex K3 surface.
By a result of [Beauville and Voisin, On the Chow ring of a K3 surface], it is possible to find a point $y\in X$ such that whenever $C\_1$, $C\_2$ are curves on $X$, $C\_1\cap C\_2$ is proporti... | 9 | https://mathoverflow.net/users/2868 | 141206 | 77,052 |
https://mathoverflow.net/questions/141068 | 1 | Is the space $\psi$ (described in problem **5I** of *L. Gillman and M. Jerison, Rings of continuous functions, Springer Verlag, 1976*) a F-Z-space (i.e, space with $cl(X-Z(f))$ is a zero set for every $f$ in $C(X)$)?
$\textbf{Clarification}$ A collection $\mathcal{A}$ of infinite subsets of $\mathbb{N}$ is said to b... | https://mathoverflow.net/users/38926 | The space $\psi$ | As I commented above, I think the answer is that a $\psi$-space need not be an FZ-space, and that a counterexample may be constructed from a Luzin gap. Here are the details, which did not fit into the comment.
We first construct the MAD family, which will give the counterexample. Start by splitting ${\mathbb N}$ int... | 4 | https://mathoverflow.net/users/25700 | 141210 | 77,055 |
https://mathoverflow.net/questions/141194 | 4 | For a non-constant polynomial $A \in \mathbb{Z}[x]$, let $\mathcal{P}(A)$ denote the set of prime numbers $p$ which divide $A(n)$ for some integer $n$. If $\mathcal{P}(A) \subseteq \mathcal{P}(B)$ for some $A,B$, does there necessarily exist $C \in \mathbb{Q}[x]$ such that $A|B\circ C$? (Here, $B \circ C = B(C(x))$ is ... | https://mathoverflow.net/users/14456 | Set of primes dividing polynomials and composition | I guess this answer complements Gene's answer above. Here is an example to think about. Let
$$
A=(x^2-2)(x^2-17)(x^2-34).
$$
It's an easy exercise in quadratic reciprocity to show that $\mathcal{P}(A)$ is the set of all primes. Let
$$
B=(x^2-2)(x^2-41)(x^2-82).
$$
In the same way $\mathcal{P}(B)$ is the set of all prim... | 9 | https://mathoverflow.net/users/4140 | 141211 | 77,056 |
https://mathoverflow.net/questions/141184 | 8 | An $n$-dimensional submanifold $L$ of a symplectic manifold $(M^{2n}, \omega)$ is called Lagrangian if $\omega|\_L = 0$. I want to get some feeling about how many Lagrangian submanifolds are.
For each $\alpha \in H\_n(M)$, is there a Lagrangian submanifold representing $\alpha$? Maybe it's not a good idea to distingu... | https://mathoverflow.net/users/11846 | How many Lagrangian submanifolds? | The answer to the first question is 'no'. For example, if $M = S^2\times S^2$ is given the product symplectic structure, then there is no Lagrangian submanifold in the homology class of $S^2\times\{x\}$.
The answer to your second question is 'yes, the notion exists', but the question is whether this set can be endowe... | 6 | https://mathoverflow.net/users/13972 | 141213 | 77,058 |
https://mathoverflow.net/questions/141202 | 7 | I [asked this question on math.se](https://math.stackexchange.com/questions/475664/is-an-ideal-generated-by-multilinear-polynomials-of-different-degrees-always-rad) and someone even put a bounty on it, yet there was no answer. Hence, I am asking here. Assume $\Bbbk$ to be a field of characteristic zero.
>
> **Defin... | https://mathoverflow.net/users/9947 | Is an ideal generated by multilinear, irreducible, homogeneous polynomials of different degrees always radical? | One general fact that comes to mind: If an ideal $I\subset \mathbb{k}[x\_1,\dots,x\_n]$ contains an element of the form $f = gx\_1 + h$ where $g,h$ don't use $x\_1$, and $g$ is a nonzerodivisor mod $I$, then the primary components of $I\cap \mathbb{k}[x\_2,\dots,x\_n]$ and $I$ are in bijection. This is *birational proj... | 12 | https://mathoverflow.net/users/5495 | 141215 | 77,059 |
https://mathoverflow.net/questions/141225 | 3 | In a paper of Heier and Wong, It is written that from a pointwise argument due to Berger does follow that the scalar curvature (and thus also the total scalar curvature) of a Kaehler metric of positive holomorphic sectional curvature is also positive.
I am looking for the paper of Berger ("Sur les varietέs d'Einstein c... | https://mathoverflow.net/users/39546 | holomorphic sectional curvature and total scalar curvature | The answer to your question is exactly the same of the answer to [this](https://mathoverflow.net/questions/42051/negative-holomorphic-sectional-curvature) older question of mine. Enjoy!
| 3 | https://mathoverflow.net/users/9871 | 141227 | 77,063 |
https://mathoverflow.net/questions/141234 | 14 | Salem numbers and Lehmer's minimum height problem are venerated not only in number theory and diophantine analysis, where they are considered naturally interesting for their own sake, but also in fields such as hyperbolic geometry and holomorphic dynamics. As is so well known, the least known Salem number is a root $1.... | https://mathoverflow.net/users/26522 | Occurrences of D. H. Lehmer's 10-th degree polynomial | Here's a paper of McMullen where the Lehmer polynomial shows up (see Theorem 1.2 there):
<http://www.math.harvard.edu/~ctm/papers/home/text/papers/blowup/blowup.pdf>
| 7 | https://mathoverflow.net/users/38624 | 141235 | 77,064 |
https://mathoverflow.net/questions/141177 | 15 | Proposition 6.3.2.18 of Higher Algebra identifies $Mod\_{Sp}(Pr^L)$, the symmetric monoidal category of right modules over the monoidal category $Sp$ of spectra in $Pr^L$ the category of presentable categories, with the full subcategory $Pr^L\_{St}$ of stable presentable infinity categories. In particular Lurie proves ... | https://mathoverflow.net/users/333 | Lower Algebra: Modules over the monoidal category of abelian groups | A locally presentable category $\mathcal{C}$ has a (unique) structure of an $Ab$-module if and only if it is additive. Such a category need not be abelian.
This is one reason to prefer the setting of stable $\infty$-categories to the theory of abelian categories. The identification of presentable stable infty-categor... | 19 | https://mathoverflow.net/users/7721 | 141239 | 77,066 |
https://mathoverflow.net/questions/141236 | 10 | Let $e$ be an index of an oracle Turing machine program and $k$ be some natural number. Let us say that a subset of $\mathbb N$ is *arithmetic* if it is definable in the model $\langle \mathbb N,+,\cdot,<,0,1\rangle$. Now suppose that there is a non-arithmetic oracle $A$ such that $\Psi\_e^A(k)\uparrow$. Is it possible... | https://mathoverflow.net/users/5984 | If an oracle Turing machine halts with every infinite arithmetic oracle, can it fail to halt with some non-arithmetic oracle? | Consider an oracle Turing machine $M$ which enumerates its oracle $A=\{n\_0<n\_1<n\_2<\dots\}$ by querying every $n\in\omega$ in increasing fashion, and whenever it hits a new element $n\_k\in A$, it halts unless all the following conditions are met:
* $n\_k$ is the Gödel number of a finite set $T\_k$ of arithmetic s... | 9 | https://mathoverflow.net/users/12705 | 141242 | 77,068 |
https://mathoverflow.net/questions/141241 | 10 | Let $X\_r\subset Mat\_{n\times n}$ denote the matrices of rank at most
$r$, and let $S\_{\pi}C^n$ denote the irreducible $GL\_n$-module corresponding
to the partition $\pi$.
One can check that
degree($X\_r$)=dim($S\_{(n-r)^{n-r}}C^n)$=$\Pi\_{i=0}^{n-r-1}\frac{(n+i)!i!}{(r+i)!(n-r+i)!}$
Does anyone have a geometric (o... | https://mathoverflow.net/users/21399 | Why does the degree of the variety of rank at most $r$ $n\times n$ matrices equal dim$S_{(n-r)^{n-r}}C^n$? | There are a number of better statements (i.e. yes, this is "well-known").
To begin with, wouldn't you rather have the *character* of this representation, instead of just its dimension? You can get that as the class $[X\_r] \in H^\*\_{T^n}(Mat\_{n\times n})$. This improves on your statement, because for $X \subseteq V... | 14 | https://mathoverflow.net/users/391 | 141285 | 77,085 |
https://mathoverflow.net/questions/141267 | 5 | Background
----------
Consider $BU=colim \, BU\_k$ where we take $BU\_k$ to be the specific model of classifying space for the group $U(k)\subseteq O(2k)$ given by the quotient space of the infinite real Stiefel manifold $V\_{2k}$ by the action of $U(k)$. The spaces $BU\_k$ as described come with maps $f\_k : BU\_k \... | https://mathoverflow.net/users/4517 | $(BU,f)$ structures on manifolds via stable normal bundles and stable tangent bundles | This is a question about general vector bundles, not tangent/normal bundles. So let $X$ be a finite complex, and $V,W \to X$ be two real vector bundles such that $V \oplus W$ is trivial*ized* and $n$-dimensional.
If a stable complex structure on $V$ is given, pick an embedding $\iota:V \oplus \mathbb{R}^r \to X \times ... | 4 | https://mathoverflow.net/users/9928 | 141296 | 77,092 |
https://mathoverflow.net/questions/141284 | 8 | I have the indefinite quadratic form $q(x,y,z) = 19 x^2 + 5 y^2 - z^2.$ It's not my fault. I find, on reflection, that I have no idea how to describe the orthogonal group of this over the integers. The thing is isotropic in $\mathbb Z,$ for example $19 + 45 = 64, $ and $76 + 5 = 81.$ Even if that were not the case, the... | https://mathoverflow.net/users/3324 | Integral orthogonal group for indefinite ternary quadratic form | Edit : this is a new answer, after more computations.
Let $H$ be the subgroup of your orthogonal group that preserve globally each connected component of the (two-sheeted) space $q(x,y,z)=-1$.
Up to this action, there is a single isometry class of isotropic vectors.
One representant is $(-1\ -3\ 8)$ and its stab... | 7 | https://mathoverflow.net/users/39552 | 141300 | 77,093 |
https://mathoverflow.net/questions/141276 | 2 | Let $r(t), t\in [0,1]$ be a continuous piecewise $C^1$ curve on the plane where $r(0)=(0,0)$ and $r(1)=(1,0)$. The distance $|r(t)|$ is a non-decreasing function and the distance $|r(t)-(1,0)|$ is a non-increasing function. $r\_m(t), t\in [0,1]$ defined below (a semi vesica piscis) is such a curve.
$$
\begin{equation}
... | https://mathoverflow.net/users/32660 | Is vesica piscis a maximal length curve constrained to two points? | The length of these curves is unbounded. For a positive integer $n$ consider a [triangular wave](https://en.wikipedia.org/wiki/Triangle_wave)
$f\_n:[0,1]\to\mathbb{R}$ with support on $[1/3,2/3]$, making $n$ (isosceles) triangular impulses on $[1/3,2/3]$ with $\|f\_n(x)\|\_\infty=\frac{1}{6\sqrt n}$. The graph of $f\_... | 3 | https://mathoverflow.net/users/6101 | 141302 | 77,094 |
https://mathoverflow.net/questions/141301 | 5 | Let $X, Y$ be quasi-projective Noetherian schemes and $f:X \to Y$ be a projective surjective morphism. Assume that every fiber of $f$ is isomorphic to a projective space $\mathbb{P}^n$ for a fixed $n$. Is it then true that $f$ is flat?
| https://mathoverflow.net/users/38832 | Is projective morphism with projective fiber flat? | If $Y$ is non-reduced, then $X = Y\_{red} \times \mathbf{P}^{n}$ is a counterexample.
| 9 | https://mathoverflow.net/users/7721 | 141315 | 77,101 |
https://mathoverflow.net/questions/133862 | 2 | Suppose we have a typical logdet function $\mathcal{L}$ with respect to a covariance matrix $\mathbf{A}$,
$$
\mathcal{L}(\mathbf{A}) = \log\vert \mathbf{I} + \mathbf{A}\mathbf{S} \vert - \mathbf{q}^T(\mathbf{A}^{-1} + \mathbf{S})^{-1} \mathbf{q},
$$
where $\mathbf{S}$ is a Symmetric Positive Semi-Definite Matrix, $\mat... | https://mathoverflow.net/users/11273 | Hessian of function of covariance matrices | I am back with good news: you have nothing to do because there are (in general) no critical points! It is the case if, in particular, $S$ is invertible.
Recall that $\mathcal{L}(A)=log|I+AS|-q^T(I+AS)^{-1}Aq$ and (according to my previous post)
$D\mathcal{L}\_A=\mathcal{L}'(A)=H\rightarrow tr((I+AS)^{-1}HS)-q^T(-(I+AS... | 2 | https://mathoverflow.net/users/9091 | 141326 | 77,105 |
https://mathoverflow.net/questions/141259 | 0 | Are there any (there should be) references on complete intersections in Grassmanian? Especially on calculating the cohomolgy of some sheaves naturally associated to the complete intersection. For example, if $X$ is the complete intersection, how to compute $\rm{Hom}^i(X,\mathcal{O}\_X),\rm{Hom}^i(X,\mathcal{O}\_X(n)),\... | https://mathoverflow.net/users/29730 | References on complete intersections in Grassmanian | You can look at the Koszul resolution of $\mathcal O\_X$; all the terms will be direct sums of $\mathcal O\_G(-n\_i)$, where $\mathcal O\_G(1)$ is the ample generator of $\mathrm{Pic}(G)$ and $n\_i>0$ (by $G$ I denote the Grassmannian in question). The spectral sequence will be very easy to deal with thanks to Borel-We... | 1 | https://mathoverflow.net/users/29992 | 141335 | 77,109 |
https://mathoverflow.net/questions/141339 | 0 | Last year Bob Harper wrote a blog post about the failure of "Church's Law" in Extensional Type Theory[1]. However his statement of the law looks to me more like an internal version of the statement "all functions are computable" and I am not surprised that this turns out to be false.
To falsify the law I would have i... | https://mathoverflow.net/users/3676 | Does "Church's Law" really fail in Extensional Type Theory? | I think there's a terminological distinction at work here. "Church's Thesis" (also called the Church-Turing Thesis) says that the intuitive notion of computability agrees with (or "is adequately captured by") its various proposed formal versions (Turing computability, lambda definability, representability in various fo... | 7 | https://mathoverflow.net/users/6794 | 141343 | 77,111 |
https://mathoverflow.net/questions/141287 | 7 | I recently started reading Étienne Ghys slides on knots and dynamics <<http://www.umpa.ens-lyon.fr/~ghys/articles/icm.pdf>> which seem very interesting. I know this approach to knots and dynamics is not entirely new, for instance, this is from the 1980's <<http://www.math.columbia.edu/~jb/bw-KPO-I.pdf>>. This post gave... | https://mathoverflow.net/users/39229 | Knots and Dynamics. Recent breakthroughs? | It's not clear when Ghys made the slides to which you have linked. The only date I could find in those was 1963 (referring to the Lorenz equations), which would make the bound on "recent" rather generous. Here's a quick summary of relatively recent concrete activity in this area that might be interesting to you.
In 1... | 4 | https://mathoverflow.net/users/18263 | 141348 | 77,114 |
https://mathoverflow.net/questions/141346 | 2 | I am not sure where one looks up this type of fact. Google was not very helpful.
| https://mathoverflow.net/users/4002 | Do unbounded chain complexes have enough injectives? | Yes, any Grothendieck abelian category has enough injectives. I believe this goes back to Grothendieck's Tohoku paper. The category in question is Grothendieck abelian since it is equivalent to a category of additive presheaves. The domain category has objects the integers, and morphisms generated by $d\_n: n \rightarr... | 6 | https://mathoverflow.net/users/1649 | 141349 | 77,115 |
https://mathoverflow.net/questions/141360 | 1 | We can define internal categories in a monoidal category like [this](http://ncatlab.org/nlab/show/internal+category+in+a+monoidal+category). Let $C$ be a dagger symmetric monoidal category. Will $C$ be locally finitely presentable? Let $C\_{int}$ be the category of internal categories in $C$. Is $C\_{int}$ locally fini... | https://mathoverflow.net/users/10007 | lfp property for dagger symmetric monoidal categories and their internal categories | For a [dagger-category](http://ncatlab.org/nlab/show/dagger-category) $C$, we have $C^{op} \simeq C$. But the only time that a category $C$ and its opposite can both be locally finitely presentable is when $C$ is a poset (this is covered in the text by Adámek and Rosicky). So the answer to the first question is: only w... | 5 | https://mathoverflow.net/users/2926 | 141362 | 77,118 |
https://mathoverflow.net/questions/141262 | 0 | Hi I am trying to understand a proof in a paper (written by Isaac Sonin), I don't know if anyone could give me a clarification on the following:
Firstly we have a Markov chain $\{Y\_k\}$ with finite state space $X\_1$ and a probability transition matrix $P\_1=\{p\_1(x,y)\}$. Let $D \subset X\_1$ and $\tau\_1, \tau\_2, ... | https://mathoverflow.net/users/38208 | Markov Chain: state reduction | Before starting you have to note that the definition of $P\_2$ and $P\_1$ at the end of the question are not the same as the definitions at the top of the question. At the top, they're matrices, whereas at the bottom, they are probability measures on $H\_1$ and $H\_2$. More specifically, $P\_1(x,B)$ means $\mathbb P((X... | 1 | https://mathoverflow.net/users/11054 | 141366 | 77,119 |
https://mathoverflow.net/questions/138896 | 7 | **1.** *Does the following integral converge ?*
$$\int\_0^\infty \frac{b(x)}{B(x)} dx$$
*where*
$$b(x) = \sum\_{n=1}^\infty \frac{n^x}{n^n} \qquad and \qquad B(x) = \sum\_{n=1}^\infty \frac{n^x}{n!}$$
**2.** *Does it possess a closed form, or some other alternative expression ?*
| https://mathoverflow.net/users/39602 | Convergence and Closed Form of an Integral Involving Bell Numbers | The question on convergence is certainly not research level.
I'm too lazy to give a detailed answer, but here is a direction in which I believe one can obtain a proof with a little effort.
Obviously, all we need is to estimate the asymptotics of $B(x)$ and $b(x)$ for $x \to \infty$. Using a variant of [Laplace's meth... | 4 | https://mathoverflow.net/users/22758 | 141369 | 77,121 |
https://mathoverflow.net/questions/141363 | 3 | I want a reference for Stasheff operad, where operad maps are defined explicitly at the point-set level. I would also like to ask the question that what exactly do one mean by Stasheff operad? Is there a specific pointset model that one thinks or is the description in terms of trees good enough to call that a Stasheff ... | https://mathoverflow.net/users/19186 | Reference for Stasheff Operad | Why not look at Stasheff's original paper? He does give a point-set model (where $K\_{n+2}$ is a compact convex semialgebraic subset of $\mathbb{R}^n$) and describes explicitly the substitution maps $\text{sub}\_i: K\_m \times K\_n \to K\_{m+n-1}$ which are collectively tantamount to the operad structure.
* James Di... | 7 | https://mathoverflow.net/users/2926 | 141374 | 77,123 |
https://mathoverflow.net/questions/141372 | 7 | It's a consequence of the uniformization theorem for simply connected Riemann surfaces that the universal cover of $\mathbb{C}\setminus(\mathbb{Z}\oplus i\mathbb{Z})$ ($\mathbb{C}$ punctured at all the integral lattice points) is the upper half plane $\mathcal{H}$.
How should I think about this map? How does the map ... | https://mathoverflow.net/users/15242 | Universal covering map from $\mathcal{H}$ to $\mathbb{C}\setminus \mathbb{Z}\oplus i\mathbb{Z}$ (the countably punctured complex plane) | On the first question (the universal cover of the complement of a lattice). The missing points are in the image, so it is not the map that "behaves" but the inverse map.
The inverse map behaves in a very simple way: it has infinitely many "logarithmic singularities" over each missing point.
"How to think about the map... | 12 | https://mathoverflow.net/users/25510 | 141386 | 77,129 |
https://mathoverflow.net/questions/139784 | 3 | Let $f:M\to M$ be a partially hyperbolic diffeomorphism. That is, there exists a continuous splitting $TM=E^u\oplus E^c\oplus E^s$ into unstable, center and stable bundles. It is well known that there exist foliations $\mathcal{W}^u$ and $\mathcal{W}^s$ tangent to $E^u$ and $E^s$, respectively.
Let's assume $f$ is dy... | https://mathoverflow.net/users/11028 | Center-stable manifolds | No, it is in general not true. An example is the one recently constructed by Rodriguez Hertz-Rodriguez Hertz-Ures. In that example (non-dynamically coherent in dimension 3) the union of stable manifolds through a center manifold is strictly contained in the center-stable manifold (its boundary is a strong stable manifo... | 1 | https://mathoverflow.net/users/5753 | 141393 | 77,131 |
https://mathoverflow.net/questions/141317 | 9 | Let $ ~~\cup\_{k=-1}^{\infty} U\_k = \mathbb{R} $ be an open covering of
$\mathbb{R}$. It is a well known fact that partitions of unity subbordinate to
the cover exists, i.e. there exists smooth
functions $ \varphi\_{k} : U\_k \rightarrow \mathbb{R} $ with compact
support such that
$$ \sum\_{k=-1}^{\infty} \varphi... | https://mathoverflow.net/users/4463 | Do partitions of unity exist if we impose additional conditions on the derivatives? | I hope the following construction will give you what you really need. If not, you'll have to explain why.
Take any nice locally finite covering $\mathbb R\subset\cup\_j U\_j$ and take any smooth partition of unity $1=\sum\_j\psi\_j^2$ subordinated to this covering. Take any smooth positive function $F$ on $\mathbb R$... | 6 | https://mathoverflow.net/users/1131 | 141395 | 77,132 |
https://mathoverflow.net/questions/141397 | 1 | I was experimenting with various presentations for groups, and I stumbled upon $G := \langle a, b \ | \ a^2, b^3, (ab)^7, [aba,b]^6 \rangle$. I found that it has order 11741184, but the magma calculator won't give me much more than that. What I would like to know is: What are the composition factors of this group? I kn... | https://mathoverflow.net/users/38744 | Help understanding a group | Actually, I just figured it out. The groups
$H := \langle a, b \ | \ a^2, b^3, (ab)^7, [a,b]^{28}, [aba,b]^6 \rangle$ and
$I := \langle a, b \ | \ a^2, b^3, (ab)^7, [a,b]^{8}, [aba,b]^6 \rangle$ are quotients of this group, and H has composition series: PSL(2,7)-PSL(2,13), and the second has composition series PSL(2,7... | 2 | https://mathoverflow.net/users/38744 | 141407 | 77,136 |
https://mathoverflow.net/questions/140979 | 7 | A link between formal series convergence in deformation quantization (strict deformation quantization) and producing $C^\*$-algebras instead of mere $\*$-algebras (which $(\mathcal{C}^{\infty}(M)[[t]],\star)$ is) after deforming the initial commutative $C^\*$-algebra of observables is evoked here: <http://ncatlab.org/n... | https://mathoverflow.net/users/37661 | Formal series convergence in deformation quantization and $C^*$-condition | OK, let me give a try on this question. There are several problems hidden underneath which one has to address.
First, for physical reasons a formal deformation is not sufficient. $\hbar$ is a constant of nature but not a formal parameter... More severely, the formal star product algebras do not allow for a reasonable... | 6 | https://mathoverflow.net/users/12482 | 141410 | 77,138 |
https://mathoverflow.net/questions/135089 | 19 | Consider the stack $Ell$ (of groupoids) of elliptic curves. I'm interested in the autoequivalence [2-group](http://ncatlab.org/nlab/show/2-group) of $Ell$, the objects of which consists of transformations $Ell \Rightarrow Ell: Ring \to Gpd$ valued in equivalences of groupoids. The arrows are isomorphisms of such transf... | https://mathoverflow.net/users/4177 | What is $Aut(Ell)$? | The 2-group is $B\mathbb Z/2$. In other words, the automorphism $1$-group of $M\_{1,1}$ is trivial, and the identity functor $M\_{1,1} \to M\_{1,1}$ has exactly one non-identity invertible natural transformation to itself: the one which sends a family of elliptic curves $\xi \colon E \to S$ to $\xi \circ i \colon E \to... | 3 | https://mathoverflow.net/users/1310 | 141418 | 77,142 |
https://mathoverflow.net/questions/141417 | 11 | Myers-Steenrod states that the isometry group of a Riemannian manifold is a Lie group. Is that also true for pseudo Riemannian manifolds? I didn't find anything related to that.
Cheers
| https://mathoverflow.net/users/39631 | Isometry group of pseudo Riemannian manifold always a Lie group? (Myers-Steenrod) | Yes. Check out Kobayashi, **Transformation Groups in Differential Geometry**, theorem 4.1 page 16, and example 2.5 page 8. The automorphisms of a pseudo-Riemannian manifold form a Lie group, as do the automorphisms of a conformal pseudo-Riemannian manifold (in dimension 3 or more), and the automorphisms of a projective... | 14 | https://mathoverflow.net/users/13268 | 141419 | 77,143 |
https://mathoverflow.net/questions/141415 | 10 | I am interested if there is an example of an infinite finitely generated non-amenable group that is residually finite but does not contain non-abelian free subgroups.
What examples of infinite finitely gnerated perfect (non-simple) gropus are non-amenable but do not contain free subgroups?
Many thanks,
Elisabeth
| https://mathoverflow.net/users/23232 | Infinite finitely generated non-amenable groups | Osin and Luck give examples of infinite finitely generated residually finite torsion groups with positive first $\ell^2$-Betti number (and hence non-amenable) in the paper "Approximating the first L2-Betti number of residually finite groups", J. Topol. Anal. 3 (2011), no. 2, 153–160.
<http://www.worldscientific.com/d... | 9 | https://mathoverflow.net/users/6460 | 141427 | 77,146 |
https://mathoverflow.net/questions/140956 | 5 | To calculate the `between centrality` [wiki def](http://en.wikipedia.org/wiki/Betweenness_centrality):
$g(v) = \sum\_{s\neq v \neq t} \frac{\sigma\_{st}(v)}{\sigma\_{st}}$
of a node in a graph/network;$\sigma\_{st}$ is the total number of shortest paths from node to node and the $\sigma\_{st}(v)$ are the paths includi... | https://mathoverflow.net/users/19684 | Methods to approximate the betweenness centrality on large networks | Stochastic approximation methods for betweenness centrality have been studied by many people. A good reference is ["Centrality estimation in large networks" by Brandes and Pich (2007)](http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.68.9850)
For large sparse networks, exact and approximation algorithms can be... | 5 | https://mathoverflow.net/users/39635 | 141430 | 77,149 |
https://mathoverflow.net/questions/141437 | 5 | This is probably a very basic question, but I don't know the answer and I also don't see how it might be obvious (which it very well might be).
Given a topological space $X$, when is the set of all its continuous self-maps generated by the subset consisting of
* the self-homeomorphisms of prime order (i.e., there e... | https://mathoverflow.net/users/8590 | When are all continuous self-maps of a topological spaces generated by retractions and self-homeomorphisms of prime order? | This is most likely pretty rare. Let G be any finite group. Then Birkhoff proved that G is the automorphism group of a finite distributive lattice L. The G is the group of homomorphisms of the space of prime filters on L with the usual kernel-hull topology. Thus any finite group is the homeomorphism group of a finite t... | 6 | https://mathoverflow.net/users/15934 | 141441 | 77,154 |
https://mathoverflow.net/questions/141436 | 2 | Let $\pi :C\rightarrow \mathbb{P}^{1}$ be a cyclic cover of degree $m$ of $\mathbb{P}^{1}$. So $C$ has an action of $\mathbb{Z}/m\mathbb{Z}$. Let $\xi$ be a primitive $m$-th root of unity. Consider the cohomology group $V\_{\mathbb{Q}}=H^{1}(C,\mathbb{Q})$. In the book "Cylic covers, Calabi-Yau manifolds and complex mu... | https://mathoverflow.net/users/37808 | Confusion about two statements about cohomology of curves with automorphisms | I think you will find that the space is a $\mathbb Q[x]/(x^m-1)$, which is a product of fields, including $\mathbb Q(\xi)$, where $\xi$ is a primitive $m$th root of unit. Thus it decomposes as a sum of vector spaces over different fields. There is no reason that $\mathbb Q(\xi)$ should be the only field, and thus no re... | 1 | https://mathoverflow.net/users/18060 | 141446 | 77,157 |
https://mathoverflow.net/questions/141378 | 42 | Is anyone familiar with the following, or anything close to it?
Lemma. Suppose $A$, $B$ are nonzero finite-dimensional vector spaces
over an infinite field $k$, and $V$ a subspace of $A\otimes\_k B$
such that
(1) For every nonzero $a\in A$ there exists nonzero $b\in B$
such that $a\otimes b\in V$,
and likewise,
... | https://mathoverflow.net/users/39608 | Is this lemma in elementary linear algebra new? | This is a nice lemma: I know a good deal of similar results but this one is unknown to me.
I believe it is suitable, as an answer, to give a proof that works with no restriction on the cardinality of the underlying field $F$. I will frame the answer in terms of matrix spaces.
Thus, we have a linear subspace $V \sub... | 19 | https://mathoverflow.net/users/34951 | 141448 | 77,159 |
https://mathoverflow.net/questions/141298 | 5 | I allready asked this on MO, but did not get any answer.
**Given a finite quiver with relations. When is the path algebra modulo relations hereditary?**
If the path algebra is finite dimensional or there are no relations, the answer is well known. **What happens if the path algebra is infinite dimensional and there... | https://mathoverflow.net/users/32972 | When are infinite dimensional path algebras hereditary? | The algebra is $A=kQ/I$ with the assumptions $I \subseteq J^2$ and $I/(JI+IJ) \neq 0$, where $J$ is the ideal in $kQ$ generated by the arrows.
Assume $A$ is hereditary. Then $J/I$ is a projective $A$-module and the exact sequence of $A$-modules $$0 \rightarrow I/IJ \rightarrow J/IJ \rightarrow J/I \rightarrow 0$$ mus... | 2 | https://mathoverflow.net/users/18756 | 141449 | 77,160 |
https://mathoverflow.net/questions/141355 | 12 | It follows from [this question](https://mathoverflow.net/questions/140200/on-super-connected-graphs)
and the corresponding answers, that the complete graphs and the cycles are precisely the graphs
$G$ having the property that, for every spanning tree $T$ of $G$, the set of leaves of $T$ is a
clique in $G$.
Motiva... | https://mathoverflow.net/users/18117 | Graphs in which every spanning tree is an independency tree | A graph G has all spanning trees independency if and only if G does not contain two adjacent vertices v and w, neither of degree one, such that the graph G' formed by removing v and w and all their incident edges is connected. (I think this is the same as what McKay said in a comment.)
For, if G' is connected, then a... | 11 | https://mathoverflow.net/users/440 | 141457 | 77,162 |
https://mathoverflow.net/questions/141382 | 13 | Let $n\geq 2$. Is it true that any $n\times n$ matrix with entries from a given ring (with identity) can be written as a sum of two invertible matrices with entries from the same ring ?
| https://mathoverflow.net/users/nan | Writing a matrix as a sum of two invertible matrices | **The answer is negative.**
There is a nice theorem of **M. Henriksen** which says that **If $n\geq 2$ then every element of $M\_n(R)$ is a sum of three units** also he proves that there are non-unit matrices in $\bf{M\_2(\Bbb{Z}\_2[x\_1,x\_2])}$ that can not be written as a sum of two units. You can find a copy of t... | 19 | https://mathoverflow.net/users/nan | 141458 | 77,163 |
https://mathoverflow.net/questions/141275 | 1 | For numerically solving a partial differential equation (PDE) what advantage does operational calculus (OC) has over common methods like finite difference (FD), and finite element (FE)?
I mean OC in the sense of Heaviside, as developped by Mikusinski.
I know that the choice between FE and FD, is a matter of taste,... | https://mathoverflow.net/users/nan | What are the advantage of using operational calculus for numerically solving pde compared to FE or FD? | In my humble opinion, the use of HOC (Heaviside Operationnal Calculus) is limited to equations with constant coefficients witch is a drastic restriction if one wants to analyse a physical phenomenon with great precison. In Oprationnal methods by Maslov a description of a special OC witch could be used to tract PDE with... | 0 | https://mathoverflow.net/users/36539 | 141465 | 77,165 |
https://mathoverflow.net/questions/141387 | 21 | $\newcommand{\N}{\mathbb{N}}$My question, more precisely, is:
**Question.** Is there a set $B\subset \N\times\N$, such that the set of indices where it is arithmetically definable, that is, $\{ n\in\N \mid B\_n\text{ is arithmetic}\}$, is not first-order definable in the structure $\langle\N,{+},{\cdot},0,1,{\lt},B\r... | https://mathoverflow.net/users/1946 | Is there a subset of the natural number plane, which doesn't know which of its slices are arithmetic? | There are $B$ with this property.
Lets first recall some definitions and notation. Suppose $X \in 2^\omega$ (which we identify with subsets of $\mathbb N$ via characteristic functions). Then $X'$ is the Turing jump of $X$, and $X^{(n)}$ is the nth iterate of the Turing jump of $X$. $X$ is said to be $n$-generic (i.e.... | 16 | https://mathoverflow.net/users/6151 | 141479 | 77,171 |
https://mathoverflow.net/questions/140690 | 4 | Let $\mathcal{K} = \mathbb{C}((t)), \mathcal{O}=\mathbb{C}[[t]]$, $G=SL\_2$ (or any semisimple group), and $\text{Gr}\_G=G(\mathcal{K})/G(\mathcal{O})$; there is a left action of $G(\mathcal{O})$ on $\text{Gr}\_G$. Let $X\_\*(T)=\text{Hom}(\mathbb{C}^{\times},T)$ (note that there is a natural embedding of $X\_\*(T)$ in... | https://mathoverflow.net/users/2623 | Orbits on the affine Grassmanian, and closure ordering | My knowledge about this is cumming from p-adic groups and not from ind/prog groups, so the following (especially part III) might be inaccurate or incomplete.
I. Proof that $\bigcup Gr\_\lambda=Gr$ for the $GL\_n$ (or similarly $SL\_n$) case.
This is basically Gauss elimination posses.
We have to proof that any mat... | 2 | https://mathoverflow.net/users/4690 | 141494 | 77,175 |
https://mathoverflow.net/questions/141501 | 22 | Definition. A topological space $X$ has the Fixed Point Property (FPP) if every continuous self-map $X\to X$ has a fixed point.
Question. If $X$ and $Y$ are homotopy-equivalent compact metrizable spaces and $X$ has the FPP, does it follow that $Y$ also has FPP? Another way to put it: Can one force a fixed point for ... | https://mathoverflow.net/users/21684 | fixed point property for maps of compacts | Lovely question! Sadly, the answer is "no" in the sense that the fixed point property is not homotopy-invariant even in the category of finite polyhedra. In fact, it is also not invariant under the operations of taking products or suspensions.
See the [three page paper](http://projecteuclid.org/DPubS?service=UI&vers... | 22 | https://mathoverflow.net/users/18263 | 141509 | 77,182 |
https://mathoverflow.net/questions/141519 | 4 | I was reading a proof of $9g-9$ theorem which states that $9g-9$ length parameters are sufficient the parametrize the Teichmuller space of a closed surface of genus $g$. The proof uses the following fact.
---
>
> **Theorem:** Let $f:\mathbb{R}^m\times \mathbb{R}^n\rightarrow \mathbb{R}$ be a strictly conves fun... | https://mathoverflow.net/users/9485 | Convexity of a minimum function | This is a standard result in convex analysis. See for example, $\S$3.2.5 of [*Convex Optimization*](http://www.stanford.edu/~boyd/cvxbook/) by Boyd and Vandenberghe (just slightly modify their proof to conclude strictness).
| 7 | https://mathoverflow.net/users/8430 | 141521 | 77,186 |
https://mathoverflow.net/questions/141483 | 10 | Let $m>n$ and consider the Set
$$S\_{m,n}=\{A \in \mathbb{R}^{m \times n}\lvert A^TA=I\_n \}.$$
Does the function $d\colon S\_{m,n} \times S\_{m,n} \rightarrow \mathbb{R}$ defined by
$$d(A,B)=\sqrt{1-\det(A^TB)}$$
define a pseudometric on $S\_{m,n}$? (A pseudometric satisfies all conditions of a metric except that two ... | https://mathoverflow.net/users/35593 | Is this a metric on the Grassmannian Manifold? | **EDIT** Actually, Cauchy-Binet suffices as the OP notices in the comments. I'll leave my overkill proof here for your amusement.
---
The proof below appeals to a famous result of Schoenberg (I've simplified the statement a bit), and basic linear algebra.
>
> **Schoenberg's theorem** (see e.g., [Prop. 3.2, 1]... | 9 | https://mathoverflow.net/users/8430 | 141535 | 77,193 |
https://mathoverflow.net/questions/141421 | 3 | The Baire-Space is the set of all infinite sequences of integers, i.e.
$$
\mathcal N = \omega^{\omega}.
$$
On this space usually the following metric is given
$$
d(\alpha, \beta) = \left\{ \begin{array}{ll} 0 & \textrm{if } \alpha = \beta \\
\frac{1}{\min\{ n : \alpha(n) \ne \beta(n)\} + 1} & \textrm{if } \alpha \ne... | https://mathoverflow.net/users/37580 | Different Metrics for Baire Space and their induced Topologies | I am going to answer the question that you all ask whenever you see a metric space "is it complete?"
$\mathbf{Proposition}$ The metric $d'$ on $\mathbb{N}^{\mathbb{N}}$ is not a complete metric.
$\mathbf{Proof}$ Let $f\_{n}:\mathbb{N} \rightarrow \mathbb{N}$ denote the function where $f\_{n}(n)=1$ and $f\_{n}(m)=0... | 3 | https://mathoverflow.net/users/22277 | 141537 | 77,194 |
https://mathoverflow.net/questions/141539 | 6 | Let $f \in S\_2(\Gamma\_1(N))$ be an eigenform. By a theorem of Shimura, there are associated "periods" $\Omega\_f^\pm$ such that, after normalizing by these periods, the L-function associated to $f$ takes algebraic values.
If $\chi$ is a Dirichlet character, one can form the twist $f\_\chi$ of $f$.
How are the per... | https://mathoverflow.net/users/10547 | Periods of Twists of Modular Forms | By a famous theorem of Manin, one can define $\Omega^{\pm}$ such that $L(f\otimes\chi,j)\in \Omega^{\epsilon}\_{f}\mathbb Q$ with $\chi(-1)(-1)^{j}=\epsilon$. So the period depends on $\chi$ only insofar as you need to know $\chi(-1)$ to determine if you should choose $\Omega\_f^{+}$ or $\Omega\_f^{-}$.
This result i... | 7 | https://mathoverflow.net/users/2284 | 141540 | 77,196 |
https://mathoverflow.net/questions/90975 | 11 | I've been reading this really nice paper by Alper <http://math.columbia.edu/~jarod/good_moduli_spaces.pdf>, and there's a question that doesn't seem to be answered (perhaps it's not relevant).
Any stack F has a corresponding `sheaf of connected components' (or sheaf of isomorphism classes), by taking $\pi\_0^{pr}(F)(... | https://mathoverflow.net/users/16857 | coarse moduli space and $\pi_0$ | You have probably already come up with the answer yourself, but I just thought the question shouldn't
hang around unanswered in the forum.
What you call "the sheaf of connected components", I would call the *coarse sheaf* of the stack or the
*sheaf associated to the stack*. It is usually not representable by an alge... | 7 | https://mathoverflow.net/users/1084 | 141542 | 77,197 |
https://mathoverflow.net/questions/141543 | 8 | The question has relevance for constructing Scott sets with certain extra desirable properties.
Suppose that $\mathfrak X$ is a countable *arithmetically closed* family of subsets of $\mathbb N$: whenever $B\in \mathfrak X$ and $C$ is definable in $\langle \mathbb N,+,\cdot, <,B, 0,1\rangle$, then $C\in\mathfrak X$.
... | https://mathoverflow.net/users/5984 | A well-behaved $A$ that is almost contained in every element of some filter for a countable arithmetically closed family $\mathfrak X$ | In my paper, *A variant of Mathias forcing that preserves $\mathsf{ACA}\_0$* [Archive for Mathematical Logic 51 (2012), 751–780; [arXiv:1110.6559](http://arxiv.org/abs/1110.6559), [doi:10.1007/s00153-012-0297-4](http://dx.doi.org/10.1007/s00153-012-0297-4)], I show that $F\_\sigma$-Mathias forcing preserves $\mathsf{AC... | 6 | https://mathoverflow.net/users/2000 | 141544 | 77,198 |
https://mathoverflow.net/questions/141547 | 2 | Suppose $M \cong \mathbb{Z}^n$ is a rank $n$ lattice, with dual lattice $N$. Suppose $\Delta$ is a full dimensional lattice polytope (i.e. convex hull of finite lattice points) in $M$. Then $\Delta$ is a reflexive polytope if and only if its dual polytope $\Delta^\vee =\{y \in M \otimes\_\mathbb{Z} \mathbb{R} \mid \lan... | https://mathoverflow.net/users/29730 | Estimates on the number of vertices of reflexive polytopes | A "cube" $[-1,1]^n$ has $2^n$ vertices and is reflexive.
| 6 | https://mathoverflow.net/users/38468 | 141548 | 77,201 |
https://mathoverflow.net/questions/141522 | 16 | Let $(E, \|\cdot\|)$ be a real normed vector space such that for any $a,b\in E$,
$$ \|x +y\|^2 + \|x-y\|^2 \geq 4 \|x\|\cdot \|y\| $$
I want to show that the norm is induced by an inner product. Any suggestion or references would be helpful.
| https://mathoverflow.net/users/nan | A property that forces the NORM to be induced by an INNER PRODUCT | If $E$ is to be a Hilbert space, a proof must establish more or less directly that the inequality implies the parallelogram law $\lVert x + y\rVert^2 + \lVert x-y\rVert^2 = 2\lVert x\rVert^2 + 2\lVert y\rVert^2$ for all $x,y \in E$. Since both, your hypothesis and the parallelogram law, are conditions on all $2$-dimens... | 23 | https://mathoverflow.net/users/29555 | 141553 | 77,202 |
https://mathoverflow.net/questions/141530 | 5 | Let $k$ be an arbitrary field, and let $\varphi:A\to B$ be a morphism of abelian varieties over $k$.
If $k$ has characteristic zero, then $\varphi(A)$ has the structure of an abelian subvariety of $B$ which is defined over $k$.
Question: For an arbitrary field $k$, has $\varphi(A)$ the structure of an abelian subva... | https://mathoverflow.net/users/36759 | Image of abelian varieties | In general, if $f:G \rightarrow H$ is any homomorphism between smooth group schemes of finite type over a field $k$, the image $f(G)$ is always a smooth closed $k$-subgroup of $H$. (This is a special case of general results in SGA3, but it seems more instructive to give the direct argument in this case rather than wade... | 9 | https://mathoverflow.net/users/39487 | 141556 | 77,203 |
https://mathoverflow.net/questions/141489 | 12 | Let $G=H\times J$, where $H\cong J\cong C\_2$ (cyclic group of order 2). Let $M \cong \mathbb{Z}$ be a $G$-module via "trivial $H$-action and negation $J$-action". My question is "What are the group cohomologies $H^\*(G,M)$?"
I tried to compute them via the Hochschild-Serre spectral sequence $E\_2^{p,q}=H^p(J,H^q(H,M... | https://mathoverflow.net/users/39666 | A question on some computation of group cohomologies | See [Kuenneth-formula for group cohomology with nontrivial action on the coefficient](https://mathoverflow.net/questions/75472/kuenneth-formula-for-group-cohomology-with-nontrivial-action-on-the-coefficient)
Let $C=C\_2$ be the cyclic group of order two, $\def\ZZ{\mathbb Z}\ZZ$ the trivial module over $C$ and $S$ the... | 7 | https://mathoverflow.net/users/1409 | 141557 | 77,204 |
https://mathoverflow.net/questions/141552 | 2 | I have also posted this question at <https://math.stackexchange.com/questions/486917/simple-approximation-to-a-sum-involving-stirling-numbers>. I have an exact answer to a problem, which is the function:
$f(x,y)=\frac{1}{y^x}\sum\_{i=1}^{x-1}{[i\binom{y}{x-i}(x-i)!S(x,x-i)]}$ where $S(x,x-i)$ is Stirling number of th... | https://mathoverflow.net/users/39687 | Simple approximation to a sum involving Stirling numbers? | Consider all functions from an $x$ element set to a $y$ element set. Then your $f(x,y)$ gives the expected value of $x$ minus the size of the image of the function.
So all you need is to find the expected value of the image. This is easy since the
probability that an element is not in the image is just $((y-1)/y)^x$.... | 11 | https://mathoverflow.net/users/38624 | 141563 | 77,206 |
https://mathoverflow.net/questions/141561 | 1 | 1, Why do people pay special attention to Q/Z in the definition of cofree modules instead of ordinary abelian groups?
2, Over a PID, is every injective module cofree? Just like the relationship between projective module and free module. If not, please give out a example, and give out the dual notion of free module.
... | https://mathoverflow.net/users/39695 | cofree modules and dual | As perhaps you surmise, the relation between "free" and "cofree" is not one of a formal duality. Ordinarily, a module is said to be "free" if it occurs in the essential image of the left adjoint to the forgetful functor $\text{Mod}\_R \to \text{Set}$. There is no formal dual of this notion because this forgetful functo... | 10 | https://mathoverflow.net/users/2926 | 141589 | 77,215 |
https://mathoverflow.net/questions/141580 | 0 | Let $G$ be a Lie Group and $\mathfrak{g}$ be its lie algebra.
Let $\mathfrak{g}$ is semisimple or reductive lie algebra, then prove that $\mathfrak{g}^\*$ (dual of $\mathfrak{g}$)is invariant under $Ad(G)$?
| https://mathoverflow.net/users/nan | when $g^*$ is invariant under $Ad(G)$? | The question you want to ask, in order to understand remark 1, page 3 of Kirillov's book, is why, if $\mathfrak{g}$ is a reductive or semisimple Lie group, for every representation of $\mathfrak{g}$, every $\mathfrak{g}$-invariant subspace has a $\mathfrak{g}$-invariant complement. In particular, Kirillov is assuming t... | 2 | https://mathoverflow.net/users/13268 | 141592 | 77,216 |
https://mathoverflow.net/questions/141026 | 3 | I asked this question in math.stackexchange few days ago.
Unfortunately, I haven't seen any simple answer.
One can say that the Stiefel-Whitney classes is dual classes to the locus of linearly dependence of generic sections. What means "generic"? I want to see some relation in local coordinates.
The same questio... | https://mathoverflow.net/users/37807 | Stiefel classes and generic sections | Consider the vector fields (let say $r$ vector fields on an $n$ dimensional manifold $M$) as a map from the trivial $r$-dimensional vector bundle $\varepsilon^r$ over $M$ into the tangent bundle $TM.$
It gives a section $\alpha$ of the bundle $HOM(\varepsilon^r, TM).$ The fiber over $x \in M$ of this later bundle is th... | 3 | https://mathoverflow.net/users/36950 | 141593 | 77,217 |
https://mathoverflow.net/questions/117036 | 42 | What does the Pontryagin class detects or is an obstruction to? Please avoid any answer using that it's the even Chern class of the complexified bundle or any interpretation that relies on the complexified bundle.
As related question might be the following: when one defines the obstruction classes on a rank $4$ vect... | https://mathoverflow.net/users/18974 | What is geometrically the Pontryagin class? | Pontryagin's original definition for his classes was an obstruction cycle as follows:
On the $n$ dimensional manifold $M$ take $(n-2i) +2$ vector fields in general position, and consider the points $x$ where they span a subspace (in $T\_xM$) of dimension less or equal to $n-2i$. The set of such points $x$ form a cycl... | 30 | https://mathoverflow.net/users/36950 | 141595 | 77,218 |
https://mathoverflow.net/questions/141598 | 11 | Let $R$ be a commutative ring with identity and let $S$ be a multiplicative subset of $R$. Is it true that for any injective $R$-module like $M$, $S^{-1}M$ (as the $S^{-1}R$-module) is also injective ?
| https://mathoverflow.net/users/nan | Is it true that if $M$ is injective then $S^{-1}M$ is also injective | No, this is false in general. I quote the Mathematical Review of Dade, Everett C.
*Localization of injective modules*.
J. Algebra 69 (1981), no. 2, 416–425.
>
> Localization of modules over a commutative ring *R* with respect to a multiplicatively closed subset *S* of *R* is an exact functor with a large number of... | 16 | https://mathoverflow.net/users/1310 | 141599 | 77,219 |
https://mathoverflow.net/questions/140673 | 21 | A language is said to have *quantifier elimination* if every first-order-logic sentence in the language can be shown to be equivalent to a quantifier-free sentence, i.e., a sentence without any $\forall$s or $\exists$s. An example is the theory of real closed fields (such as $\mathbb{R}$), considered with the four basi... | https://mathoverflow.net/users/398 | Main open computational problems in quantifier elimination? | The problem of the current approach in this area, exemplified by the book by [Basu, Pollack, Roy, "Algorithms in Real Algebraic Geometry"](http://perso.univ-rennes1.fr/marie-francoise.roy/bpr-ed2-posted1.html) is that one ends up with the simplest case: checking non-emptiness of a real algebraic set $S$, and the only w... | 14 | https://mathoverflow.net/users/11100 | 141604 | 77,222 |
https://mathoverflow.net/questions/141590 | 9 | Theorem 1.53 (3) in page 227 of Hajek and Pudlak's book, Metamathematics of First-Order Arithmetic, says:
Theorem. If $M$ is a countable model of $I\Delta\_{0}$ such that $M$ has a proper elementary end extension, then $M\models PA$.
Is the above theorem still true if we drop the countability assumption of $M$?
| https://mathoverflow.net/users/27034 | Uncountable model of bounded arithmetic with an elementary end extension | The countability assumption cannot matter. The reason is that any uncountable model $M$ is countable in a forcing extension of the set-theoretic universe. If $M$ has a proper elementary end-extension in the original universe, then this end-extension still exists in the forcing extension. So we may apply the theorem as ... | 7 | https://mathoverflow.net/users/1946 | 141607 | 77,224 |
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