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/103519 | 3 | I'm reading a paper in which the following construction appears:
Let $(\mathcal{C}, \otimes, I)$ be a (dagger) monoidal category, and $(X, \triangledown, \bot)$ be a monoid in this category.
Step 1. *Externalise* the monoid by turning it into a monoid $(Hom(I,X),\cdot,\bot)$ where for $a,b : I \rightarrow X$ define $... | https://mathoverflow.net/users/nan | Externalising monoids using the Yoneda-embedding and relation to Kleisli categories | For any monoidal category $\mathbb{C}$ there exists the "underlying" monoidal functor $\hom(I, -) \colon \mathbb{C} \rightarrow \mathbf{Set}$. As is the idea of a monoidal functor, it preserves structures defined by monoidal operations. Particularly, $\hom(I, -)$ lifts to the "underlying" 2-functor from the 2-category ... | 3 | https://mathoverflow.net/users/13480 | 103559 | 60,022 |
https://mathoverflow.net/questions/103553 | 7 | Let $(a\_n)$ be a sequence of non-negative real numbers and assume that the resulting power series defines a rational function
$$\sum\_{n=0}^\infty a\_n x^n = \dfrac{f(x)}{(1-x^{k\_1})\cdots (1-x^{k\_d})}$$
where $k\_1,...,k\_d>0$ are integers and $f(x)$ is a real polynomial s.t. $f(1) \neq 0$.
It is not hard to sh... | https://mathoverflow.net/users/18571 | Asymptotics for the coefficients of a rational function | We have the partial fraction decomposition $$\dfrac{f(x)}{(1-x^{k\_1}) \ldots (1 - x^{k\_d})} = \text{polynomial}(x) + \sum\_\omega \sum\_{j=1}^{d(\omega)} b\_{\omega,j} ( \omega-x)^{-j}$$
where $\omega$ are the roots of $(1 - x^{k\_1})\ldots(1 - x^{k\_d})$ and $d(\omega)$ is the multiplicity of the root $\omega$. Then... | 8 | https://mathoverflow.net/users/13650 | 103562 | 60,024 |
https://mathoverflow.net/questions/103583 | 8 | Let $X$ be an uncountable set, and let $\Omega$ be the power set of $X$, viewed as a $\sigma$-algebra. Does there exist a positive $\sigma$-additive measure of finite total mass on $(X, \Omega)$ such that each point of $X$ has measure zero?
| https://mathoverflow.net/users/13410 | Finite measure on the power set | I assume you mean a $\sigma$-additive measure. This is Ulam's *measure problem*. A positive answer is closely tied up to the existence of real-valued measurable cardinals, so it is equiconsistent with the existence of a measurable cardinal, which is a large cardinal assumption significantly beyond the usual axioms of s... | 14 | https://mathoverflow.net/users/6085 | 103585 | 60,034 |
https://mathoverflow.net/questions/103586 | 5 | Take an exotic $\mathbb{R}^4$ i.e. $V = (\mathbb{R}^4,d)$ such that $V$ is not diffeomorphic to $\mathbb{R}^4$ with standard metric.
Is it true (obvious?) that any two smooth maps $f\_1, f\_2: S^k \to V$ are equivalent via **smooth** homotopy?
edited: (for any k)
| https://mathoverflow.net/users/14925 | smooth homotopy on exotic R^4 | Yes, since smooth maps which are (continuously) homotopic are always smoothly homotopic. See Kosinski's "Differential Manifolds", Theorem III.2.5 and Corollary III.2.6.
| 9 | https://mathoverflow.net/users/8103 | 103589 | 60,035 |
https://mathoverflow.net/questions/103584 | 7 | Let R be an associative ring with a unit, and consider the standard projective model structure of non-negatively graded (left) R-module, $Ch\_R$. A map $f:M\to N$ in $Ch\_R$ is a *weak equivalence* if it induces isomorphisms $H\_kM\to H\_kN$ on all homologies, and a *cofibration* if each $f\_k:M\_k\to N\_k$ is a monomo... | https://mathoverflow.net/users/17083 | On the difference between a projective chain complex and a level-wise projective chain complex | **Edit:** I put my answer into a broader perspective.
In the following [We] refers to Weibel's "An Introduction to Homological Algebra".
Recall that a chain complex $C$ is *split exact* if it is acyclic and $Z\_n$ (the cycles) is a direct summand of $C\_n$ for each $n$ [We, Def. 1.4.1, Ex. 1.4.2]. Moreover, let $... | 9 | https://mathoverflow.net/users/10194 | 103591 | 60,036 |
https://mathoverflow.net/questions/103587 | 4 | Can anyone tell me the German term for "restricted Lie algebra" ? Many thanks in advance ! Kind regards, Stephan Kroneck.
| https://mathoverflow.net/users/13462 | German term for "restricted Lie algebra" ? | This is called: Lie-$p$-Algebra
See for example, Strade "Einige Vereinfachungen in der Theorie der modularen Lie Algebren" Abh. Math. Sem Hamburg 1984, starting:
>
> 1937 hat N. Jacobson den Begriff der Lie-$p$-Algebra (englisch: restricted Lie algebra) über Körpern der Characteristik $p>0$ eingeführt.
>
>
> ... | 10 | https://mathoverflow.net/users/nan | 103594 | 60,038 |
https://mathoverflow.net/questions/103598 | 7 | Fix a ground scheme $S$ (a field say).
By atlas for an algebraic stack I mean a smooth and surjective morphism $Y \to X$ from a scheme (or algebraic space or affine scheme) $Y$.
If the stack $X$ is smooth then all atlases are smooth schemes over $S$.
What about when $X$ is smooth and proper? Can I find a proper atlas... | https://mathoverflow.net/users/25442 | If $X$ is a smooth and proper stack, does it admit a smooth and proper atlas? | The answer is "no". Consider a smooth, proper $1$-dimensional Deligne-Mumford stack over $\mathbb{C}$ which has coarse moduli space $\mathbb{P}^1$ and which has a single "stacky" point (with any nontrivial stabilizer group at that point).
**Edit.** A more precise definition of the stack is as follows. Let $m$ be an ... | 11 | https://mathoverflow.net/users/13265 | 103599 | 60,040 |
https://mathoverflow.net/questions/103565 | 8 | The group $\mathrm{PSL}\_2(\mathbf{Q})$ of fractional linear transformations $x \mapsto (ax+b)/(cx+d)$ such that $a,b,c,d \in \mathbf{Q}$ and $ad-bc = 1$ acts on $\mathbf{Q} \cup \lbrace \infty\rbrace$. The weakest version of my question is the following:
>
> Is there a short proof that no element of $\mathrm{PSL}\... | https://mathoverflow.net/users/7709 | Orbits of the projective special linear group on $\mathbf{Q} \cup \{\infty\}$ | If your matrix is scalar it acts trivially on the projective line, so assume it's not scalar; hence it's conjugate in $\text{GL}\_2(\mathbf{Q})$ to a companion matrix $A=\pmatrix{0 & -1 \cr 1 & z}$.
If $z$ is integer with $|z|\ge 3$ then the dynamics of $A$ on the real projective line $\mathbf{P}^1(\mathbf{R})$ has ... | 11 | https://mathoverflow.net/users/14094 | 103607 | 60,044 |
https://mathoverflow.net/questions/103540 | 22 | Suppose you have a triangular chessboard of size $n$, whose "squares" are ordered triples $(x,y,z)$ of nonnegative integers that add up to $n$. A rook can move to any other point that agrees with it in one coordinate -- for example, if you are on $(3,1,4)$ then you can move to $(2,2,4)$ or to $(6,1,1)$, but not to $(4,... | https://mathoverflow.net/users/8604 | Hexagonal rooks | Here is a paper about this problem: ["Non-attacking queens on a triangle"](http://www.cin.ufpe.br/~pcp/nonattacking_queens.pdf).
And here's another one ["Putting Dots in Triangles"](http://arxiv.org/abs/0910.4325)
| 13 | https://mathoverflow.net/users/10095 | 103610 | 60,046 |
https://mathoverflow.net/questions/103612 | 2 | Suppose that $F$ is a finite field and $S\subset F[t]$ is a (finite) set of primes. Is is true that any ring automorphism of $R:=F[t][S^{-1}]$ has finite order?
A ring automorphism of $R$ is uniquely determined by an automorphism of $F$ ($F$ is the set of ring elements which are algebraic over the prime ring, so $F$... | https://mathoverflow.net/users/3969 | Order of ring automorphisms of localizations of polynomial rings over finite fields | Every such automorphism is contained in the automorphism group of the field or rational functions $F(t)$ over $F$, which equals $\mathrm{PGL}\_2(F)$, and so is a finite group.
[Edit:] upon further reflection, my answer is incomplete, because $\mathrm{PGL}\_2(F)$ is the group of isomorphisms of $F(t)$ as an $F$-algebr... | 7 | https://mathoverflow.net/users/4790 | 103617 | 60,050 |
https://mathoverflow.net/questions/103622 | 2 | Consider a holomorphic function $f: S \to \mathbb{C}$ where $S$ is a path connected open subset of $\mathbb{C}$ (not necessarily simply connected). Is it then possible to determine if $f$ contains a zero?.
I would suspect that the answer is that it is not possible in general, but has this ever been proven? From Matiy... | https://mathoverflow.net/users/93606 | Computability of finding roots in holomorphic functions. | For example, let $S$ be the open unit disk and $f(z) = -1/2 + \sum\_{j \in A} z^j$ where $A$ is some subset of $\mathbb N$. Then $f$ has a zero if and only if $A$ is nonempty. $A$ could be, say, the set of $n$ such that a given Turing machine with a given input halts in time $\le n$, and there is no algorithm to determ... | 5 | https://mathoverflow.net/users/13650 | 103627 | 60,054 |
https://mathoverflow.net/questions/103623 | 7 | Which man came up with the solution to the basic Calculus of Variations problem first?
<http://en.wikipedia.org/wiki/Euler-Lagrange>
Makes it sound like Lagrange got it first in 1755, then sent it to Euler.
But
<http://en.wikipedia.org/wiki/Calculus_of_variations>
makes it sound like Euler got it in the 1730s... | https://mathoverflow.net/users/25451 | Who came up with the Euler-Lagrange equation? | According to Giaquinta and Hildebrandt (Calculus of Variations I, p. 70): "Euler's differential equation was first stated by Euler in his *Methodus inveniendi* [2], Chapter 2, no. 21. Quite often, one speaks of Lagrange's differential equation, or the Euler-Lagrange equations. Yet Lagrange himself attributes this equat... | 13 | https://mathoverflow.net/users/13972 | 103629 | 60,056 |
https://mathoverflow.net/questions/103613 | 3 | I have a bunch of inequalities, and I'm trying to see if another inequality can be deduced from the first bunch. For example, assuming that $a \leq b$ and $c \leq d$, we can deduce that $a + c \leq b + d$, but we cannot deduce that $a + d \leq b + c$. Is there some software that allows me to take some inequalities whic... | https://mathoverflow.net/users/25446 | Is there a software to prove or deduce symbolic inequalities? | As @Robert Israel points out, the linear case is MUCH easier than the general case, and the really general case (arbitrary inequalities of the form $f(\mathbf{x}) \geq 0$) is clearly undecidable, but if the inequalities are polynomial, this is the problem of "quantifier elimination over real closed fields", which you c... | 6 | https://mathoverflow.net/users/11142 | 103630 | 60,057 |
https://mathoverflow.net/questions/103638 | 2 | I have a problem. I don't know if every continuos function is necessarily a quasi-isometry. I was trying to prove that, but failled for now. I also can't find a counterexample. If you have any hints, i would be very happy if you could help me. Thank you.
| https://mathoverflow.net/users/25452 | Continuous function and quasi-isometry | The map $\mathbb{R}\rightarrow \mathbb{S}^1$ given by $f(x) = \exp(2\pi i x)$ is continuous, yet not a quasi-isometry.
The map from the open semicircle $(\-\pi/2, \pi/2)$ to $\mathbb{R}$ given by $f(\theta) = \tan(\theta)$ is a homeomorhpism, but not a quasi-isometry.
Any map (not necessarily continuous) between co... | 2 | https://mathoverflow.net/users/11142 | 103640 | 60,064 |
https://mathoverflow.net/questions/103636 | 9 | I am looking for a modern proof of the so-called "Theorem of the Base"--that the Neron-Severi rank of a smooth projective variety is finite. One can prove this for varieties over $\mathbb{C}$ easily via transcendental methods---however, [the most recent proof](http://books.google.com/books?id=vRhtM0UMUh4C&pg=RA1-PA221&... | https://mathoverflow.net/users/6950 | Modern Proof of the Theorem of the Base | SGA 6, Exp. XIII
| 8 | https://mathoverflow.net/users/13265 | 103642 | 60,066 |
https://mathoverflow.net/questions/103543 | 2 | For this question, we say that a zero-dimensional space $X$ is $\omega$-pseudocompact if every partition of $X$ into clopen sets is finite. In other words, a zero-dimensional space $X$ is $\omega$-pseudocompact if and only if every continuous mapping $f:X\rightarrow\mathbb{N}$ is bounded.
One can easily show that a zer... | https://mathoverflow.net/users/22277 | Is every zero-dimensional space with no infinite clopen partition pseudocompact? | I think the answer is yes. to see the sketch of proof you could suppose on the contrary that, $f$ is not bounded and choose $$x\_1, x\_2,...,x\_m,... \in X$$ and$$r\_1,r\_2,...r\_m,...\in \mathbb{R}^+$$ so that $$r\_1< |f(x\_1)|< r\_2<|f(x\_2)|< ... r\_m< |f(x\_m)|< ...$$
Because $X$ is zero-dimensional and $f$ is cont... | 1 | https://mathoverflow.net/users/23317 | 103644 | 60,068 |
https://mathoverflow.net/questions/98392 | 3 | What kind of singularity is commonly meant by $A\_{\infty}$?
| https://mathoverflow.net/users/4096 | $A_{\infty}$ singularity | For $\mu \ge 0,$ the $A\_{\mu}$ series of map germs $f : (\mathbb{R}^n,0) \to (\mathbb{R},0)$ is, for $\varepsilon\_i = \pm 1$, given by $f(x) = \varepsilon\_1x\_1^2 + \cdots + \varepsilon\_{n-1}x\_{n-1}^2 \pm x\_n^{\mu +1}$. These all have algebraically isolated singularities at $0 \in \mathbb{R}^n$. I this setting, f... | 3 | https://mathoverflow.net/users/20443 | 103648 | 60,070 |
https://mathoverflow.net/questions/103411 | 16 | Let $\alpha\in\mathbf{Ord}$ and $n\in\mathbb{N}^+$.
Let $F\_\alpha(n)$ be the number of distinct values taken by ordinal exponentiation $\underbrace{\alpha \hat{\phantom{\hat{}}} \alpha \hat{\phantom{\hat{}}} \dots \hat{\phantom{\hat{}}} \alpha}\_\text{n times}$ with parentheses inserted in all possible ways.
The ... | https://mathoverflow.net/users/9550 | Number of distinct values taken by $\alpha$ ^ $\alpha$ ^ $\dots$ ^ $\alpha$ with parentheses inserted in all possible ways, $\alpha\in\mathbf{Ord}$ | The sequence $F\_{\alpha}(n)$ indeed does not depend on $\alpha$ when $\alpha \geq \omega$. This question is closely related to [this other question](https://mathoverflow.net/questions/103486), and I will refer to my answer in that question.
Let $S\_\alpha(n)$ be the set of ordinals defined by $\alpha ^ {\alpha ^ {\c... | 10 | https://mathoverflow.net/users/23338 | 103653 | 60,072 |
https://mathoverflow.net/questions/88592 | 0 | What systems of wavelets provide a discrete frame for $L^2[0,\infty)$?
Specifically, I need a mother wavelet $\psi(x)$ that has a continuous second derivative, such that the system of wavelets $\{\psi\_{jk}(x)=\psi(2^jx-k)|j,k\in\mathbb{Z}\}, k\geq0$ forms a basis (or generally a frame) for $L^2[0,\infty)$. Preferab... | https://mathoverflow.net/users/16615 | Is there a wavelet frame for $L^2[0,\infty)$? | There is a general construction by Dahmen-Kunoth-Urban that adapts Cohen-Daubechies-Feauveau wavelets to $[0,\infty)$ or $[0,1]$ satisfying various boundary conditions.
Reference: Wolfgang Dahmen, Angela Kunoth, Karsten Urban,
*Biorthogonal Spline-Wavelets on the Interval - Stability and Moment Conditions*
It can b... | 2 | https://mathoverflow.net/users/824 | 103656 | 60,075 |
https://mathoverflow.net/questions/103651 | 9 | This is related to my previous question: [When is a quantum affine space $\mathbb{A}^{n}$ Calabi-Yau?](https://mathoverflow.net/questions/102628/when-is-a-quantum-affine-space-mathbban-calabi-yau) I now would like to know the global dimension of the ring $R=\mathbb{C}\langle x\_1,\dots,x\_n\rangle/I$, where I is the tw... | https://mathoverflow.net/users/50973 | Global dimensions of non-commutative rings | Marc Wambst constructed a resolution for your algebra $A$, which he calls naturally enough a *quantum Koszul complex*. This is a projective resolution of $A$ as an $A$-bimodule and it looks like the usual bimodule Koszul resolution of polynomial rings sprinkled with $q$'s all over the place.
His quantum Koszul comple... | 12 | https://mathoverflow.net/users/1409 | 103660 | 60,076 |
https://mathoverflow.net/questions/103658 | 5 | I believe $\aleph\_{\omega}$ has more than $\aleph\_{\omega}$ countable subsets but I do not see the proof. I fear it is obvious, but not to me today.
| https://mathoverflow.net/users/38783 | Why does $\aleph_{\omega}$ have more than $\aleph_{\omega}$ countable subsets? | Let $\{A\_\alpha:\alpha<\omega\_\omega\}$ be all countable subsets of $\omega\_\omega$. We build one that is not among them, giving the contradiction. Pick $x\_0\in\omega\_1$ wich is not in $\bigcup\{A\_\alpha:\alpha<\omega\}$. Then choose $\omega\_1\leq x\_1<\omega\_2$ which is not in $\bigcup\{A\_\alpha:\alpha<\omega... | 14 | https://mathoverflow.net/users/6647 | 103661 | 60,077 |
https://mathoverflow.net/questions/103664 | 3 | The definition of inversion vector for permutation is very well defined. Each permutation can be mapped to a unique inversion vector. So is there a well defined inversion vector for each multiset permutation, which maintains such one-to-one correspondence? I'm particularly interested in the multiset permutations where ... | https://mathoverflow.net/users/11105 | inversion vector for multiset permutation | One can define the inversion vector in the same way and the same proof goes through. Given a multiset $M$ and a multipermutation $\pi=\pi\_1\cdots \pi\_n$, define its inversion vector to be $i(\pi)=(i\_1,i\_2,\dots,i\_n)$, where
$$i\_k=\left|\lbrace j \text{ such that } j>k \text{ and } \pi\_{k}>\pi\_j\rbrace\right|.$$... | 3 | https://mathoverflow.net/users/2384 | 103670 | 60,080 |
https://mathoverflow.net/questions/103645 | 1 | Let $A$ be a ring, Noetherian or even of finite type over a field if necessary. Let $I$ be an ideal in $A$, $\widehat{A}$ the formal completion of $A$ along $I$, $M$ an $A$-module, finitely generated if necessary, and $\widehat{M}$ the formal completion. Then the sheaf cohomology $H^{i}(Spec(A),M)=0$ for $i > 0$.
Is... | https://mathoverflow.net/users/6254 | Cohomology vanishing for formal completions of modules? | Yes --- this follows from [EGA3 I, Chapter 0, Proposition 13.3.1](http://www.numdam.org/numdam-bin/feuilleter?id=PMIHES_1961__11_) . This general result gives conditions under which it is possible to conclude that
$H^i( X, \lim\limits\_\longleftarrow \mathcal{F}\_k )$
is isomorphic to
$\lim\limits\_\longleftarrow... | 2 | https://mathoverflow.net/users/6827 | 103674 | 60,082 |
https://mathoverflow.net/questions/103676 | 5 | Hi, Let $A$ be a division algebra over a local field $F$ of dimension $n^2$ and $K$ be an extension of $F$ of degree $n$. Then if follows from COROLLARY 2 in page 225 of Weil's Basic Number Theory that $A$ is split over $K$. Now my question is
1. Does $A$ contain a subfield isomorphic to $K$ ? Why?
2. Could you desc... | https://mathoverflow.net/users/4245 | Maximal subfields in a division algebra over a local field | Corollaries 3.4 and 3.7 on pages 130 and 131 of Milne's notes (jmilne.org/math/CourseNotes/CFT.pdf) will give you the answers.
| 8 | https://mathoverflow.net/users/10427 | 103679 | 60,085 |
https://mathoverflow.net/questions/103684 | 3 | Let $A$ be an abelian variety over a field $k$ of dimension $g\geq 2$.
There exists a finite morphism $A\to \mathbf{P}^g\_k$. Here's the question.
Does there exist a finite morphism $A\to \mathbf{P}^g\_k$ of degree two?
Can we say something about the minimal degree of a finite morphism $A\to \mathbf{P}^g\_k$?
| https://mathoverflow.net/users/22189 | Are abelian varieties degree two covers of some projective space | For a very general, principally polarized Abelian variety $(A,\Theta)$ of dimension $g$ over $\mathbb{C}$, every Cartier divisor $D$ on $A$ is numerically equivalent to $m\Theta$ for some integer $m$. In particular, the intersection number $D^g$ is $m^g \Theta^g$. So the minimal degree of an effective, nonzero divisor ... | 11 | https://mathoverflow.net/users/13265 | 103685 | 60,087 |
https://mathoverflow.net/questions/103688 | 0 | Let $A/k$ be a simple abelian variety.
Does there exist a non-simple abelian variety $B/k$ and a finite homomorphism $f:B\to A$ over $k$?
I don't need $f:B\to A$ to be etale.
| https://mathoverflow.net/users/22189 | Is any simple abelian variety covered by a non-simple abelian variety | No. If $End\_A$ and $End\_B$ are the endomorphism rings then an isogeny $B\to A$ will give an isomorphism $End\_A\otimes\mathbb Q\to End\_B\otimes\mathbb Q$. But the first of these algebras is a division algebra and the second is not.
| 7 | https://mathoverflow.net/users/8726 | 103690 | 60,089 |
https://mathoverflow.net/questions/103686 | 8 | (1) Let $X$ be a projective (integral) curve over $\mathbb{C}$ and let $P$ be a singular point of $X$. Is there always a Cartier divisor whose support is exactly $P$ (set-theoretically)?
The following may be two related questions:
(2) Let $f: Y \to X$ be the normalisation of the curve $X$. Then we have a pushforwar... | https://mathoverflow.net/users/11599 | Cartier divisors on singular curves | Ok, the first thing you have to specify is what does it mean for to talk about a Cartier divisor (or Weil divisors). There are a number of options here. The one I like best for *Cartier divisors* would be invertible subsheaves of $K(X)$, the fraction field of $X$. This coincides with the one in Fulton's intersection th... | 13 | https://mathoverflow.net/users/3521 | 103691 | 60,090 |
https://mathoverflow.net/questions/31580 | -1 | Otherwise, if all the elements in a set can be represented by a at most n symbols (finite Kolmogorov complexity), I could count them by creating a n dimensional pairing function. Or atleast, that is my assumption.
Any thoughts?
Edit: as @Carl has pointed out the correct term for sequences that have no finite repres... | https://mathoverflow.net/users/7168 | Do all uncountable sets contain elements with infinite Kolmogorov complexity? | Contrary to what some of the commenters said, there's no great difficulty in generalizing Kolmogorov complexity to infinite strings. For example, given a language L⊆{0,1}\*, we could let K(L) be the length of the shortest program that decides L, or K(L)=∞ if L is undecidable. (Or we could also talk about programs that ... | 8 | https://mathoverflow.net/users/2575 | 103693 | 60,092 |
https://mathoverflow.net/questions/103695 | 0 | Let me be more precise than the title. (This will be my last attempt to do something with abelian varieties. Sorry for all the basic questions. The answers have been great!)
Let $A$ be a simple abelian variety over a field $k$. Let $g\geq 2$ be the dimension of $A$.
Does there exist an integer $n\geq 1$ such that $... | https://mathoverflow.net/users/22189 | Does the self-product of a $g$-dimensional abelian variety contain an abelian variety of dimension smaller than $g$ at some point | No (I suppose that $k$ is algebraically closed). This is because Poincaré's complete reducibility theorem contains a unicity statement for the intervening factors (up to isogeny). See Mumford, Abelian varieties, p. 173-174.
| 7 | https://mathoverflow.net/users/17308 | 103697 | 60,094 |
https://mathoverflow.net/questions/103353 | 3 | Let $k, N, m \in \mathbb{N}$ such that $k \leq N$.
What is the maximal number $e$ of strings $\sigma\_1, \sigma\_i, \dots, \sigma\_e$, each of length $N$ such that
$$
\forall j < k, \left(\sum\_{i=1}^e \sigma\_i[j]\right) \leq 2^{N-k}(m-1)
$$
For example if $m=3$, $k=4$, $N=5$, we have $e = 14$. An example of such... | https://mathoverflow.net/users/25373 | Maximal number of binary strings given constraints | This is a counterexample to the formula $e = 2^{N-k} + 2^{N-m}m$ for the maximum number of strings satisfying the given constraints. The parameters of the example are $N = 6$, $k = 5$, and $m = 4$. The constraint is that the columns sum to no more than $6$. The conjectured formula predicts that 18 is the maximum number... | 2 | https://mathoverflow.net/users/22379 | 103699 | 60,095 |
https://mathoverflow.net/questions/103666 | 5 | My question is:
For given k-dimension manifold M, Is there any way to produce a manifold N so that there exist a map $f:M\rightarrow N$, $degf=1$
The trivial method I know:
The only trivial method I can image is when $M=M\_1\#M\_2$, then we can produce a deg=1 map from $M$ to $M\_1$. Or for the torus, we can produc... | https://mathoverflow.net/users/25054 | Create examples of deg=1 maps | In dimension 2, [Edmonds proved](http://www.jstor.org/stable/1971246?origin=crossref) that every degree one map between closed orientable surfaces is homotopic to a "pinch map", which takes a subsurface which is bounded by one component, and sends it to a disk. This is a special case of the connect sum method you descr... | 9 | https://mathoverflow.net/users/1345 | 103700 | 60,096 |
https://mathoverflow.net/questions/103523 | 7 | Let $X$ be a variety over an algebraically closed field with null characteristic. Let $C$ be a smooth subvariety of $X$ of dimension 1, and let $x$ be a point of $C$. We assume that $X$ is analytically irreducible at $x$.
I consider the normalization $X'$ of $X$, and more precisely the morphism $C'\to C$ where $C'$ i... | https://mathoverflow.net/users/19205 | Relation between blowup and normalization | The answer for the four question is **no**, as this example shows.
Consider $X$ the surface defined by $x^2-(t^3+y)y^2 = 0$. Let $A$ denote the associated ring.
Then $X' = \operatorname{Spec} A[\frac xy]$.
The curve $D$ is given by $u^2-t^3$, with $u = \frac xy$.
The blowup (an affine chart of) at the origin of $X$... | 1 | https://mathoverflow.net/users/19205 | 103703 | 60,098 |
https://mathoverflow.net/questions/103687 | 5 | I have a sequence of sets of random numbers, with each set generated
by an unknown random number generator. I am assuming that in the
sequence, the random number generator is the same one for a certain
duration, such that a few sets in a row are from the same random
number generator. In essence the sequence looks like ... | https://mathoverflow.net/users/25463 | How to test if two sets of random numbers might be from the same random number generator? | Some non-trivial information may be acquired, especially with not-so-great pseudo-random number generators, by applying standard compression algorithms (e.g., gzip) to chunks of output. If the ratio of compressed to not-compressed are significantly different, one infers that the sources are different, etc. Obvious benc... | 5 | https://mathoverflow.net/users/15629 | 103704 | 60,099 |
https://mathoverflow.net/questions/103663 | 0 | Help me prove
$$
\sum\_{n=1}^\infty \frac{\sigma\_a(n)\sigma\_b(n)}{n^s} = \frac{\zeta(s) \zeta(s-a) \zeta(s-b) \zeta(s-a-b)}{\zeta(2s-a-b)} $$
| https://mathoverflow.net/users/25458 | Ramanujan Divisor Function | The functions involved are "multiplicative". Therefore it is enough to prove the "local result":the same sum running over the non-negative powers of the same prime on the left anf the product of local factors of Riemann's zeta function on the right. To deal with the resulting power series on the left, use the formula f... | 1 | https://mathoverflow.net/users/25465 | 103706 | 60,100 |
https://mathoverflow.net/questions/103671 | 3 | Let $D \subset \mathbb{C}^n$ be a bounded symmetric domain. It is known that $D$ can be realized as the unit ball of some complex norm $||\cdot||$. Using the Bergman metric on $D$, one can define a triple product on $\mathbb{C}^n$ which enjoys many nice properties. In the paper <http://www.springerlink.com/index/FN6613... | https://mathoverflow.net/users/36038 | Automorphisms of bounded symmetric domains | In S. Helgason's oldest large book, "Differential geometry and symmetric spaces" (?) from the 1960s, symmetric spaces and bounded symmetric domains are discussed in detail. Also, I. Satake's book from c. 1980, I think in a style of the sort you're wanting.
| 3 | https://mathoverflow.net/users/15629 | 103707 | 60,101 |
https://mathoverflow.net/questions/103698 | 7 | If $P$ and $P'$ are partial orders, a *strictly order preserving* map from $P$ to $P'$ is an $f:P\to P'$ satisfying that $x\lt y$ implies $f(x)\lt f(y)$ for all $x,y\in P$.
An *interval* in $P$ is a set of the form ${}[x,y]=\{z\mid x\le z\le y\}$, where $x\le y$. Note that points in an interval need not be comparable... | https://mathoverflow.net/users/6085 | Strictly order preserving maps into the integers | The construction of Farley and Schroeder that you linked to essentially gives an example of size the bounding number $\mathfrak b$. So it's at least consistent with $\neg$CH to have an example of size $\aleph\_1$. Details: Take an unbounded family $\mathcal B$ of functions $\omega\to\omega$, and form the analog of the ... | 5 | https://mathoverflow.net/users/6794 | 103709 | 60,103 |
https://mathoverflow.net/questions/103430 | 3 | * Cotorsion modules
A module M is called cotorsion if for all flat modules X, $Ext\_R^1(X,M)=0$ .
* Strongly cotorsion modules
M is called strongly cotorsion if for all modules X of finite flat dimension, $Ext\_R^1(X,M)=0$ .
* Questions
By
definitions it is easy to observe that strongly cotorsion modules a... | https://mathoverflow.net/users/5775 | Do the classes of cotorsion and strongly cotorsion modules coincide? | Let me raise an example:
Let $R = k[[t]]$, $k$ be a field. $R$ is a regular ring as well as a Dedekind ring.
1. $R$ is regular hence every $R$-module $X$ has finite flat dimension. Thus if $M$ is strong cotorsion, then $Ext^1\_R(X,M) = 0$ for all $X$, so $M$ is injective.
2. By (1) we have $k$ is not strong cotors... | 2 | https://mathoverflow.net/users/17901 | 103713 | 60,104 |
https://mathoverflow.net/questions/103705 | 5 | For an $2n$-dimensional complex manifold $M$, and a smooth vector bundle $E$ over $M$, it is well-known (see Voisin, Huybrechts) that there exists an operator $\overline{\partial}$, built locally from the usual anti-holomorphic derivative, that acts on $\Gamma^{\infty}(E) \otimes\_{C^{\infty}} \Omega^{(0,\bullet)}(M)$ ... | https://mathoverflow.net/users/1648 | Global Definition of the Dolbeault Complex of a Vector Bundle | To get the Dolbeault complex, you need a choice of *holomorphic* structure on $E$, not just a smooth one. If $\mathcal{E}$ is the locally free sheaf of $\mathcal{O}\_M$-modules corresponding to $E$, then $\mathcal{E}\subset \mathcal{A}^0(\mathcal{E})=\mathcal{C}^\infty\_M\otimes\_{\mathcal{O}\_M}\mathcal{E}$ and
$\ove... | 7 | https://mathoverflow.net/users/6278 | 103714 | 60,105 |
https://mathoverflow.net/questions/103731 | 5 | Let $X$ be a metric space. Recall that a function $f: X \rightarrow X$ is **contractive** if there exists $C \in (0,1)$ such that for all $x,y \in X$, $d(f(x),f(y)) \leq C d(x,y)$; a function $f$ is **weakly contractive** if for all $x \neq y \in X$, $d(f(x),f(y)) < d(x,y)$.
Also let us say that an **attracting point... | https://mathoverflow.net/users/1149 | An extension of Edelstein's Attraction Theorem? | **Q1.**
Take $X=\{\\,z\in\mathbb C\mid 1<|z|<2\\,\}$ and $f(z)=-\tfrac{z}{\sqrt{|z|}}$.
It seems to answer the other questions.
You may take a spiral in X which is $f$-invariant. This produce a simply connected example.
| 8 | https://mathoverflow.net/users/1441 | 103739 | 60,118 |
https://mathoverflow.net/questions/103736 | 13 | There are 3 standard proofs of Bertrand's Postulate:
(1) Chebyshev's original proof
(2) Ramanujan's simplification of Chebyshev's proof
(3) Erdos's proof
I recently learned about the very simple proof that if the Goldbach conjecture is true, then Bertrand's postulate follows (see [here](http://www.proofwiki.org... | https://mathoverflow.net/users/15915 | Are there any interesting or lesser known proofs related to Bertrand's Postulate | Bertrand's Postulate follows as a direct consequence of the following theorem of J.J. Sylvester:
**Theorem (Sylvester, 1892):** Let $k$ be a positive integer. Then at least one of any $k$ consecutive integers greater than $k$ is divisible by a prime greater than $k$.
(For comparison: Chebyshev's analytic proof dat... | 20 | https://mathoverflow.net/users/22971 | 103744 | 60,121 |
https://mathoverflow.net/questions/103754 | 0 | Is a C^1-function in a bounded domain $\Omega\subset R^n;$ an element of the Sobolev space $W^{2,\infty}(\Omega)$ ?
| https://mathoverflow.net/users/25248 | Derivable functions & Sobolev spaces | No. Consider $x^{3/2}$ on $(-1,1)$. It has Holder continuous but not Lipschitz derivative. See Gilbarg-Trudinger Ch. 4 for some related exercises (e.g. a C^1 function with derivative continuous but not Holder continuous for any $\alpha$).
| 2 | https://mathoverflow.net/users/16659 | 103758 | 60,129 |
https://mathoverflow.net/questions/103751 | 15 | It may be a easy question for experts.
The definition of the Weyl group of a complex semisimple Lie algebra $\mathfrak{g}$ is well-known: We first $\textbf{choose}$ a Cartan subalgebra $\mathfrak{h}$ and we have the root space decomposition. The Weyl group now is the group generated by the reflections according to r... | https://mathoverflow.net/users/24965 | About the intrinsic definition of the Weyl group of complex semisimple Lie algebras | Probably the earliest intrinsic definition of *Weyl group* occurs in section 1.2 of the groundbreaking paper "Representations of Reductive Groups Over Finite Fields" by Deligne and Lusztig (*Ann. of Math.* 103, 1976, available at JSTOR). This is done elegantly in the closely related but more general setting of a reduct... | 16 | https://mathoverflow.net/users/4231 | 103770 | 60,132 |
https://mathoverflow.net/questions/103767 | 2 | Hi there! I have a very simple question, which requires an expert in multilinear algebra.
$V$ is an $n$-dimensional vector space, and $\omega\in V^\ast\wedge V^\ast$ is a skew-symmetric form on it. Then, mimicking symplectic geometry, I call *isotropic* a subspace $L\subseteq V$ such that $\omega|\_L\equiv0$.
**QUE... | https://mathoverflow.net/users/22606 | How to characterize the dual of an isotropic hyperplane? | Your guess is right. Knowing that insertion of vectors acts as a derivation you get
$i\_X i\_Y (\alpha\wedge\omega)=i\_Y\alpha\cdot i\_X\omega+i\_X\alpha\cdot i\_Y\omega +\alpha\cdot i\_Xi\_Y\omega$
If $X,Y\in \ker \alpha$ then the right hand side reduces to $\alpha \cdot i\_X i\_Y\omega$ and if you assume $\alpha ... | 2 | https://mathoverflow.net/users/745 | 103771 | 60,133 |
https://mathoverflow.net/questions/103711 | 7 | The title sounds a bit philosophical, but it is
about mathematics. Let me explain.
Consider a first order theory $T$, which is an extension
of Peano Arithmetic. Call this theory "good" if it is
consistent and satisfies the following
*Property*:
For any $\phi\in\Sigma^0\_{n+1}$ such that
$T\vdash \phi$ there... | https://mathoverflow.net/users/9833 | Are all the theorems true? | $\def\zfc{\mathrm{ZFC}}\def\pa{\mathrm{PA}}$First, there is no consistent recursively axiomatizable theory extending Robinson’s arithmetic which has the property of having existential witnesses as described by Sridhar Ramesh. Let $\pi=\forall x\,\theta(x)$ be a true but $T$-unprovable $\Pi^0\_1$ sentence with $\theta$ ... | 11 | https://mathoverflow.net/users/12705 | 103776 | 60,135 |
https://mathoverflow.net/questions/103759 | 7 | Consider the claim:
(C) There is a transitive set $S \in V$ such that the structure $(S, \in)$ is an elementary submodel of $(V,
\in)$.
Obviously, this claim cannot be a theoreom of ZFC, by Godel's 2nd Incompleteness Theorem. But does (C) follow from ZFC+CON(ZFC)? Or are large cardinals needed to prove (C)? More g... | https://mathoverflow.net/users/25479 | Elementary submodels of V | A cardinal $\delta$ is [correct](http://cantorsattic.info/Correct#Reflection_and_correctness) if the theory $V\_\delta\prec V$ holds, that is, if $V\_\delta$ is an elementary substructure of $V$. This theory is expressible as a scheme in first order logic in the language of set theory augmented by a constant symbol for... | 13 | https://mathoverflow.net/users/1946 | 103779 | 60,136 |
https://mathoverflow.net/questions/103768 | 6 | Let $X$ be an irreducible algebraic variety over some field $k$. It is well known, that if $X$ is smooth and of dimension $d$, then the tangent bundle of $X$ is smooth, irreducible and of dimension $2d$. There is no difficulty in defining the tangent bundle $T\_X$ in the non-smooth case. It is defined as the (global) S... | https://mathoverflow.net/users/21885 | Dimension of irreducible components of tangent bundle | How about the union in $\mathbb{A}^9$ of the three $5$-planes, $\Lambda\_1 = \text{span}(\mathbf{e}\_1,\mathbf{e}\_6,\mathbf{e}\_7,\mathbf{e}\_8,\mathbf{e}\_9)$, $\Lambda\_2=\text{span}(\mathbf{e}\_2,\mathbf{e}\_4,\mathbf{e}\_5,\mathbf{e}\_8,\mathbf{e}\_9)$ and $\Lambda\_3=\text{span}(\mathbf{e}\_3,\mathbf{e}\_4,\mathb... | 4 | https://mathoverflow.net/users/13265 | 103780 | 60,137 |
https://mathoverflow.net/questions/103787 | 4 | We begin with $2n+1$ cards, each with a distinct number from $-n$ to $+n$ on it, face up in between the two players of the game. The players take turns selecting a card and keeping it. The first player to collect three cards that sum to zero wins the game. If the cards are exhausted and neither player has won, a draw i... | https://mathoverflow.net/users/3106 | Generalized tic-tac-toe | First player wins for $n$ at least five. First turn, name $0$. They name a number, say $-a$. Choose two numbers $b$ and $c$ such that neither $b$, $c$, nor $b+c=a$. Then name $b$, forcing them to name $-b$, then $c$, forcing them to name $-c$, then $-b-c$, winning. You can always choose two such numbers, since each pos... | 9 | https://mathoverflow.net/users/18060 | 103794 | 60,142 |
https://mathoverflow.net/questions/100893 | 5 | In his "Murphy's Law" paper, Vakil showed that every "singularity type" (with a precise meaning) occurs on certain Hilbert schemes; for instance, the Hilbert scheme of *nonsingular* curves in projective space. He also gives a method for constructing such singularities; however, the process to construct even, say, a sin... | https://mathoverflow.net/users/5094 | Examples of nice reduced singularities on Hilbert schemes--Edited | Once again, I apologize for my previous comment! The new formulation of the problem is definitely more fun. Here is one solution which only uses smooth, rational curves.
Let $n$ be an integer, $n\geq 4$ (in fact, $n=2$ and $n=3$ make sense if you use the Kontsevich spaces instead of Hilbert schemes).
Start with $\mat... | 9 | https://mathoverflow.net/users/13265 | 103802 | 60,146 |
https://mathoverflow.net/questions/84123 | 7 | I have a couple of questions about known theorems for GCH+Kurepa families.
Definition first: Let $\kappa$ be a infinite cardinal. A $\kappa^+$ Kurepa family is a family $F$ of subsets of $\kappa^+$ such that $F$ has size $>\kappa^+$ and for every $a<\kappa^+$, the set $\lbrace X\cap \alpha|X\in F\rbrace$ has size $\l... | https://mathoverflow.net/users/13694 | GCH+ Kurepa Families | Hi Ioannis! I guess you might know the answer by now; if we suppose for 1 that there is a class of inaccessible cardinals in the ground model and force with an Easton support product of Lévy collapses between the inaccessibles, we obtain a model of GCH where $KH(\kappa^+)$ fails for all $\kappa$; the argument is the sa... | 7 | https://mathoverflow.net/users/21255 | 103804 | 60,148 |
https://mathoverflow.net/questions/103659 | 1 | Let $V$ and $W$ be two complex vector spaces with an action of a finite group $G$. The $G$-equivariant polynomial maps from $V$ to $W$ are finitely generated as a module over the ring of $G$-invariant polynomials on $V$. In other words, there exist $G$-equivariant polynomial maps $p\_1,\dotsc,p\_n$ so that any $G$-equi... | https://mathoverflow.net/users/20919 | finite generation of $G$-equivariant holomorphic maps by polynomials? | I believe the answer is yes for $C^\infty$ maps and actions of compact (not necessarily finite) Lie groups. I think it is due to Poénaru and can be found in his book
*Singularités $C^\infty$ en présence de symétrie*
Lecture Notes in Mathematics, Vol. 510.
See also Lemma 6.6.1 in *Dynamics and symmetry* by Michael F... | 1 | https://mathoverflow.net/users/25355 | 103805 | 60,149 |
https://mathoverflow.net/questions/103795 | 4 | Let $P$ be a compact, convex subset of $\mathbb{R}^n$ (infinite-dimensional generalisations welcome, but not necessary). Let's say that disjoint subsets $W\_1$, $W\_2$ $\subset P$ are *opposed* if there exist parallel hyperplanes $H\_1$, $H\_2$ supporting $P$, such that $W\_i \subset H\_i \cap P$.
Let $F\_1$ and $F\_... | https://mathoverflow.net/users/14891 | Are faces of a compact, convex body "opposed" iff their extreme points are pairwise "opposed"? | **NO.**
Let $P$ be the convex hull of two parabolic arcs,
say
$$\{\,(x,0,z)\in \mathbb R^3\mid 1\ge z=x^2\,\}$$
and
$$\{\,(0,y,z)\in \mathbb R^3\mid 0\le z=-y^2+\varepsilon\cdot y+1\,\}.$$
Take
$$F\_1=\{\,(x,0,1)\in \mathbb R^3\mid |x|\le 1\,\}$$
and
$$F\_2=\{\,(0,y,0)\in \mathbb R^3\mid 0\le -y^2+\varepsilon\cdot... | 2 | https://mathoverflow.net/users/1441 | 103806 | 60,150 |
https://mathoverflow.net/questions/103807 | 2 | The following is supposed to be "clear" according to Kueker, but I could not see why. Can anyone help?
Let $A$ be a countable structure with uncountable many automorphisms. Then for every $\vec{a}\in A^{<\omega}$, $(A,\vec{a})$ has a non-trivial automorphism, i.e. there exists some $f:A\rightarrow A$ such that $f\neq... | https://mathoverflow.net/users/13694 | Countable structures with uncountable many automorphisms | Let $\vec{a} \in A^{<\omega}$. There are uncountably many automorphisms $f$ and only countably many possible values for $f(\vec{a})$, so there must be two different automorphisms $f\_1$ and $f\_2$ with $f\_1(\vec{a}) = f\_2(\vec{a})$. Then $f\_2^{-1} \circ f\_1$ is a nontrivial automorphism fixing $\vec{a}$.
In fact,... | 11 | https://mathoverflow.net/users/1682 | 103809 | 60,152 |
https://mathoverflow.net/questions/103817 | 1 | Hello,
I am searching for a reference on a result I know must exist proving regularity for weak solutions of a (nonlinear, but well-behaved) elliptic homogeneous PDE. Working over say a bounded open subset of $\mathbb{R}^n$ would be fine, and I don't need to deal with very nonlinear PDE - quasilinear is enough for m... | https://mathoverflow.net/users/25490 | Seeking reference on regularity theory for nonlinear elliptic PDE | "Elliptic Partial Differential Equations of Second Order" by David Gilbarg and Neil S. Trudinger
| 3 | https://mathoverflow.net/users/24313 | 103819 | 60,156 |
https://mathoverflow.net/questions/103766 | 12 | Are there any good tricks to construct a heart of a t-structure? (I'm thinking on the derived category of coherent sheaves of some variety)
I'll start with the only one I know. If $(T,F)$ is a [torsion pair](http://www.math.uoi.gr/~abeligia/torsion.pdf) on an abelian category $A$ then you can form the *tilt* inside $... | https://mathoverflow.net/users/25442 | got any tricks to build up t-structures on derived categories? | This will be a short overview on techniques I am familiar with. For simplicity, I will talk about bounded t-structures, which are determined by their heart $\mathcal{A} = D^{\le 0} \cap D^{\ge 0}$; and on the bounded derived category of coherent sheaves $D^b(X)$ on a variety/stack.
1. Tilting is in principle extremel... | 20 | https://mathoverflow.net/users/7437 | 103822 | 60,158 |
https://mathoverflow.net/questions/103810 | 4 | A well known result about the natural numbers $\mathbb{N}$ says that for any finite subset $A \subset \mathbb{N}$ there exists $R \ge 0$ such that if $n$ is in the subgroup of $\mathbb{Z}$ generated by $A$ and if $n \ge R$ then $n$ is in the semigroup generated by $A$.
Are there any references to a higher dimensiona... | https://mathoverflow.net/users/20787 | Reference for subsemigroups of $\mathbb{N}^n$ | This combines comments of myself and Lee Mosher.
Exercise 7.15 of Miller and Sturmfels combinatorial commutative algebra book proves the following. Let $N=C\_U\cap G\_U$. Then $N$ is a finitely generated semigroup and there exists according to exercise 7.15 an element $a$ of $S\_U$ with $a+N\subseteq S\_U$. Now take ... | 6 | https://mathoverflow.net/users/15934 | 103827 | 60,160 |
https://mathoverflow.net/questions/103832 | 7 | Suppose $\kappa$ is a measurable cardinal and $U$ is a normal measure on $\kappa$. $M\_1$ is the ultrapower of the universe $V$ constructed form $U$,$j\_{01}$ is the induced elementary embedding, $M\_2$ is the ultrapower of $M\_1$ constructed from $j\_{01}(U)$...We can iterate this process. We know $M\_0,M\_1,M\_2,...$... | https://mathoverflow.net/users/22635 | Definability of the direct limit | There are in fact many different definable ways to represent the direct limit. One particularly concrete representation of the direct limit is to use maximal *threads*, that is, maximal sequences $\langle x\_i \mid j\leq i\lt\omega\rangle$, where $x\_{i+1}=j\_{i,i+1}(x\_i)$, where $j\_{i,i+1}:M\_i\to M\_{i+1}$ is the u... | 6 | https://mathoverflow.net/users/1946 | 103833 | 60,164 |
https://mathoverflow.net/questions/103835 | 8 | By second order arithmetic I mean the axiomatic theory $Z\_2$, that is Peano arithmetic extended by second order variables with the full comprehension axiom, and not defined semantically using power set in ZF. By third order arithmetic I mean that extended by third order variables and the comprehension axiom. And so on... | https://mathoverflow.net/users/38783 | Does higher order arithmetic interpret the axiom of choice? | There is quite a bit of this in Simpson's book *Subsystems of Second Order Arithmetic* in the specific context of second-order arithmetic. Here are three relevant results:
>
> Corollary VII.5.11 (conservation theorems). Let $T\_0$ be any one of the $L\_2$-theories
> $\Pi^1\_\infty\text{-CA}\_0$, $\Pi^1\_{k+1}\text... | 7 | https://mathoverflow.net/users/5442 | 103839 | 60,165 |
https://mathoverflow.net/questions/103837 | 33 | Let $X$ be a set and $(T,\cdot)$ an abelian group. Is there a *category of $T$-dynamical systems on $X$* which yields useful information about $X$ and $T$?
More precisely, is there a category whose objects are dynamical systems, i.e., $\phi:X \times T \to X$ such that $\phi(x,t + s) = \phi(\phi(x,t),s)$? And if so,
... | https://mathoverflow.net/users/18263 | Is there a categorical treatment of dynamical systems? | For every category $\mathfrak C$ and every monoid $\mathcal S$, you can define the category of $\mathcal S$-flows on $\mathfrak C$ as follows:
-- Objects are pairs $(X, \alpha:\mathcal S\to \mathrm{End}(X))$, where $\alpha$ is a monoid homomorphism (that is, multiplication in $\mathcal S$ becomes composition of morph... | 27 | https://mathoverflow.net/users/24891 | 103840 | 60,166 |
https://mathoverflow.net/questions/103844 | 1 | Define the seminorm on the space $S=[0,1]\times[0,T]$
$$\mid u\mid\_{\alpha} = \sup\frac{|u(x, t) - u(y,s)|}{(|x-y|^2 + |t-s|)^{\frac{\alpha}{2}}}.$$
Define the norms on the same space
$$\lVert u \rVert\_{C^{0, \alpha}} = \lVert u \rVert\_{C^0} + \mid u\mid\_{\alpha}$$
and
$$\lVert u \rVert\_{C^{2, \alpha}} = \lVert u ... | https://mathoverflow.net/users/25266 | How to show this Holder bound? | If $f,g\in C^{\alpha}(S)$, then for all $(x,t),(x',t')\in S$, we have
\begin{align}
|f\cdot g(x,t)-f\cdot g(x',t')|&=|f(x,t)(g(x,t)-g(x',t'))+g(x',t')f(x,t)-f(x',t')g(x',t')|\\\
&\leq \lVert f\rVert\_{\infty}|g(x,t)-g(x',t')|+\lVert f\rVert\_{\infty}|f(x,t)-f(x',t')|,
\end{align}
hence
\begin{equation}\[f\cdot g\]\_{... | 1 | https://mathoverflow.net/users/17118 | 103847 | 60,169 |
https://mathoverflow.net/questions/103848 | 10 | Let $n>2$ and let $k$ be either $\bf Q$ or a finite field whose characteristic is prime to $n$. Let $A\_{g,n}$ be the moduli scheme, which represents the functor, which with every $k$-scheme $S$
associates the set of principally polarized abelian schemes $\cal A$ over $S$, together with a symplectic isomorphism $({\bf... | https://mathoverflow.net/users/17308 | Mordell-Weil group of the universal abelian scheme | For $g=1$ there is a classical paper of Shioda (one of the two cited below) that proves that in char. zero, the group is what you expect but in char. p there are situations in which you get sections of infinite order.
On rational points of the generic elliptic curve with level N structure over the field of modular fu... | 10 | https://mathoverflow.net/users/2290 | 103856 | 60,174 |
https://mathoverflow.net/questions/103846 | 14 | Dear everyone,
**Motivation :**
From the past few days, I have been reading about the Galois Representations . I was really amazed to see that every seminal idea in the theory of elliptic curves have Galois Representation as an ingredient !!
**I would be very happy listening to :**
* What made Galois represen... | https://mathoverflow.net/users/24713 | Why are Galois Representations so important in Number theory ? | For abelian number fields, the Dedekind zeta function factors into Dirichlet $L$-functions. Hecke characters are one-dimensional representations. For possible generalizations to non-abelian field extensions, you will require higher dimensional Galois representations and Artin $L$ functions.
(Maass 1949) Maass wave fo... | 10 | https://mathoverflow.net/users/10400 | 103862 | 60,176 |
https://mathoverflow.net/questions/103874 | 2 | "If $U$ is an open subset of the complex plane, then matrices $X\in\textrm{M}(n,\mathbb C)$ all of whose eigenvalues belong to $U$ make up an open subset of $\textrm{M}(n,\mathbb C)$." Trying to prove this by using the argument principle, I was led to the following question: do approximately the same polynomials have a... | https://mathoverflow.net/users/19142 | Do approximately the same polynomials have approximately the same roots? | I think so, yes. Suppose all the roots lie *inside* the unit disc. By compactness $|p(z)|$ takes a positive minimum $\delta$ on the contour (the unit circle). For $q(z)$ in the epsilon-neighbourhood, $q(z)-p(z)$ is a polynomial with epsilon coefficients, so we can take epsilon small enough so that on the contour $|q(z)... | 3 | https://mathoverflow.net/users/12647 | 103875 | 60,184 |
https://mathoverflow.net/questions/103860 | 7 | Related to [Jon's question](https://mathoverflow.net/questions/103306/strong-monotone-limits-and-dense-subalgebras-of-von-neumann-algebras-again), I have two questions. Let $\mathcal{A}$ be a concrete $C^\*$-algebra on a Hilbert space $\mathcal{H}$. For any selfadjoint subset $S$ of $\mathbb{B}(\mathcal{H})$, let $S^m$... | https://mathoverflow.net/users/25499 | The monotone closure of a $C^*$-algebra | Edited. As Masayoshi points out, my reading of Hamana's paper was incorrect. I'm quite sure question 1 is false in general but I don't have a reference. (Masayoshi, did you look in Pedersen's book? I feel the answer may be there but I don't have access to it right now.)
I guess I'd better be more explicit about quest... | 8 | https://mathoverflow.net/users/23141 | 103876 | 60,185 |
https://mathoverflow.net/questions/95677 | 6 | It is well known that the function $f(z)=2\cos(\sqrt {-z})$ (or more accurately the entire function $f(z)=2\sum\_{n=0}^\infty \frac{z^n}{(2n)!}$) satisfies such a functional equation, i.e. $f(4z)= f(z)^2-2$ ; it is not hard to show that this is the unique solution with $c=-2$ and $f'(0)=-1$. Patient calculations (see f... | https://mathoverflow.net/users/17164 | Mandelbrot set and analytic functions such that $f(az)=f(z)^2+c$ | Such functions were studied, since Poincare.
They are called Poincare functions, or "global linearizers".
Let $f$ be a rational function, and consider the functional equation
$F(kz)=f(F(z))$. Suppose $F$ is analytic at $0$. Then $a=F(0)$ must be a fixed point of $f$.
If $F'(0)$ not $0$, then $k=f'(0)$ by the chain rule... | 8 | https://mathoverflow.net/users/25510 | 103890 | 60,192 |
https://mathoverflow.net/questions/79265 | 4 | Hi,
Let $f$ be an entire function. The *spherical derivative* $\rho(f)$ is defined by
$$\rho(f)(z):= \frac{|f'(z)|}{1+|f(z)|^2}.$$
A result from Clunie and Hayman states that if $\rho(f)$ is bounded, then $f$ is of exponential type. The proof uses the machinery of Nevanlinna's theory of value distribution.
My que... | https://mathoverflow.net/users/1162 | Bounded spherical derivative implies finite order | Yes, of course. A bound on spherical derivative immediately gives T(r)=O(r^2)
where T is the Nevanlinna characteristic. And that finite order of T implies
finite order of f is proved in the beginning pages of any book on Nevanlinna theory.
BTW. Your idea on a simple proof of Picard theorem is not new.
A proof based o... | 3 | https://mathoverflow.net/users/25510 | 103891 | 60,193 |
https://mathoverflow.net/questions/103892 | 15 | **Modified Birthday Problem:** a bunch of people line up, and the winner is the first person who shares their birthday with someone lined up ahead of them. What position in the line is optimal?
**Three (similar) approaches:** Recall the well known [Birthday Problem](https://en.wikipedia.org/wiki/Birthday_problem), an... | https://mathoverflow.net/users/22971 | Solving a modified birthday problem at a glance | The approaches in the OP already seem quite straightforward to me, and as is clear from the answer by Arthur B, doing the calculation exactly is not too complicated.
Anyway, here is my suggestion (after a bit of scribbling) for how one might have seen the solution at a glance.
All we have to do is compare the winn... | 7 | https://mathoverflow.net/users/14302 | 103903 | 60,198 |
https://mathoverflow.net/questions/103783 | 10 | What is the "no small subgroups" argument for $GL(n,\mathbb R)$? That is, how do we show that in $GL(n,\mathbb R)$ there exists a neighborhood of the identity which contains no subgroup other than the trivial one? I had some scribbling (for the $n=2$ case) but could not arrive at a clean proof.
| https://mathoverflow.net/users/19142 | "No Small Subgroups" Argument | Here is Asaf's agrument expanded a bit. It has the advantage of working for all Lie groups simultaneously.
Given a Lie group $G$ with Lie algebra $\mathfrak{g}$, consider the exponential map $\exp:\mathfrak{g}\rightarrow G$. It is known that it is a diffeomorphism on a small enough open set $U\subseteq\mathfrak{g}$.
... | 21 | https://mathoverflow.net/users/1708 | 103906 | 60,199 |
https://mathoverflow.net/questions/103896 | 9 | Do there exist known examples of predicates $P$ (possibly functional) such that
1) $P$ admits a first-order definition in the language ${\Bbb N}(+,\times,0,1)$;
2) $P$ admits no definition that does not involve both $+$ and $\times$;
3) the theory of ${\Bbb N}(+,P,0,1)$ is decidable?
(Or
3') the theory of ${... | https://mathoverflow.net/users/10909 | Beyond Presburger Arithmetic | From a [paper by Françoise Point](http://www.math.osu.edu/~friedman.8/pdf/0AppB072710.pdf), "On the expansion $(N, +; 2^x)$ of Presburger arithmetic," I learned of a much more general result of Semenov, “Logical theories of one-place functions on the set of natural numbers”, Izv. Akad. Nauk SSSR Ser. Mat., 47:3 (1983),... | 7 | https://mathoverflow.net/users/3545 | 103914 | 60,203 |
https://mathoverflow.net/questions/103919 | 4 | Seemingly the same fact goes under several names: MacMahon master theorem, Wronski relation, unnamed fact about symmetric functions - I wonder what is history and what should be ``correct name'' ?
Let me remind the facts and formulate questions more precisely:
Denote by $e\_k$ elementary and $S\_k$ complete symmetr... | https://mathoverflow.net/users/10446 | Historical question about MacMahon theorem, Wronski relation etc | I will answer only the second question. Full disclosure: I wrote much of the WP [MacMahon Master Theorem](http://en.wikipedia.org/wiki/MacMahon_Master_theorem) article.
**No**, you should not call a trivial symmetric function identity after MacMahon (which was already used by Euler and Newton, see e.g. [Newton's ide... | 6 | https://mathoverflow.net/users/4040 | 103929 | 60,212 |
https://mathoverflow.net/questions/73553 | 2 | Let F be meromorphic function,with what properties may it expanded as power series with coefficients of integers in such a form:
$$F=\sum\_0^{\infty}a\_i x^i,a\_i\in \mathcal{N} \bigcup 0,\exists M \space a\_i \leq M^i$$.
and when the coefficients consist of a sequence of computably enumerable relation.
If the qu... | https://mathoverflow.net/users/14024 | When may function (meromorphic) be expanded as power series with coefficients of integers | This was a large research subject in 1930-s. The key authors are G. Polya, Ch. Pisot
and Raphael Robinson.
The book of Bieberbach, Analytische Fortsetzung (in German, there is a Russian translation) contains a chapter with a survey
of these results.
The general spirit of these results is the following: if you have a ... | 12 | https://mathoverflow.net/users/25510 | 103930 | 60,213 |
https://mathoverflow.net/questions/103926 | 4 | Consider the differential equation $ -y''(x)+x^{4}y(x)-E\_{n}y(x)=0 $ with the boundary conditions $ y(0)=0=y(\infty).$
How can I use the shooting method or other numerical methods to solve this equation? My only idea is to set $ R=10000 $ for example and solve $ y(0)=0=y(R) $,
Of course I also could make the subst... | https://mathoverflow.net/users/23964 | How can I solve a boundary value numerically on an infinite interval? | You're facing a singular Sturm-Liouville eigenvalue problem. One way you may be able to use the shooting method is to get an asymptotic expansion of your solution at the $x=\infty$ (Frobenius or WKB methods should work). Using the asymptotics to evaluate $y(x)$ and $y'(x)$ at some finite $x$ and then shoot from there t... | 2 | https://mathoverflow.net/users/2622 | 103931 | 60,214 |
https://mathoverflow.net/questions/103913 | 2 | Let $\mathcal{D} \approx \mathbb{P}^{\delta\_d}$ be the space of homogeneous degree $d$
polynomials in three variables (up to scaling), where $\delta\_d = \frac{d(d+3)}{2}$.
Define $\mathcal{A}$ to be space of degree $d$ curves with a strict node at
the point $[1,0,0]$, ie
$$ \mathcal{A} := \{ f \in \mathcal{D}: ... | https://mathoverflow.net/users/4463 | General position argument | Dear Ritwik -- This is not a precise reference for the result you state. However, standard dimension results, i.e., Exercise II.3.22, p. 95 of Hartshorne's *Algebraic Geometry*, lead to a proof very quickly (and I would consider this as a standard application of dimension theory). Let $\overline{\mathcal{C}}\subset \ov... | 3 | https://mathoverflow.net/users/13265 | 103947 | 60,225 |
https://mathoverflow.net/questions/91199 | 4 | If $f$ and $g$ are two elliptic functions with the same periods, then there exists an algebraic relationship of the form $P(f,g)=0$, where $P$ a polynomial of two variables with constant coefficients.
This is a well known property. As a special case we have $P(f, f^{\prime})=0$, which is satisfied by the Weierstrass ... | https://mathoverflow.net/users/43162 | Algebraic relationships between elliptic functions | There exists an algebraic relation between two elliptic functions if and only if their
lattices are commensurable. This means that every period of one function is a rational
multiple of periods of another.
Proof. Let $f$ and $g$ be two elliptic functions, let $F(f,g)=0$ be an algebraic relation.
Choose a point $a$ su... | 7 | https://mathoverflow.net/users/25510 | 103955 | 60,231 |
https://mathoverflow.net/questions/103911 | 8 | Suppose $E$ is an elliptic curve over $\mathbb{Q}\_p$ with good ordinary reduction. Can someone please tell me how to compute the associated $(\phi,\Gamma)$-module of the Tate module of $E$, or give me a reference to where it is computed?
| https://mathoverflow.net/users/25515 | (phi, Gamma) module of ordinary elliptic curve | If your elliptic curve has good ordinary reduction, then the attached Galois representation is reducible : it is an extension of $\eta\_2$ by $\eta\_1 \chi$ where $\eta\_{1,2}$ are unramified characters and $\chi$ is the cyclotomic character. The $(\phi,\Gamma)$-modules of $\eta\_2$ and $\eta\_1 \chi$ are easy to compu... | 10 | https://mathoverflow.net/users/5743 | 103959 | 60,232 |
https://mathoverflow.net/questions/103909 | 10 | Let ${\mathbb T}$ denote the ordinary $\Lambda$-adic Hecke algebra of say tame level $N$. If I specialize ${\mathbb T}$ to a classical weight $k \geq 2$, then it is proven by Hida that the result is the Hecke algebra of classical $p$-ordinary forms of weight $k$ and level $pN$. (Here by specialize I mean mod out by the... | https://mathoverflow.net/users/25513 | Non-classical specializations of Hida families | I don't know if this is always true, and I don't think this is known in all cases, even after inverting $p$.
What is known and what is more or less formal to prove (it depends of your starting point, and in particular, of how you define the "$\Lambda$-adic Hecke algebra of tame level $N$) is
that there is a natural s... | 5 | https://mathoverflow.net/users/9317 | 103961 | 60,233 |
https://mathoverflow.net/questions/103916 | 3 | In my research work, I recently have come across the following system of three linear first order pde's whose characteristic polynomial has one real and two complex conjugate zeros.
$3u\_{1,1}+u\_{3,1}+2u\_{2,2}=0,$
$u\_{1,2}+2u\_{2,1}+3u\_{3,2}=0,$
$\cos \phi\ (u\_{1,2}-u\_{2,1})-\sin \phi\ (u\_{1,1}+u\_{2,2})-... | https://mathoverflow.net/users/25516 | Classification of a certain System of Linear First Order PDEs whose characteristic polynomial has one real and two complex conjugate zeros | Your system of PDE's appears to be of real principal type, as defined by Hormander. Hormander studied the singularities of distributional solutions to such a PDE and how they propagate. This in turn leads to an a regularity theorem for a compactly supported distributional solution on a bounded open domain. Since the ad... | 4 | https://mathoverflow.net/users/613 | 103965 | 60,235 |
https://mathoverflow.net/questions/103904 | 5 | Let $K$ be a field s.t. $charac(K)\not= 2$. Let $A\in\mathcal{M}\_n(K)$ be such that $rank(A)\geq n/2$. Can one find a matrix $B$ such that $B$ is similar to $A$ and $A+B$ is invertible ?
| https://mathoverflow.net/users/9091 | Invertible matrix | Initially my assumption was that the question was primarily about finite fields $K$, but after reading the comments it appears to me that there is some confusion about the infinite case as well. So I'll prove the claim for $|K|=\infty$. I would guess the assertion is also true if $K$ is finite, but I don't know how to ... | 3 | https://mathoverflow.net/users/17498 | 103970 | 60,236 |
https://mathoverflow.net/questions/103973 | 8 | For a compact Hausdorff topological space $X$, its K-theory $K^0(X)$ is defined to be the Grothendieck group of the isomorphism classes of finite dimensional vector spaces on $X$. For example $K^0(\text{pt})=\mathbb{Z}$.
My question is: is there a categorification of K-theory, just as we can categofify natural number... | https://mathoverflow.net/users/24965 | Is there a categorification of topological K-theory? | The abelian group $K^0(X)$ has a natural $(\infty,0)$-categorification'', meaning a spectrum $K(X)$ whose $\pi\_0$ recovers $K^0(X)$: take $K(X)$ to be the spectrum of maps from (the suspension spectrum of) $X$ to the (complex or real, depending) K-theory spectrum. Or we could try a different variant, the spectrum give... | 10 | https://mathoverflow.net/users/3931 | 103975 | 60,238 |
https://mathoverflow.net/questions/103968 | 2 | Hello,
This question is a follow-up from [About Goldbach's conjecture](https://mathoverflow.net/questions/61842/about-goldbachs-conjecture).
As $N\_{2}(n)=\sum\_{r\leq n}1\_{\mathbb{P}}(n-r)1\_{\mathbb{P}}(n+r)$, Chebotarev's theorem allows to write:
$$\dfrac{N\_{2}(n)}{\pi(n)}\sim \dfrac{N\_{1}(n)}{\varphi(P\_{... | https://mathoverflow.net/users/13625 | About Extended Goldbach's conjecture | This is from Section 4 of Hardy and Littlewood's "[On some problems of partitio numerorum III: On the expression of a number as a sum of primes](http://www.springerlink.com/content/d700t434065k0230/?MUD=MP)" 1923 Acta Math. 44: 1–70.
| 2 | https://mathoverflow.net/users/630 | 103977 | 60,240 |
https://mathoverflow.net/questions/103966 | 2 | Let $H\_{d\_1,g\_1}, H\_{d\_2,g\_2}$ be two Hilbert schemes of curves in $\mathbb{P}^3$ with degrees $d\_1, d\_2$ and genus $g\_1, g\_2$. Denote by $H:=H\_{d\_1,g\_1}\times H\_{d\_2,g\_2}$
where an element in $H$ is a pair of curves $(C\_1, C\_2)$ with $C\_i \in H\_{d\_i,g\_i}$.
As far as I understand, a generic elem... | https://mathoverflow.net/users/9164 | General degree $d$ surface in $\mathbb{P}^3$ | By the Noether-Lefschetz theorem, a very general surface $X\subset \mathbb{P}^3$ of degree $d\geq 4$ has $\mathrm{Pic}(X) \cong \mathbb Z$, generated by a hyperplane section $H$. But then if $C \sim aH$ and $C'\sim bH$ on $X$, we have $C\cdot C' = abd^2 > 0$, and so $C,C'$ must intersect in at least one (and genericall... | 1 | https://mathoverflow.net/users/7399 | 103979 | 60,242 |
https://mathoverflow.net/questions/103971 | 5 | Let $H=K\_1\oplus K\_2$ be infinite dimensional Hilbert spaces. Voiculescu's free Gaussian functor gives us free group factors $L(H)$, $L(K\_1)$, $L(K\_2)$ acting on the full Fock space $\Gamma(H)$ and, moreover, $L(H)=L(K\_1)\*L(K\_2)$.
But what is known about the relative commutant $L(K\_1)'\cap L(H)$?
Certainly... | https://mathoverflow.net/users/10779 | How well do we know relative commutants in $L(\mathbb{F}_\infty)$? | Given any two tracial von Neumann algebras $(N\_1, \tau\_1)$ and $(N\_2, \tau\_2)$ the $L^2$ space of the free product $(N\_1 \* N\_2, \tau)$ canonically decomposes as
$$
L^2(N\_1 \* N\_2, \tau) = \mathbb C \oplus\_{n \in \mathbb N} \bigoplus\_{i\_1 \not= i\_2, i\_2 \not= i\_3, \ldots, i\_{n - 1} \not= i\_n } \overlin... | 10 | https://mathoverflow.net/users/6460 | 103980 | 60,243 |
https://mathoverflow.net/questions/103967 | 10 | I am helping a friend of mine, that works in history of mathematics. She is studying the story of the solution of the cubic equation by Cardano. Sometimes she asks me some mathematical questions, that are very hard to motivate from a modern point of view, but that were interesting to Cardano. So please do not ask for m... | https://mathoverflow.net/users/7845 | When is the degree of this number 3? | Edit: The takeaway is that there is another, exclusive way to generate $a$ and $b$ that given an equation of degree $3$. First, chose parameters $a$ and $s$ such that $2a/s(s^2-3)$ is not a perfect cube and $1-4/s^2(s^2-3)^2$ is not a perfect square. This is most such values of $s$. Then choose $b$ according to the for... | 7 | https://mathoverflow.net/users/18060 | 103991 | 60,247 |
https://mathoverflow.net/questions/103825 | 3 | I'm reading a proof of Arnold's theorem about analytic linearization of analytic circle diffeomorphisms. The following result is used in the proof and I don't see why it should be true.
Let $S\_\sigma= \{ z:|\Im z|<\sigma \}$. We have a function $h$ that is holomorphic on $S\_{\sigma-\delta}$, and is real-valued on t... | https://mathoverflow.net/users/17086 | Holomorphic image of a strip in complex plane | I think the easiest argument is using Rouché's theorem, which is itself proved using the argument principle. Let $a \in S\_{\sigma-3\delta}$, let $K \subset S\_{\sigma-2\delta}$ be the circle about $a$ of radius $\delta$, and let $f(z)=z-a$. Then for $z \in K$ we have $|h(z)|<\delta=|f(z)|$, so by Rouché's theorem $f$ ... | 1 | https://mathoverflow.net/users/17086 | 103995 | 60,250 |
https://mathoverflow.net/questions/103924 | 1 | Let $f \in \mathbb{C}[x,y,z]$ be a polynomial which defines an isolated singularity $0 \in D:= (f=0) \subset \mathbb{C}^3$.
Assume that $\mathcal{D}:= (f+tx =0) \subset \mathbb{C}^3 \times \mathbb{C}$ defines a smoothing of $D$ over a small disk.
Let $g \in \mathbb{C}[x,y,z]$ be a polynomial which has a zero of order... | https://mathoverflow.net/users/12390 | Adding singular equations to a smoothing of a hypersurface singularity | Yes.
Call $d$ the maximum between the degrees of $f$ and $g$. Let us homogenize the equations to homogeneous equations of degree $d$ in four variables. The polynomial $f + t(x + g)$ corresponds to a linear system of curves $C\_t$ of degree $d$ in $\mathbb P^3$. By Bertini's theorem $C\_t$ is smooth away from the base... | 3 | https://mathoverflow.net/users/4790 | 103997 | 60,252 |
https://mathoverflow.net/questions/104003 | 10 | Recall that if $X$ is a projective algebraic complex manifold, $L$ is a semisimple $\mathbb C$-local system on $X$ *of geometric origin* (roughly speaking, this means that $L$ is a cohomology sheaf $R^if\_\*\mathbb C$ for some algebraic morphism $f:Y\to X;$ see BBD for the precise definition), and $\eta\in H^2(X,\mathb... | https://mathoverflow.net/users/370 | Is there an explicit example of a coefficient sheaf for which hard Lefschetz fails? | The good news (or is it bad news?) is that there is no counterexample in the semisimple case. Simpson proved that hard Lefschetz holds for any semisimple local system on a smooth complex projective variety. To be more precise, apply theorem 1 and lemma 2.6 of his paper on *Higgs bundles and local systems*. More recentl... | 16 | https://mathoverflow.net/users/4144 | 104011 | 60,259 |
https://mathoverflow.net/questions/85127 | 4 | We know that Hardy-Littlewood maximal function is $(p,p)$ for any $p>1$. But one proves first that it is weak type $(1,1)$ and then use interpolation. I am just curious to know: is there a way of proving the $(p,p)$ property directly, i.e. without using interpolation?
| https://mathoverflow.net/users/17822 | Hardy-Littlewood maximal function | Here is an old trick. Use the Poisson kernels instead of balls to define the central maximal function. Suppose that $p=2$. Then $Mf(x)=\int P\_{a(x)}(x-y)f(y)dy$ for some function $a(x)>0$.
Consider the adjoint operator $Vg(y)=\int P\_{a(x)}(x-y)g(x)dx$ with some positive $g\in L^2$. We have $Vg(y)^2=\iint P\_{a(x)}(x... | 6 | https://mathoverflow.net/users/1131 | 104013 | 60,260 |
https://mathoverflow.net/questions/103728 | 12 | Let **X** be **a topological space** that is not too bad (let's say "not too bad" = "compactly generated Hausdorff"), and let ∼ be an equivalence relation such that **X /∼ is compact Hausdorff**.
Does there exist a **compact subspace A⊂X** that meets every equivalence class of ∼?
(This would then imply that **A /∼... | https://mathoverflow.net/users/5690 | compact quotient | The answer is positive if $X$ is second countable and locally compact, and $X/\sim$ is first countable (in addition to being compact Hausdorff). Proof: we claim that any point in $X/\sim$ has a neighborhood which is contained in the image of a compact subset of $X$. Given this, the rest is easy: use compactness of $X/\... | 6 | https://mathoverflow.net/users/23141 | 104014 | 60,261 |
https://mathoverflow.net/questions/103868 | 5 | Does there exist an example of an abelian category A and *abelian subcategory* B, where B is wide but not full (as a subcategory of A)?
| https://mathoverflow.net/users/10355 | Does there exist a wide but not full abelian subcategory of an abelian category? | Take $\textrm{Mod-}A$, where $A$ is a finite dimensional local $k$-algebra ($k$ a field). Let $\mathcal C$ be the category whose objects are the same as those of $\textrm{Mod-}A$, but whose homomorphisms are replaced by all $k$-vector space homomorphsims, i. e. $\textrm{Hom}\_{\mathcal C}(X,Y) := \textrm{Hom}\_k(X,Y)$.... | 13 | https://mathoverflow.net/users/17498 | 104024 | 60,269 |
https://mathoverflow.net/questions/104029 | 0 | My question is:
How to find out all the finite subgroup of SO(n)? Or just for the simple case SO(4) SO(5)?
With more discribe:
If $S^n\backslash \Gamma$ is a manifold,
I just want to know that how many subgroups of SO(n) can be realized as the $\Gamma$ above?
If for generic n, it may be difficult, can we jus... | https://mathoverflow.net/users/25054 | A Question about SO(n) | There are a pair of double covers $\text{SU}(2) \times \text{SU}(2) \to \text{SO}(4) \to \text{SO}(3) \times \text{SO}(3)$, and the first resp. the second more or less reduces the classification of finite subgroups of $\text{SO}(4)$ to the classification of finite subgroups of $\text{SU}(2)$ resp. $\text{SO}(3)$ (these... | 5 | https://mathoverflow.net/users/290 | 104033 | 60,273 |
https://mathoverflow.net/questions/104006 | 1 | Does anybody suggest how to face the inhomogeneous Bernoulli differential equation
$y'+P(x)y=Q(x)y^n+f(x)$
for the simple case $f=const.$ and for the generic case.
I would like to know about techniques of approximation, bounds, asymptotic limit, numerical techniques etc.
Thank you
Roberto
| https://mathoverflow.net/users/12585 | Inhomogeneous Bernoulli Equation | I presume your $f(t)$ should be $f(x)$. This is Chini's equation. See e.g.
<http://www.maplesoft.com/support/help/Maple/view.aspx?path=odeadvisor/Chini>
| 1 | https://mathoverflow.net/users/13650 | 104039 | 60,276 |
https://mathoverflow.net/questions/104043 | 14 | Let $N$ be a positive integer. Thanks to [the Miller-Rabin test](http://en.wikipedia.org/wiki/Miller%E2%80%93Rabin_primality_test) and the work of [Agrawal, Kayal and Saxena](http://en.wikipedia.org/wiki/AKS_primality_test), these days people have much much faster algorithms for testing whether $N$ is prime or composit... | https://mathoverflow.net/users/14901 | Saying things rapidly about integer factorisations | An expert on computational number theory will be able to provide much, much more detail than I can, but the short answer is that as far as anyone knows, primality-testing is the rare exception, and almost all of these problems are hard (for a classical computer)! That is, the detailed properties of the prime factorizat... | 8 | https://mathoverflow.net/users/2575 | 104045 | 60,281 |
https://mathoverflow.net/questions/104031 | 3 | Why unknowns are usually denoted by "X" ?
More precisely: is [this answer](http://www.ted.com/talks/terry_moore_why_is_x_the_unknown.html) really a serious answer or might be a 1 April joke ?
Let me sketch it. But please watch it, it is really fun and cool and < 6 min.
(Are there any alternative versions?)
**Ans... | https://mathoverflow.net/users/10446 | Why unknowns are usually denoted by "X" ? | Interesting. Cajori says: "Nor is there historical evidence to support the statement found in Noah Webster's Dictionary, under the letter x, to the effect that 'x was used as an abbreviation of Ar. shei (a thing), something, which, in the Middle Ages, was used to designate the unknown, and was then prevailingly transcr... | 4 | https://mathoverflow.net/users/24792 | 104046 | 60,282 |
https://mathoverflow.net/questions/104030 | 3 | Does anyone know a good book or article on the History of the Fourier transform? It's first appearance (of the transform) and use in particular? Or at least some source with some historical information on that matter.
There is plenty about the Fourier series but I couldn't find much illuminating about the transform y... | https://mathoverflow.net/users/10893 | History of the Fourier transform | I. Grattan Guinness, Joseph Fourier 1768-1830. A survey of his life and work, based on a critical edition of his monograph on the propagation of heat, presented to the Institut de France in 1807.
Fourier transform, unlike Fourier series, was really invented by Fourier, when he studied the
heat equation.
| 6 | https://mathoverflow.net/users/25510 | 104050 | 60,285 |
https://mathoverflow.net/questions/90637 | 9 | It is well-known that Chebyshev polynomials are the polynomials of minimal $L\_\infty$ norm on [-1,1] with leading coefficient 1. But what if you want the minimal $L\_\infty$ polynomial on two disjoint intervals $[-1,1]$ and $[a,b]$?
An upper bound can be obtained by seeking the minimal $L\_\infty$ norm polynomial o... | https://mathoverflow.net/users/11444 | polynomials with minimal $L_\infty$ norm on multiple disjoint intervals | There exists a complete theory for minimal $L^\infty$ norm polynomials on two intervals.
It is due to N. I. Akhiezer. You can look at his books: Lectures on Approximation theory
or, another book, Elliptic functions.
There is also a nice survey paper on this problem for any number of intervals,
by Sodin and Yuditskii, F... | 3 | https://mathoverflow.net/users/25510 | 104055 | 60,289 |
https://mathoverflow.net/questions/102076 | 4 | I need to answer (affirmatively, I hope) the following question:
>
> In a Lie group $G$ whose Lie algebra $\mathfrak{g}$ is equipped with an $\mathrm{Ad}$-invariant scalar product, is the open subset
> $$\{g \in G \mid \operatorname{ker} (1 + \operatorname{Ad}\_{g}) = \{0\}\}$$
> dense in $G$?
>
>
>
In the ... | https://mathoverflow.net/users/17294 | Is the condition ``adjoint action does not have eigenvalue $-1$" dense in a Lie group? | [This is an embellished version of the answer I gave in the comment section]
The key to the solution lies in recalling that when one is playing with Lie groups, the functions that one encounters are not mere $C^\infty$ functions but are actually real-analytic. There is some discussion of this in the second edition of... | 7 | https://mathoverflow.net/users/425 | 104065 | 60,295 |
https://mathoverflow.net/questions/103843 | 5 | Given a compact oriented aspherical $3$--manifold $M$ with torus boundary $\partial M\simeq T^2$ (*e.g.* a knot complement), the condition that the images in $\pi\_1 M$ of basis $x,y\in \pi\_1 T^2$ under the inclusion generate the boundary of a compact oriented $3$-manifold with fundamental group $\pi\_1 M$ (the $3$-ma... | https://mathoverflow.net/users/2051 | What is a higher genus analogue of the Pontryagin product? | An obvious necessary condition is that the pair $(M,\Sigma)$ satisfies Lefschetz duality. Since $M$ and $\Sigma$ are aspherical, this can be reformulated in terms of group cohomology.
Let $\Gamma=\pi\_1(M)$ and $S=\pi\_1(\Sigma)$. Assume that the inclusion $\Sigma\subset M$ is $\pi\_1$-injective, so that $S$ is a sub... | 3 | https://mathoverflow.net/users/8103 | 104090 | 60,304 |
https://mathoverflow.net/questions/104097 | 4 | I find myself staring blankly at a system of PDEs in $n$ dimensions which has "one equation per component" of the Hessian of the unknown function - that is, it specifies the Hessian in terms of the derivative and the value of the function:
$\qquad \partial\_i \partial\_j f + B\_{ij}^{\ \ \ k} \partial\_k f + A\_{ij} ... | https://mathoverflow.net/users/25517 | Integrability conditions for 'componentwise' systems of linear PDEs | The answer is, in general, yes. There are further integrability conditions. What you should be considering is the system of $1$-forms on $\mathbb{R}^n\times \mathbb{R}\times \mathbb{R}^n$ (with coordinates $(x^i,u,p\_i)$ given by
$$
\begin{align}
\theta &= du - p\_i\ dx^i\\\
\theta\_i &= dp\_i + (B^k\_{ij}p\_k + A\_{ij... | 5 | https://mathoverflow.net/users/13972 | 104103 | 60,311 |
https://mathoverflow.net/questions/104066 | 12 | In Chapter III, Theorem 7.4 of *The Arithmetic of Elliptic Curves* (first edition), Silverman gives the following lemma and proof:
>
> **Lemma**: Let $M \subset Hom(E\_1, E\_2)$ be a finitely generated subgroup, and let $M^{div} = \{ \phi \in Hom(E\_1, E\_2) : [m] \circ \phi \in M$ for some integer $m \geq 1\}$. Th... | https://mathoverflow.net/users/1050 | Proving finite generation by tensoring with $\mathbb{R}$ | How do you prove that the ring of integers in a number field is finitely generated? One usually embeds them in $\mathbb{R}^{r\_1}\times\mathbb{C}^{r\_2}$. How do you prove that the units in a number field are finitely generated? One generally embeds them in a hyperplane in $\mathbb{R}^{r\_1+r\_2-1}$. Then one shows tha... | 20 | https://mathoverflow.net/users/11926 | 104110 | 60,316 |
https://mathoverflow.net/questions/104109 | 3 | I have the following iterative process
$$a\_n=a\_{n-1}(1-\phi(a\_{n-1})),\quad 0< a\_0<1,$$
where $\phi(x)$ is a continuous increasing function, $\phi(0)=0$, and if $x\in(0,1)$ then $0< \phi(x)<1$. For instance, if $\phi(x)=x$, then this is just the logistic map.
Clearly, $a\_n$ converges to $0$ as $n\to\infty$ for a... | https://mathoverflow.net/users/7646 | Convergence rate of an iterative process | The standard techniques is as follows. At first, we guess the growth rate by replacing the discrete dynamical system to ODE: $a\_n-a\_{n-1}=-a\_{n-1}\phi(a\_{n-1})$ is replaced to $A'(n)=-A(n)\phi(A(n))$, where function $A(n)$ imitates $a\_n$. Nex, we solve this equation, rewriting it as $g(A(n))'=1$, where $g'(t)=-1/(... | 3 | https://mathoverflow.net/users/4312 | 104113 | 60,319 |
https://mathoverflow.net/questions/104114 | 1 | I have a set of 3 quadratic equtions:
$A\_1µ\_1^2 + B\_1µ\_1µ\_2 + C\_1µ\_2^2 + D\_1µ\_1 + E\_1µ\_2 + F\_1 = 0$
$A\_2µ\_2^2 + B\_2µ\_2µ\_3 + C\_2µ\_3^2 + D\_2µ\_2 + E\_2µ\_3 + F\_2 = 0$
$A\_3µ\_3^2 + B\_3µ\_3µ\_1 + C\_3µ\_1^2 + D\_3µ\_3 + E\_3µ\_1 + F\_3 = 0$
I want to write a C-code, that find solutions for $µ\_1... | https://mathoverflow.net/users/25557 | Simultaneous quadratic equations | The magic word is "elimination". The obvious thing to do is to solve the first equation for $\mu\_2$ in terms of $\mu\_1.$ You will have two roots. Plug in each one into the second equation, expressing $\mu\_3$ in terms of $\mu\_1,$ then you will have eight possible equations satisfied by $\mu\_1.$ The equation will be... | 6 | https://mathoverflow.net/users/11142 | 104116 | 60,320 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.