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/92967 | 5 | Let $d(n)$ be the number of divisors function, i.e., $d(n)=\sum\_{k\mid n} 1$ of the positive integer $n$. The following estimate is well known
$$
\sum\_{n\leq x} d(n)=x \log x + (2 \gamma -1) x +{\cal O}(\sqrt{x})
$$
as well as its variability, e.g., the lim sup of the fraction
$$
\frac{\log d(n)}{\log n/\log \log n}
... | https://mathoverflow.net/users/17773 | A truncated divisor function sum | The key is to count integers with a given number of prime factors: if $\omega(n)=\sum\_{p|n}1$ and $\Omega(n)=\sum\_{p^a|n,\,a\ge1}1$, then $2^{\omega(n)}\le\tau(n)\le2^{\Omega(n)}$ and there are results that control the number of integers with a given value of $\omega(n)$, or of $\Omega(n)$. The simplest one of them i... | 6 | https://mathoverflow.net/users/4003 | 93279 | 54,806 |
https://mathoverflow.net/questions/93282 | 15 | This was sparked because I wanted to compute $\pi\_2(Sym^2(\Sigma\_2))$ via $Sym^2(\Sigma\_2)\approx \mathbb{T}^4$# $\bar{\mathbb{C}P}^2$.
We know how to compute $\pi\_1$ of $M$ # $N$ via van-Kampen's theorem. But what about higher homotopy groups? I looked in the literature and google without luck, and so I am wond... | https://mathoverflow.net/users/12310 | Homotopy Groups of Connected Sums | The 2nd homotopy group of a connect sum is fairly reasonable to compute. $\pi\_i X$ is isomorphic to $\pi\_i \tilde X$ provided $i \geq 2$ and $\tilde X$ indicates any covering space of $X$, so we might as well take the universal cover. By the Hurewicz theorem, $\pi\_2 \tilde X$ is isomorphic to $H\_2 \tilde X$. In the... | 33 | https://mathoverflow.net/users/1465 | 93285 | 54,808 |
https://mathoverflow.net/questions/92515 | 13 | Regarding the hyperfinite $II\_{1}$ factor $R$ as $C^{\*}$-algebra, is it known whether any two irreducible representations of $R$ are unitarily equivalent? If it is known that there exists a pair of irreducible representations that are not unitarily equivalent, please provide a reference.
(Note: I am not requiring t... | https://mathoverflow.net/users/6269 | Does the hyperfinite II_1 factor admit two irreducible representations that are not unitarily equivalent? | There are $2^c$ mutually non-equivalent irreducible representations. Since $\ell\_\infty(N)$ has $2^c$ many pure states (there are $2^c$ many ultrafilters on $N$), any $C^\*$-algebra containing $\ell\_\infty(N)$ has at least as much pure states. Since $R$ has $c$ many unitary elements, there are $2^c$ many mutually non... | 18 | https://mathoverflow.net/users/7591 | 93286 | 54,809 |
https://mathoverflow.net/questions/93289 | 17 | $\DeclareMathOperator{\Spec}{Spec}$
[Edit] Martin pointed out that $\dim A = 0$ does not imply that $\Spec A$ is discrete. Therefore I changed the wording of question 2.[/Edit]
---
With dimension of a ring I mean the Krull-dimension.
---
It is well-known that for a commutative ring $A$ the following are equ... | https://mathoverflow.net/users/21815 | A $0$-dimensional ring that is not noetherian | Take any compact totally disconnected Hausdorff space $X$ (for example the Cantor set, or the one-point compactification of $\mathbb{N}$). Then $\mathcal{C}(X,\mathbb{F}\_2)$ is a ring whose spectrum is homeomorphic to $X$. In particular, this ring is zero-dimensional, but this ring is noetherian iff $X$ is finite.
M... | 29 | https://mathoverflow.net/users/2841 | 93292 | 54,813 |
https://mathoverflow.net/questions/91930 | 0 | I posted this question a few days back at math.SE but could not get help. So I was kind of forced to ask here. Please excuse me if this question is not suitable here.
Let $V$ be a locally convex space, and let $K$ be convex compact set in $V$. Define $A(K)\subset C(K)$ as
$$ A(K)=\{ \phi:K\rightarrow \mathbb{C}\; ... | https://mathoverflow.net/users/21884 | Affine Homeomorphism between a compact set K and the state space on A(K) | I believe that this is Theorem 7.1 in Ken Goodearl's book "Partially ordered abelian groups with interpolation."
| 2 | https://mathoverflow.net/users/22052 | 93308 | 54,820 |
https://mathoverflow.net/questions/93310 | 8 | The title question is too vague so let me be specific.
Much of modern finite semigroup theory uses profinite semigroups and properties of profinite semigroups that depend on the existence of prime ideals in Boolean algebras, which is a choice principle weaker than AC. Often results are proved first via profinite meth... | https://mathoverflow.net/users/15934 | Is there a general result that theorems about finite structures proved in ZFC can be proved in ZF? | As long as your varieties have reasonable definitions, the axiom of choice will be eliminable from proofs like these. The point is that any finite semigroup has an isomorphic copy whose underlying set consists of natural numbers, and that copy will be in Gödel's constructible universe $L$, where the axiom of choice hol... | 8 | https://mathoverflow.net/users/6794 | 93313 | 54,821 |
https://mathoverflow.net/questions/93315 | 3 | How many simple closed curves can be put on a orientable genus $g$ surface $\Sigma\_g$ such that the following are true:
1. The curves are pairwise non-homotopic
2. The curves are pairwise set-theoretically disjoint.
I am trying to give an upper bound on this number.
| https://mathoverflow.net/users/nan | Number of non-intersecting non-homotopic simple closed curve | This is Lemma 3.2 of this [paper](http://www.fmf.uni-lj.si/~mohar/Reprints/1996/BM96_JCT68_Juvan_SystemsofCurves.pdf) by Juvan, Malnic and Mohar.
**Theorem.** Let $F$ be a family of non-null homotopic closed curves on a surface $\Sigma$ with $b \geq 0$ boundary components which are pairwise non-homotopic and pairwis... | 1 | https://mathoverflow.net/users/2233 | 93318 | 54,823 |
https://mathoverflow.net/questions/93312 | 0 | Let $y(z) = \sum\_k y\_k z^k$ be a holomorphic function in a vicinity of the point $z=0$. Define its Borel transform $By$ as a function $By(z) = \sum\_k \frac {y\_k}{k!} z^k$.
The well-know formula allows to reconstruct $y(z)$ from $By(z)$. It is given by an application of a simple integral operator. Namely
$$y(z) = ... | https://mathoverflow.net/users/13921 | An operator realizing the Borel transform | Well, your formula is essentially a Laplace transform:
$$
F(p)=\int\_0^\infty f(y) e^{-yp} dy
$$
Change variables and write $p=1/z$, assuming $p>0$, to get
$$
\frac{1}{z}F\left(\frac{1}{z}\right) = \int\_0^\infty f(t z) e^{-t} dt
$$
So you need to look up the literature on the "inverse Laplace transform".
| 1 | https://mathoverflow.net/users/454 | 93319 | 54,824 |
https://mathoverflow.net/questions/93324 | 9 | The "conservative" class theory, NBG, proves no new theorems about sets (with respect to ZFC). The choice function used here is set choice, and it's not too hard to prove (if M is a ctm for ZFC, then D(M) is a ctm for NBG and has the same set universe).
However, if we add the axiom that there is a bijection (in the c... | https://mathoverflow.net/users/15735 | Source for NBG+Equipollence conservative over ZFC? | Yes, NBG + Equipollence, which is equivalent to NBG + Global Choice, is a conservative extension of ZFC; this was independently discovered by many people (at least Cohen, Felgner, Grishin, Jensen, Kripke, and Solovay). A detailed proof can be found in
>
> Ulrich Felgner: *Comparison of the axioms of local and unive... | 11 | https://mathoverflow.net/users/12705 | 93331 | 54,829 |
https://mathoverflow.net/questions/93325 | 2 | Is there a formula for solving problems such as: If there are n coin denominations x1,x2...xn that total p cents, what is the possible number of combinations of coins that total less than or equal to p.? Where n and p are positive real numbers, of course.
On a side note, wikipedia says: "There is an explicit formula ... | https://mathoverflow.net/users/22173 | Variations of the Frobenius coin problem | Presumably the algorithm to compute the number of ways to change exactly $p$ cents (call it $n\_p$) is to write down the generating function:
$G(z) = \prod\_{i=1}^\infty \dfrac{1}{1-z^{x\_i}},$ then $n\_p$ is then simply the coefficient of $z^p.$ To get the answer to the question you are asking, just sum the coeffici... | 2 | https://mathoverflow.net/users/11142 | 93334 | 54,831 |
https://mathoverflow.net/questions/93330 | 151 | Consider a compact Riemann surface $X$ of genus $g$.
It is well-known that its fundamental group $\pi\_1(X)$ is the free group on the generators $a\_1,b\_1,...,a\_g,b\_g$ divided out by the normal subgroup generated by the single relator $[a\_1,b\_1]\cdot \ldots\cdot [a\_g,b\_g]$.
(This has of course nothing to d... | https://mathoverflow.net/users/450 | Why is the fundamental group of a compact Riemann surface not free ? | As per Theo's request, I'm posting this as an answer, though it's largely an expansion on Vitali's comment. Let $F\_n$ be the free group on $n$ letters; $K(F\_n, 1)$ is a wedge of $n$ circles and so has vanishing cohomology in degrees $>1$. On the other hand, if $X$ is a compact Riemann surface of genus $g>1$, $X$ is a... | 59 | https://mathoverflow.net/users/6950 | 93340 | 54,835 |
https://mathoverflow.net/questions/93307 | 3 | Assume Q is a quantum Lie group which allows a R matrix (with the usual
quantum Yang-Baxter equation).
Take the Jordan normal form J of R. Does a Q exist with offdiagonal J elements
(i.e. R has defective eigenvalues)?
| https://mathoverflow.net/users/11504 | A property of quantum group R matrices? | What is meant by a Quantum Lie Group here?
If we are talking about the quantized enveloping algebras of complex semisimple lie algebras $U\_q(\mathfrak{g})$ then it is a fact that the action of the universal $R$-matrix on the tensor product of finite dimensional representations is semisimple i.e. its Jordan form is d... | 2 | https://mathoverflow.net/users/22705 | 93348 | 54,840 |
https://mathoverflow.net/questions/93242 | 26 | Assuming the axiom of choice does not hold we have that there is a vector space without a basis. The situation can be, in some sense, worse. It is consistent that there are vector spaces that have two bases with completely different cardinalities.
Is anything known on when a vector space is spanned by sets of differ... | https://mathoverflow.net/users/7206 | Sizes of bases of vector spaces without the axiom of choice | Yes, ZF+BPIT implies that vector space dimension is well-defined.
[Edit: some Googling shows that James Halpern gave the same answer [back in the 1960s](http://www.ams.org/journals/proc/1966-017-03/S0002-9939-1966-0194340-1/S0002-9939-1966-0194340-1.pdf).]
Working in ZF+BPIT, fix a field $F$ and an $F$-vector space $... | 14 | https://mathoverflow.net/users/12106 | 93359 | 54,848 |
https://mathoverflow.net/questions/93362 | 23 | Over the last several years, the college-level promotion & tenure committee at my university has increasingly been seeking to apply "objective" criteria for assessing the impact of candidates' research. Journal impact factors have been a favored metric, and we have tried to argue that these are not a reliable measure, ... | https://mathoverflow.net/users/18048 | How to assess research "impact" for tenure/promotion committees | We have produced a list of top 10 journals for each area of mathematics represented in department plus a list of top 10 general subject journals so our candidates for tenure/promotion need to have publications in one of these journals. However I know for a fact that this has not stopped the administration from using im... | 3 | https://mathoverflow.net/users/20302 | 93368 | 54,853 |
https://mathoverflow.net/questions/58325 | 16 | I remember reading somewhere that the complex Fourier coefficients were introduced by an engineer sometime around 1900, but I can't find anymore this information.
Does anyone know the name of this person and where I can find a reference to it?
EDIT: I state the question more clearly: "Who was it that first wrote a ... | https://mathoverflow.net/users/13108 | complex fourier coefficients, introduced by? | In the book *Fourier Series and Wavelets*, J.P. Kahane, P.G. Lemarié-Rieusset, Gordon and Breach Publishers, 1995, pp. 1 (available as a *Publications Mathématiques d'Orsay* [here](http://portail.mathdoc.fr/PMO/afficher_notice.php?id=PMO_1994_A1)), the authors state that
"The subject matter of Fourier series consist... | 34 | https://mathoverflow.net/users/22714 | 93375 | 54,856 |
https://mathoverflow.net/questions/85606 | 12 | Color the positive integers using just two colors. By van der Waerden's theorem, we can find a $k$-term arithmetic progression as long as we consider a long interval.
I imagine it is possible to find a $k$-term arithmetic progression so that the terms in the progression have minimal gaps by possibly taking an even lo... | https://mathoverflow.net/users/3199 | Density in van der Waerden's theorem | They grow very fast. Denote by N(l) the largest integer such that we can color the numbers with two colors from 1 to N(l) without an l-long arithmetic progression. Now if you use this coloring for (k-1)/2-long sequences and put such colorings after each other, then any k-long arithmetic progression will be longer than ... | 6 | https://mathoverflow.net/users/955 | 93380 | 54,858 |
https://mathoverflow.net/questions/93295 | 10 | (I asked this on [math.stackexchange](https://math.stackexchange.com/questions/126023/separating-vectors-for-c-algebras), without response).
Let $A$ be a C$^\*$-algebra, concretely acting on a Hilbert space $H$. Suppose that $\xi\_0\in H$ is cyclic and separating for $A$ (that is, the map $A\rightarrow H, a\mapsto a(... | https://mathoverflow.net/users/406 | Separating vectors for C$^*$-algebras | The answer is yes for trace vectors, but no in general. Take a closed nowhere dense subset $C\subset[0,1]$ with positive measure and consider the state $\phi$ on $C([0,1],M\_2)$ defined by $$\phi(f)=\int\_C f(x)\_{11}\, dx + \int\_{[0,1]\setminus C}\mathrm{tr}f(x)\,dx.$$ Here $f(x)\_{11}$ is the $(1,1)$-entry of $f(x) ... | 12 | https://mathoverflow.net/users/7591 | 93383 | 54,860 |
https://mathoverflow.net/questions/93382 | 1 | Consider a two player game. There are N balls marked 1 to N. A move consists of removing a ball n and all other balls which are divisors of n (including 1). The players alternate the moves. The one who takes the last ball wins the game.
Eg. [1 ,2 ,3 , 4, 5]--[Player 1 (4, 2, 1)]-->[3, 5]--[Player 2 (3)]-->[5]--[Play... | https://mathoverflow.net/users/22693 | A modified divisor game | One can indeed prove that the first player has a winning strategy using the classical strategy stealing argument. If the second player had a winning strategy, the first player would just play "1" and then follow the second player's winning strategy. This works since any move by the second player would also remove the n... | 3 | https://mathoverflow.net/users/1061 | 93384 | 54,861 |
https://mathoverflow.net/questions/93391 | 8 | What is the fastest decay possible for the Fourier transform of a generalized function with compact support and finite value at the origin? I know that regular functions cannot attain exponential decay, due to the Paley-Wiener Theorem and that a short paper by Ingham (1934) suggests that the fastest decay in that case ... | https://mathoverflow.net/users/22718 | Fastest decay of Fourier Transform for Generalized Functions of compact support | I think this question (at least for usual functions) is related to what is known as the "Beurling-Malliavin multiplier theorem". A recent survey is "Beurling-Malliavin multiplier theorem: the seventh proof", by Mashreghi, Nazarov and Havin, St Petersburg Math J. 17 (2006), 699-744 (see <http://www.ams.org/journals/spmj... | 6 | https://mathoverflow.net/users/21428 | 93397 | 54,869 |
https://mathoverflow.net/questions/93389 | 4 | Suppose, for each $n>0$, I have two complexes of abelian groups $(A\_{n}, d)$ and $(B\_{n},d')$
and a quasi-isomorphism
$$f\_{n}: A\_{n} \rightarrow B\_{n}.$$
Furthermore, suppose I have maps of complexes $A\_1 \rightarrow A\_2 \rightarrow A\_3 \dots$
and similarly for $B\_n$, compatible with $f\_n$.
Is it true that... | https://mathoverflow.net/users/21787 | Direct limits and quasi-isomorphism | Yes indeed. You may assume that the complexes belong to a Grothendieck category if you wish, or just stick with abelian groups. (Grothendieck category: abelian category with exact filtered direct limits that possesses a generator). In the countable case, you have Milnor's exact sequence
$$
\oplus\_{n} A\_n \longright... | 5 | https://mathoverflow.net/users/6348 | 93402 | 54,871 |
https://mathoverflow.net/questions/93407 | 2 | how can i solve $ -ixDG(x,s)-iG(x,s)/2= \delta ( \frac{x}{s}-1) $
i do not know , since it is a first odrder differntial operator, the formal solution i've found would be
$ G(x,s)= \sum\_{n} \frac{u\_{n}(x)u\_{m}(s)}{\lambda \_{n}} $
where $ -ixDu\_{n}(x)-iu\_{n}(x)/2= u\_{n} \lambda \_{n} $
here G(x,s) is the ... | https://mathoverflow.net/users/21933 | dilation operator green function | You have to be specific about the boundary conditions that pick out the Green function that you want. Otherwise the answer is not unique. Let me presume "retarded" boundary conditions, namely $G(x,s) = 0$ for $x<s$. Since the left hand side of the differential equation you are trying to solve is a first order different... | 0 | https://mathoverflow.net/users/2622 | 93423 | 54,881 |
https://mathoverflow.net/questions/93434 | 10 | Given $M$ and $N$, two connected orientable manifolds of the same dimension, when is $M$ # $N$ diffeomorphic to $M$ # $\overline{N}$, where $\overline{N}$ is $N$ with the orientation reversed?
If $N$ has an orientation-reversing automorphism, is this a necessary or sufficient condition for $M$#$N$ to be diffeomorphic ... | https://mathoverflow.net/users/22431 | When is the connected sum of manifolds orientation-independent? | What kind of object do you want to consider $M \# N$ to be, an oriented manifold, or unoriented? Presumably you're taking the connect sum up to some kind of equivalence. You'll also need for $M$ and $N$ to be connected if you want connect-sum to be well-defined in any sense.
In the oriented sense, $M \# N$ is well-d... | 9 | https://mathoverflow.net/users/1465 | 93451 | 54,893 |
https://mathoverflow.net/questions/93453 | 4 | I have a question that I've been thinking for a while now.
*Can you find a set of distinct positive odd integers* $n\_1, n\_2, \ldots, n\_k$ *for some finite positive integer* $k$ *such that* $\left(\frac{1}{n\_1} + \frac{1}{n\_2} + \ldots + \frac{1}{n\_k} \right)$ *is a positive integer as well?*
This statement o... | https://mathoverflow.net/users/22733 | The sum of reciprocals of odd numbers | In fact, you can choose the $n\_i$ so that the sum is $1$: see the discussion at <http://www.ics.uci.edu/~eppstein/numth/egypt/odd-one.html>
| 11 | https://mathoverflow.net/users/22368 | 93455 | 54,895 |
https://mathoverflow.net/questions/93428 | 3 | If $A$ is a Cohen Macaulay local ring, and $B$ is a quotient ring of $A$ and $B$ is also Cohen Macaulay, Then is $B$ always a quotient by a regular sequence of $A$?
| https://mathoverflow.net/users/21742 | A problem on Cohen Macaulay ring | Here is an even better example:
$A=(k[x,y,z]/(xy-z^2))\_{\mathfrak m}$ with ${\mathfrak m}=(x,y,z)$, $B=k[x]\_{(x)}$.
Here $A$ is Gorenstein and $B$ is regular. The map is given by $x\mapsto x$ and $y,z\mapsto 0$.
Geometrically, $B$ corresponds to a line going through the vertex of a quadric cone corresponding to $... | 2 | https://mathoverflow.net/users/10076 | 93460 | 54,898 |
https://mathoverflow.net/questions/93427 | 4 | Consider $n$ individuals {$1,2, \ldots, n$}. For each (unordered) pair of individuals $i \neq j$ we consider a random variable $X\_{i,j}$ that can be thought of as the distance between $i$ and $j$. Each individual kills its closest neighbour (everything happens at the same time). Can we say anything about the distribut... | https://mathoverflow.net/users/1590 | "birds on wire" type problem | The number of people who kill any given $i$ has asymptotically Poisson distribution (since the events that $j$ kills $i$ for different $j$s are almost independent). Thus the number of survivors $S$ is roughly $N/e$. Since different $i$s are also nearly independent, $S$ is roughly normal.
The oriented version, where d... | 2 | https://mathoverflow.net/users/9422 | 93473 | 54,902 |
https://mathoverflow.net/questions/93468 | 5 | I'm trying to understand analogies and disanalogies between ${\Bbb R}$, the reals numbers, and ${\bf No}$, the surreal numbers.
${\Bbb R}$ admits countable dense sets such as the rationals. This leads me to wonder whether ${\bf No}$ might contain a dense subclass somehow smaller than the whole.
On one hand, any su... | https://mathoverflow.net/users/10909 | Definable map from all the ordinals to the surreal numbers with a dense image? | The answer is that the existence of a definable class embedding
like that is independent of ZFC. In fact, it is equivalent to the
axiom V=HOD.
**Theorem.** The following are equivalent.
1. There is a definable bijection from Ord to No.
2. There is a definable surjection from Ord to No.
3. There is a definable map ... | 7 | https://mathoverflow.net/users/1946 | 93485 | 54,909 |
https://mathoverflow.net/questions/93481 | 7 | I have proved a simple Lemma that I need for a larger result, and I was wondering whether it is actually another more famous result in disguise.
The lemma says that for any set of vectors in $\mathbb{R}^n$, it is possible to choose a set of coefficients in ${-1,+1}$ such that the weighted sum of the vectors is close ... | https://mathoverflow.net/users/4997 | More general form of inequality? | I don't have access right now to search tools, but the result is is due to Steinitz and might be described in
Kadets, Mikhail I.; Kadets, Vladimir M. (1997). "Chapter 2.1 Steinitz's theorem on the sum range of a series, Chapter 7 The Steinitz theorem and B-convexity". Series in Banach spaces: Conditional and uncondi... | 10 | https://mathoverflow.net/users/2554 | 93500 | 54,919 |
https://mathoverflow.net/questions/93501 | 5 | Let p>3 be a prime. For every a=1,2,...,p-1 let x(a) be the number in {1,...,p-1} such that ax(a) is congruent to 1 mod p (its inverse mod p). Let S be the sum of ax(a)^2 and let T be the sum of x(a) (in both cases sum over all a=1,2,...,p-1). Is it true that S-2T is always divisible by p^2? I noted this by computing f... | https://mathoverflow.net/users/22750 | sums modulo p^2 | This follows from well-known results (although perhaps not in a totally straightforward way).
Let us write $x(a) = a^{-1} + p y(a)$, where $a^{-1}$ denotes the inverse of $a$ modulo $p^2$, and $y(a) \in \{0, \dots, p-1\}$. Then we find that
$$ a x(a)^2 = a^{-1} + 2p y(a) \pmod{p^2} $$
so
$$ S = \sum\_{a = 1}^{p-1} a... | 11 | https://mathoverflow.net/users/2481 | 93505 | 54,922 |
https://mathoverflow.net/questions/91790 | 4 | Suppose $X$ and $Y$ are topological spaces. Let's define the join $X\ast Y$ as the quotient space $X\times Y\times [0,1]/\sim$, where $\sim$ is the equivalence relation generated by $(x,y,0)\sim(x,y',0)$ and $(x,y,1)\sim(x',y,1)$. In particular, define the cone over $X$, $Cone(X)$, as the join of $X$ with a point. Is i... | https://mathoverflow.net/users/19671 | Cone over the Join of two topological spaces | If you use initial topologies to define the join, as in Section 5.7 of my book "Topology and groupoids", then the result you want is exactly 5.7.4 on p. 174, and the picture for it is as suggested by Tyler (Fig. 5.7).
Of course it can't be true generally with quotient topologies, as products don't preserve quotients... | 3 | https://mathoverflow.net/users/19949 | 93507 | 54,924 |
https://mathoverflow.net/questions/93394 | 3 | Hello,
I hope that you can help me with this.
Let P be a set of points in the plane, such that |p|=n, what is the maximal number of open discs containing atleast k points for some k, two discs are equivalent if they contain the same points.
| https://mathoverflow.net/users/22722 | upper bound for the number of open discs containing k points in the plane | Hmm, i think i have an idea, the number of distinct open discs containg atleast k points, for k>2 is bounded by ${n \choose 3}$, since every disc is uniquely defined by the 3 points contained in it and closest to its boundry. every 3 points form a triangle bounded by some disc.
suppose two diffrent discs have the same ... | 0 | https://mathoverflow.net/users/22722 | 93508 | 54,925 |
https://mathoverflow.net/questions/67418 | 5 | I have proved a few years ago the following proposition:
>
> There exists $f: [0,1] \to [0,1]$ with Darboux property such that there exist $A,B \subset[0,1]$ with $A\cap B=\emptyset,\ A \cup B=[0,1]$ with $f(A)\subset B$ and $f(B)\subset A$. (of course $A,B\neq \emptyset$)
>
>
>
A function $f : I\subset \Bbb{R... | https://mathoverflow.net/users/13093 | Darboux function on $[0,1]$ with interesting property | Just a comment, and a hint. The property of the function $f$ that you consider, that is, *The interval $[0,1]$ admits a partition in two sets that are mapped into each other by $f$,* may be restated equivalently as: *No iterate of $f$ of odd order has a fixed point (or also, $f$ has no periodic point of odd period*). T... | 4 | https://mathoverflow.net/users/6101 | 93510 | 54,927 |
https://mathoverflow.net/questions/91271 | 10 | Consider Leech lattice definition provided by Wilson (Octonions and Leech lattice, 2008).
There are 819 E8 sublattices defined by
$ (2\lambda, 0, 0); $
$ (\lambda \overline{s}, (\lambda \overline{s}) j, 0); $
$ ( (\lambda s)j, \lambda k, (\lambda j) k ) $
where $\lambda$ span 240 vectors of E8 lattice, j,k are 16... | https://mathoverflow.net/users/nan | Octonionic reflection groups | Yes, such groups are interesting, see this paper by Daniel Alcock, "Reflection groups on the octave hyperbolic plane," <http://www.ma.utexas.edu/users/allcock/research/oh2.pdf>
If you write to Daniel, he will probably give you more references.
| 8 | https://mathoverflow.net/users/21684 | 93520 | 54,929 |
https://mathoverflow.net/questions/93408 | 7 | Geometric topology is more motivated by objects it wants to prove theorems about. Geometric topology is very much motivated by low-dimensional phenomena -- and the very notion of low-dimensional phenomena being special is due to the existence of a big tool called the Whitney Trick, which allows one to readily convert c... | https://mathoverflow.net/users/22728 | Reference on Geometric Topology | Geometric topology is not really a unique field in the way that algebraic topology and general topology are. Its various subareas may share something of a common feel (and indeed an arxiv category), but are often too diverse to have any common techniques. Those areas include, for instance:
Low-dimensional topology (c... | 16 | https://mathoverflow.net/users/10819 | 93523 | 54,930 |
https://mathoverflow.net/questions/93521 | 4 | Take a convex polygon $P$ in the plane, and find its minimum area bounding rectangle, $R$. I'm interested in the ratio of the area of $R$ to the area of $P$. The ratio has a minimum of $1$ for rectangles and is $2$ for triangles. I want to say that $2$ is the maximum ratio possible.
Is this known? Is there a nice pro... | https://mathoverflow.net/users/19029 | Area ratio of a minimum bounding rectangle of a convex polygon | I believe this is easy to prove. Assume you have at least four points and take the longest chord $C$ of the polygon. Now choose your rectangle such that two edges are parallel and the same length as $C$. Now observe that since each "half" of the polygon contains a triangle inscribed in the "half" rectangle (where "half... | 7 | https://mathoverflow.net/users/4580 | 93528 | 54,933 |
https://mathoverflow.net/questions/93522 | 11 | Given a compact Kaehler manifold $M$ of complex dimension $2n$, there are essentially two ways to compute its signature $\sigma(M)$, i.e. the index of the intersection form on $H\_{2n}(M,\mathbb{R})$:
1.by Hodge index theorem $\sigma(M)=\sum\_{p,q}(-1)^p h^{p,q}$, here $h^{p,q}$ stands for the Hodge numbers.
2.by H... | https://mathoverflow.net/users/22727 | Two approaches to compute the signature of a Kaehler manifold | About your last question, a recent theorem of Kotschick-Schreieder (see <http://arxiv.org/abs/1202.2676> page 2) says that a linear combination of Hodge numbers equals a linear combination of Chern numbers for all projective manifolds (modulo the usual Kähler symmetries) iff it is a linear combination of the numbers $\... | 14 | https://mathoverflow.net/users/13168 | 93534 | 54,935 |
https://mathoverflow.net/questions/93515 | 7 | Let $A(t)$ be a $n\times n$-matrix-valued continuous (plus possibly other niceness conditions; see below) curve, with the matrix entries being complex in general. If I am not mistaken, $A(t)$ generates a minimal Lie algebra $\mathfrak{g}$ of matrices, in the sense that $\mathfrak{g}$ is the intersection of all $n\times... | https://mathoverflow.net/users/22757 | Lie algebra "generated" by matrix-valued curve? | It can't always be done. Take $n=2$, and start with a smooth curve $V:[0,1]\to SL(2,\mathbb{C})$ such that $V(0) = \mathbb{1}\_2$ and
$$
V(1) = \pmatrix{-1&1\\\\ 0&-1}.
$$
Set $A(t) = V'(t)V(t)^{-1}$. Then $A:[0,1]\to{\frak{sl}}(2,\mathbb{C})$, and it satisfies $V'(t) = A(t)V(t)$ with $V(0) = \mathbb{1}\_2$, but $V(... | 8 | https://mathoverflow.net/users/13972 | 93537 | 54,938 |
https://mathoverflow.net/questions/93525 | 3 | Let $D,E \subset \mathbb{C}^3$ be prime divisors where $D$ is ***smooth*** and $E$ is **not necessarily smooth**.
Assume that $D \cap E$ has ***SNC*** support and let
$D \cap E = \bigcup \Gamma\_i$ be a decomposition into irreducible components.
**(Added)** Let $f\_1 \colon X\_1 \rightarrow \mathbb{C}^3$ be a bl... | https://mathoverflow.net/users/12390 | Blowing up intersections of two divisors make them disjoint? | I'm not exactly sure what you want (see Sasha's question above) but algorithms for embedded resolution of singularities might do what you want (for example, see the recent book of Kollár, various papers by Villamayor and co-authors, or Wlodarczyk)
Indeed, do a log resolution (ie, principalization) of the ideal $I\_D ... | 3 | https://mathoverflow.net/users/3521 | 93540 | 54,940 |
https://mathoverflow.net/questions/93516 | 10 | Consider the class of bounded sequences to which every Banach limit (non-negative shift-invariant continuous functional on $l^\infty$ taking convergent sequences in the usual sense to their limits) assigns the same limit value.
**Does a sequence belong to this class if its Cesaro means have a limit?**
Also, is th... | https://mathoverflow.net/users/18483 | Cesaro means and Banach limits | We can characterize Banach limits as continuous functionals on $\ell^\infty$ which vanish on
$$ X := \{(x\_n - x\_{n+1}): (x\_n) \in \ell^\infty\} $$
and which send the constant sequence $(1,1,\dots)$ to $1$.
Note that $X$ is a subspace.
The Hahn-Banach Theorem tells us that we are asking: if $(y\_n) \in \ell^\infty$... | 13 | https://mathoverflow.net/users/22052 | 93547 | 54,942 |
https://mathoverflow.net/questions/93549 | 7 | It is easy to visualize some Morse functions on surfaces (such as the torus) via the height function, but this seemingly doesn't work for 3-manifolds. I am looking for an explicit one on the 3-torus $\mathbb{T}^3$. In paritcular:
It is true that every nice-enough 3-manifold admits a self-indexing Morse function $f:M... | https://mathoverflow.net/users/12310 | Morse Function on 3-Torus from Heegaard Splitting | The Morse function on $(S^1)^3$ that gives the genus 3 Heegaard splitting (the standard one) is basically just a smoothed "distance from the 1-skeleton function".
So if you think of $S^1$ as the unit circle in $\mathbb C$, then
$$f : (S^1)^3 \to \mathbb R$$
is given by
$$f(z\_1,z\_2,z\_3) = |z\_1-1|^2 + |z\_2... | 13 | https://mathoverflow.net/users/1465 | 93552 | 54,944 |
https://mathoverflow.net/questions/93545 | 4 | As the subject says, I have some questions about some Ruzsa-type inequalities for additive-energy. It would be wonderful if anyone could shed some light.
First, the definition: let $E(A,B)$ be the number of all quadruples $(a,a',b,b') \in A \times A \times B \times B$ such that $a+b = a' + b'$ denote the additive ene... | https://mathoverflow.net/users/16321 | Ruzsa-type inequalities for additive energy | 1) No. Let $A=B=C$ be a set that is a union of interval $\{1,\dots,N\}$ and $N$ random elements from $\{1,\dotsc,N^2\}$. Then $F(A,A)=KN$ for some constant $K$. On the other hand, $A+A$ is basically the interval $\{1,\dotsc,N^2\}$. So, $F(A,A+A)\approx N^2$.
2) Example above show that it fails for $k=1$ and $l=2$ (f... | 5 | https://mathoverflow.net/users/806 | 93555 | 54,946 |
https://mathoverflow.net/questions/88507 | 3 | I am studying an article: *The parametric problem of capillarity: the case of two and three fluids*, by U. Massari. In one of his proofs, he uses an inequality I can't manage to prove. It is like this:
>
> Let $\Omega \subset \Bbb{R}^n$ be an open, bounded set, with Lipschitz boundary with constant $L$. The followi... | https://mathoverflow.net/users/13093 | Inequality involving perimeter and area | I have found an article which deals with this kind of inequalities. It is available in the following link: [Funzioni BV e Tracce](http://archive.numdam.org/ARCHIVE/RSMUP/RSMUP_1978__60_/RSMUP_1978__60__1_0/RSMUP_1978__60__1_0.pdf)
| 4 | https://mathoverflow.net/users/13093 | 93559 | 54,948 |
https://mathoverflow.net/questions/93012 | 0 | We have $k$ different types of coupons (with replacement).If we collect at least $l$ different coupons, we win a prize. We can only afford to collect $m$ coupons.
Let's say we take all those $m$ coupons, and we collect exactly $L$ different coupons. Then the probability that we win a prize is $P(L \geq l)$
Let $T$ ... | https://mathoverflow.net/users/20886 | Rigorous proof of the duality of Coupon collector's problem and Occupancy problem | To specify the probability space is rarely a good idea but since you insist on it, a possible choice would be $\Omega=K^\mathbb N$, where $K=\{1,2,\ldots,k\}$ denotes the set of types of coupons, endowed with the sigma-algebra ${\mathcal K}=2^K$ and the discrete measure $p=(p\_i)$ on $(K,\mathcal K)$ describing the pro... | 6 | https://mathoverflow.net/users/4661 | 93561 | 54,950 |
https://mathoverflow.net/questions/87425 | 1 | Consider the following definition.
Let $C$ be a cycle of a simple graph $G$. We say that $C$ is *convex* if for any pair of distinct vertices $u,v \in V(C)$ $$ d\_C(u,v) < d\_{G-C}(u,v).$$
Is there any other name for such cycles? I was trying to find out some references/literature presenting results related to such... | https://mathoverflow.net/users/1737 | Definition of convex cycles | In the paper entitled Convex cycle bases and Cartesian products
by Hellmuth, Leydold and Stadler, I found the following characterization of convex cycles
Let $G$ be a simple graph and $C \subseteq G$ a cycle. If $|C|$ is odd then $C$ is convex if and only if for every edge $e = xy \in C$ there exist a unique vertex... | 1 | https://mathoverflow.net/users/1737 | 93564 | 54,952 |
https://mathoverflow.net/questions/93541 | 2 | I'm currently studying the implicitization of bezier curves (that is, finding a function that *f(x, y) = 0* for any x and y pairs of a curve *p(t)*) as part of an algorithm for curve intersection. The approach I'm taking involves taking the resultant of two functions known as "moving lines." For a cubic curve, the two ... | https://mathoverflow.net/users/21860 | The resultant of two degree n and n - 1 functions in two variables of t | I don't really understand the question. You seem to want to eliminate the variable $t,$ which you can do with a single application of the Sylvester resultant (that is, the determinant of the [Sylvester matrix](http://en.wikipedia.org/wiki/Sylvester_matrix)). The determinant of the two polynomials (thought of as polynom... | 3 | https://mathoverflow.net/users/11142 | 93568 | 54,954 |
https://mathoverflow.net/questions/93572 | 6 | The technique of faithfully flat descent, and, in the case of vector spaces, Galois descent has been used quite a bit in Algebraic Geometry. However, the question of whether, say, a given $k$-vector space $V$ arises from some $L$-vector space $W$ seems like it could be asked in a wide variety of settings. I'm wondering... | https://mathoverflow.net/users/21254 | Applications of Descent? | The question is a little too vague to have a really satisfactory answer, but obviously descent
is omnipresent in (modular) representation theory.
For example, the basic result that two representations of a group $G$ on some $k$-vector spaces are isomorphic iff they become isomorphic on some extension $L$ of $k$ is a... | 5 | https://mathoverflow.net/users/9317 | 93574 | 54,959 |
https://mathoverflow.net/questions/93578 | 5 | I am not sure if this problem is of the appropriate difficulty for math overflow, but here it is.
Suppose we are considering pairs $(x,y)$ with $1 \leq x,y \leq p-1$ for some prime $p$. As points over $\mathbb{F}\_p^2$, we can define the usual projective equivalence relation where we consider $(x,y)$ and $(x', y')$ t... | https://mathoverflow.net/users/10898 | A question regarding simultaneous congruences | The size of the exceptional set is $\Omega(p^2)$ and indeed, there are at most $\frac6{\pi^2}(1+o(1))p^2$ pairs, projectively equivalent to a point inside the square $Q:=[1,N]\times[1,N]$, where $N=\lfloor\sqrt p\rfloor$. To see this, notice that if a point from $Q$ is an integer multiple of another point from $Q$, the... | 8 | https://mathoverflow.net/users/9924 | 93585 | 54,965 |
https://mathoverflow.net/questions/93322 | 2 | Let $G$ be a locally profinite group and $K$ an open compact subgroup (mod the center), then Bushnell has shown that the following three statements are equivalent for a finite dimensional representation $\rho$ of $K$:
* the compact induction $ind\_K^G \rho$ is admissible
* compact induction is isomorphic to the usual... | https://mathoverflow.net/users/10400 | When is compact induction in GL(2) from an open compact group admissible? | The answer to the second question is no. Example: $G=GL\_2(F)$, $K$ stabilizer of an edge $e$ in the Bruhat-Tits tree, and $\rho$ such that the induction is irreducible $=: \pi$. Now let $K'$ be the stabilizer of a vertex of $e$, and $\rho'$ induction to $K'$ of the restriction of $\rho$
to the pointwise stabilizer of ... | 4 | https://mathoverflow.net/users/9852 | 93599 | 54,970 |
https://mathoverflow.net/questions/93603 | 13 | In category theory, a monomorphism (also called a monic morphism or a mono) is defined to be a left-cancellative morphism. It seems that this definition generalizes the definition of injections. However, even in a concrete category, a monomorphism may not be an injection. Why could this happen? I know examples of monom... | https://mathoverflow.net/users/15770 | Why the underlying function of a monomorphism may not be an injection | There are several issues here. Let $\mathcal C$ be a category.
(1) The question only makes sense when $\mathcal C$ is concrete, as otherwise "injective" has no meaning. When $\mathcal C$ is concrete, we can choose a faithful functor $\pi : \mathcal C\to Set$, and regard a morphism $f:X\to Y$ as being injective if $\p... | 19 | https://mathoverflow.net/users/7399 | 93607 | 54,974 |
https://mathoverflow.net/questions/93621 | 12 | As we know, most of the spectral sequences are doubly graded. However, this "doubly graded" condition is not a part of the formal definition of spectral sequence. Is there any useful triply (quadruply, quintuply, etc.) graded spectral sequences? If not, is there a hope that some meaningful work can be done with this to... | https://mathoverflow.net/users/15770 | Triply graded spectral sequence? | Well, there is an eponymous spectral sequence in my thesis (still never published
in full, but there is an announcement, stuff about it in Ravenel's book, and papers by
Tangora and others). Quite generally, take a connected graded algebra $A$ over a field $k$, filter it for example by the powers of its augmentation id... | 12 | https://mathoverflow.net/users/14447 | 93623 | 54,981 |
https://mathoverflow.net/questions/93610 | 11 | This is probably a silly question, but I figured that if there is a good answer, this would be a good place to ask.
Ever since I got my hands on the book "Toric Varieties" by Cox, Little and Schenck, I've been excited to learn about the different combinatorial properties of polytopes that one can deduce from the corr... | https://mathoverflow.net/users/2384 | Detecting tilings by toric geometry | A related question (but not exactly the one you asked) is:
>
> Can one tell if a convex polytope $P$ and its translations by $\mathbb Z^n$ tile $\mathbb R^n$? Which polytopes $P$ have this property?
>
>
>
Fix some positive quadratic form $q$ on $\mathbb R^n$ and the corresponding distance function. Let $P^0$ ... | 11 | https://mathoverflow.net/users/1784 | 93625 | 54,982 |
https://mathoverflow.net/questions/93562 | 7 | [Waldhausen](http://www.math.uni-bielefeld.de/~fw/LNM1126_318-419.pdf)'s definition of a category with cofibrations includes the choice of a *distinguished* zero object. Probably this also means that an exact functor should preserve zero objects *on the nose* (Waldhausen writes "takes $\\*$ to $\\*$"). Isn't this rathe... | https://mathoverflow.net/users/2841 | Choice of base point in a Waldhausen category | When Waldhausen defines $K$-theory, he considers sequences of the form
$\* \rightarrow A\_1 \rightarrow A\_2 \rightarrow \cdots \rightarrow A\_n$
for varying $n$, with all maps cofibrations. For any fixed $n$, these sequences together with the obvious morphisms again form a category, and these categories nearly ass... | 9 | https://mathoverflow.net/users/18256 | 93627 | 54,983 |
https://mathoverflow.net/questions/93602 | 1 | I am reading John McCleary's A User's Guide to Spectral Sequence and was quite confused about one result: On page 15 of the version I was reading, it says that if $E^{\star,\star}\_2$ is the bigraded vector space in Example 1.E, then $P(E^{\star,\star}\_2,t)=(1+t^{11})(1+t^4+t^8+t^{12})(1+t^3)$. I am quite confused on ... | https://mathoverflow.net/users/15770 | A Question on McCleary's book on Spectral Sequences | I think the Poincar\'{e} polynomial should be $P(E\_2^{\*,\*},t)=(1+t^{8})(1+t^3+t^6+t^9)(1+t^2)$ instead of the one you mentioned so this is an erratum in the book. Its clear from the algebra structure that there can not be any term of degree 26.
| 1 | https://mathoverflow.net/users/14124 | 93629 | 54,984 |
https://mathoverflow.net/questions/93620 | 1 | I am interested in conditions in terms of standard scales of summation methods that guarantee the existence of an averaged limit for all [almost convergent sequences](http://en.wikipedia.org/wiki/Almost_convergent_sequence). For the Cesaro summation method $(C, 1)$ [this fails](https://mathoverflow.net/questions/93516/... | https://mathoverflow.net/users/18483 | Cesaro means for $\alpha<1$ and Banach limits | The paper G.G. Lorentz: A contribution to the theory of divergent sequences; Acta mathematica, Volume 80, Number 1, 1960, 167-190; DOI: [10.1007/BF02393648](http://dx.doi.org/10.1007/BF02393648), contains several interesting results related to your questions on Banach limits.
A characterization of matrix methods that... | 3 | https://mathoverflow.net/users/8250 | 93640 | 54,987 |
https://mathoverflow.net/questions/93642 | 7 | I was wondering if it was possible to extend the De Rham theorem to differentiable spaces, as defined for instance in [this paper](http://www.jstor.org/stable/10.2307/1970846) of Chen. If it has been done, could anyone come up with a good reference?
Alternatively, I could use a reference proving the "classical" De Rh... | https://mathoverflow.net/users/21180 | De Rham theorem for differentiable spaces | You can also try Chapters 1 and 2 of
>
> Johan L. Dupont: Curvature and Characteristic Classes, Lecture Notes in Mathematics, vol. 640, Springer Verlag, 1978.
>
>
>
| 3 | https://mathoverflow.net/users/20302 | 93651 | 54,992 |
https://mathoverflow.net/questions/93618 | 5 | Dear community,
I'm looking for a table that lists the full commutation relations for the Chevalley group $G\_2$. I could do this myself (although it would take quite a while), but maybe there is already such a table out there?
Sincerely, and thank you,
Moshe Adrian
| https://mathoverflow.net/users/8891 | Commutator table for Chevalley group G_2 | It would help to clarify the context of "Chevalley group" here, but over any field one gets uniform results by starting with a Chevalley basis for the associated complex Lie algebra. The commutation relations for rank 2 groups including $G\_2$ are given explicitly in SGA3 and written down similarly in my 1975 Springer ... | 8 | https://mathoverflow.net/users/4231 | 93655 | 54,996 |
https://mathoverflow.net/questions/93638 | 23 | Let $X=C^{\alpha}(\Omega,\mathbb{R})$ be the space of Hölder continuous functions. What is its dual?
| https://mathoverflow.net/users/19874 | Dual of the space of Hölder continuous functions? | Just a few words about how to get a representation for the dual, since the details on this topic are certainly treated in the literature.
To fix notations, assume e.g. $\Omega\subset\mathbb{R}^n$ be an open neighborhood of $0$ , let $\Delta\_\Omega\subset\Omega\times\Omega$ denote the diagonal, and $\tilde\Omega:=(\... | 20 | https://mathoverflow.net/users/6101 | 93667 | 55,004 |
https://mathoverflow.net/questions/93668 | 10 | Let $R$ be a commutative ring and $M$ an $R$-module.
The module $M$ is finitely generated iff there is an exact sequence $R^{k\_0} \to M \to 0$.
Similarly, $M$ is finitely presented iff there is an exact sequence $R^{k\_1} \to R^{k\_0} \to M \to 0$.
It seems we could generalize this as follows:
for $n \in \mathbb{Z}\_{... | https://mathoverflow.net/users/22804 | Generalization of finitely generated, finitely presented modules? | All these notions have been defined and studied long time ago. Serre called a module type $FL\_n$ if it is finitely $n$-presented in your terminology. Type $FL\_\infty$ and type $FL$ is used for finitely $\infty$-presented and finitely $\omega$-presented. They are studied a lot for group rings ($R$ does not need to be ... | 10 | https://mathoverflow.net/users/8176 | 93674 | 55,007 |
https://mathoverflow.net/questions/93265 | 13 | Notation: If $G$ is a countable group and $H$ is a subgroup, for $g\in G$, let $|\mathcal{O}\_{gH}|$ be the size of the $H$-orbit of $gH$ in the $H$-set $G/H$.
Does there exist a countable group $G$ and a subgroup $H$ with $[G\colon H]=\infty$ such that:
1. There is a $g\in G$ with $|\mathcal{O} \_{gH}|\neq |\math... | https://mathoverflow.net/users/351 | Does such an infinite index subgroup exist? | I believe (1 and 2) and (3) are mutually exclusive. Here is a proof:
First, the commensurator
$$
Comm\_G(H) = \{g\in G : |\mathcal{O} \_{gH}|, |\mathcal{O} \_{g^{-1}H}|<\infty\}
$$
is a group. We will show:
>
> Lemma: $\varphi\colon Comm\_G(H)\to \mathbb{Q}\_{>0}$ by
> $g\mapsto \displaystyle\frac{[H\colon H\... | 7 | https://mathoverflow.net/users/351 | 93684 | 55,010 |
https://mathoverflow.net/questions/93435 | 3 | Consider the following impartial combinatorial game, a generalization of [Chomp!](http://en.wikipedia.org/wiki/Chomp) as mentioned in the paper by [David Gale](http://www.claremontmckenna.edu/math/moneill/Math188/papers/Gale2.pdf): At each step, we have a finite partially ordered set $S$. Player I or II chooses some $s... | https://mathoverflow.net/users/2841 | Chomp! without the law of the excluded middle | This is an attempt to answer the question, as revised by Emil to agree with Gale's theorem, assuming that the poset $S$ is finite (which I take to mean K-finite, equivalently a surjective image of $\{0,1,\dots,n-1\}$ for some natural number $n$). I do not require the order-relation or even the equality relation on $S$ ... | 5 | https://mathoverflow.net/users/6794 | 93697 | 55,016 |
https://mathoverflow.net/questions/93710 | 3 | Given a projective curve $C$, is it possible that $C$ can degenerate into union of lines i.e., does there exist a family of curves $\pi:\mathcal{C} \to B$ such that $\pi^{-1}(0)=C$ and there exists $a \in B$ such that $\pi^{-1}(a)$ is the union of lines?
| https://mathoverflow.net/users/9164 | Degeneration of projective curves | Yes, in the following sense. Pick a trivalent graph $G$ with $v$ vertices and regard it as the dual complex of the stable curve $E$ consisting of one copy of $\mathbb P^1$ for each vertex of $G$ and one node for each edge. The genus $g$ of $E$ is given by $2g-2=v$. The stack of stable curves of genus $g$ is irreducible... | 12 | https://mathoverflow.net/users/8726 | 93712 | 55,025 |
https://mathoverflow.net/questions/93716 | 17 | Background
----------
Recently, I have been writing up some notes on derived functors and I came across the Eilenberg-Watts theorems [1], which essentially explain why it is hard to find derived functors besides Ext and Tor. Before asking my question, allow me to briefly state these theorems.
Let $R$ and $S$ be rin... | https://mathoverflow.net/users/1437 | Cov. right-exact additive functors that don't commute with direct sums? | Here is a specific example, though it admits obvious generalizations. Let $R=S=\mathbb{Z}$, and consider the functor from abelian groups to abelian groups defined by
$$T(X) = \mathrm{Ext}^1\_{\mathbb{Z}}(\mathbb{Q}\_p/\mathbb{Z}\_p,X),$$
where $p$ is a prime.
This is right-exact since $\mathrm{Ext}^2\_{\mathbb{Z}}\equ... | 17 | https://mathoverflow.net/users/437 | 93720 | 55,027 |
https://mathoverflow.net/questions/72027 | 13 | **2-cocycle twists of Hopf algebras**
-------------------------------------
Let $H$ be a Hopf algebra over a field $k$. Then a (left, unital) 2-cocycle on $H$ is a map
$$ f: H \otimes H \to k$$
such that
$$ f(x\_{(1)},y\_{(1)})f(x\_{(2)} y\_{(2)}, z) = f(y\_{(1)}, z\_{(1)}) f(x, y\_{(2)} z\_{(2)}) $$
(in Sweedler not... | https://mathoverflow.net/users/703 | 2-cocycle twists of braided Hopf algebras | The two concepts - twisting a Hopf algebra one-sided to an algebra and two-sided to a new Hopf algebra - are actually intimately connected and play an important role in several areas of current research. The former are known (as always, there's restrictions on the equivalence, e.g. in infinite dimension) as **Galois ob... | 9 | https://mathoverflow.net/users/22709 | 93724 | 55,029 |
https://mathoverflow.net/questions/93274 | 6 | Wielandt wrote a paper titled "Remarks on diagonable matrices".
According to Mathematische Werke - Mathematical Works : Linear Algebra and Analysis
by Helmut Wielandt, Hans Schneider, Bertram Huppert (Editor) page 260 this paper from Wielandt remained unpublished (at least from the 1950s to the 1980s).
Does anyone ... | https://mathoverflow.net/users/3958 | Unpublished work of Wielandt | Now that the server is back up, I am posting this as a real answer.
With some work, you might be able to find the proof in Wielandt's notebooks, which were TeXxed and put online [here](http://www3.math.tu-berlin.de/numerik/Wielandt/index_en.html).
The TeX source files are also published, so you can download them an... | 4 | https://mathoverflow.net/users/1898 | 93742 | 55,039 |
https://mathoverflow.net/questions/93681 | 3 | Let G be a locally compact Hausdorff group, and C\*(G) the full group C\* algebra.
I found in some books that "representation theory of both is the same". Can this be expressed as "the categories are equal"?
(Here "representation of G" means a unitary weakly continuous representation on a Hilbert space, and represe... | https://mathoverflow.net/users/22789 | Are the categories of representations of G and C*(G) isomorphic? | Following Marc's suggestion, I'll make my comments into a more extended answer.
**The answer is yes.** Every unitrary representation of $G$ can be "integrated" to get a representation of $C^{\ast}(G)$ and every representation of $C^{\ast}(G)$ can be "separated" to get a representation of $G$ (the terms 'integrating'... | 4 | https://mathoverflow.net/users/12248 | 93746 | 55,040 |
https://mathoverflow.net/questions/93544 | 17 | [EDIT]: After getting a very nice answer by Kevin Buzzard I realize that my question was a little bit too vague and I try to restate it more precisely.
As the title says, I would like to understand an isomorphism of Hida from a more geometric perspective than what I normally read. What bothers me is that there are tw... | https://mathoverflow.net/users/18238 | Geometric interpretation of Hida isomorphism | I am not sure what your criteria would be for a proof to be given a geometric interpretation, but the reason why weights "disappear" when we take the inverse limit on the level stems from the contraction property of Hecke operators (at $p$), or informally from the fact that Hecke operators at $p$ diminish the level.
... | 7 | https://mathoverflow.net/users/2284 | 93747 | 55,041 |
https://mathoverflow.net/questions/93744 | 11 | There is a well-known estimate for the sum of all binomial coefficients $\binom{n}{k}$ satisfying $k \leq \alpha n$ for some $\alpha$ satisfying $0 < \alpha \leq 1/2$:
$$ \sum\_{k=0}^{\alpha n}\binom{n}{k} = 2^{(H(\alpha) + o(1))n}$$
where $H(\alpha) = -\alpha\log\_2(\alpha) - (1-\alpha)\log\_2(1-\alpha)$ is the bi... | https://mathoverflow.net/users/4974 | Estimating a partial sum of weighted binomial coefficients | Expanding on the previous answers. I'm taking $\lambda$ and $\alpha$ to be constants which do not vary as $n\to\infty$.
If $α<λ/(λ+1)$ then the sum is within a constant of the last term. In fact the largest terms are approximately in geometric progression so you can get it quite accurately by computing the ratio.
... | 10 | https://mathoverflow.net/users/9025 | 93756 | 55,046 |
https://mathoverflow.net/questions/93719 | 15 | What is an example of a 0-dimensional locally compact Hausdorff space $X$ for which the Stone-Čech compactification $\beta(X)$ is *not* 0-dimensional?
It is known that if $X$ is a 0-dimensional locally compact Hausdorff space which is also paracompact, then $\beta(X)$ is 0-dimensional. (Engelking 1989, Th. 6.2.9). I ... | https://mathoverflow.net/users/20300 | Is Stone-Čech compactification of 0-dimensional space also 0-dimensional? | In [this paper](http://www.sciencedirect.com/science/article/pii/0166864180900206) (*Spaces $N\cup\mathscr{R}$ and their dimensions, Topol. Appl.
11(1) 1980 93-102*) — I hope the PDF is freely available) Jun Terasawa constructs maximal almost disjoint families on $\mathbb{N}$ whose associated spaces can have any dimens... | 16 | https://mathoverflow.net/users/5903 | 93762 | 55,052 |
https://mathoverflow.net/questions/93732 | 3 | Is the holonomy group for general (not necessarily Riemannian) compact manifolds compact?
I believe this is true for Riemannian manifolds, according to Berger's classification.
Any insights would be appreciated.
| https://mathoverflow.net/users/11084 | Holonomy of compact manifolds | You are misinterpreting Berger's theorem; it's not even true for compact Riemannian manifolds. See *On compact Riemannian manifolds with noncompact holonomy groups*, Burkhard Wilking. J. Differential Geom. Volume 52, Number 2 (1999), 223-257.
What *is* true is that, for a simply-connected Riemannian manifold, the hol... | 12 | https://mathoverflow.net/users/13972 | 93766 | 55,054 |
https://mathoverflow.net/questions/93693 | 8 | A geodesic on a Riemannian manifold is generally not the shortest among nearby curves with the same endoints. But it can always be divided into parts each the shortest among nearby curves between its endpoints.
How much does that generalize? Is there some natural description of Lagrangians with the following propert... | https://mathoverflow.net/users/38783 | Extremals versus minima for variational problems | You are asking a very classical question, which is the question of sufficient conditions for a minimum in the calculus of variations. Quite a lot is known about conditions on Lagrangians that ensure minimality for local solutions.
For example, it is known that the Lagrangian for $k$-dimensional area in a Riemannian ... | 6 | https://mathoverflow.net/users/13972 | 93768 | 55,056 |
https://mathoverflow.net/questions/93771 | 4 | What are the proper compact connected subgroups of $Spin(n)$ of maximal rank where $Spin(n)$ is the spin group, that is, the universal cover of the special orthogonal group $SO(n)$?
In fact, I am only interested in the highest dimension of a compact connected subgroup of $Spin(n)$ of maximal rank. I am not sure if th... | https://mathoverflow.net/users/12486 | Proper compact connected subgroup of $Spin(n)$ | I think that the answer here is just the double cover of the obvious answer for $SO(n)$, which is $U(n/2)$ when $n$ is even and $SO(n{-}1)$ when $n$ is odd. You can double-check this by consulting the Dynkin tables of maximal subgroups.
**Added after Mikhail's comment:** Mikhail actually went to the tables and checke... | 8 | https://mathoverflow.net/users/13972 | 93779 | 55,060 |
https://mathoverflow.net/questions/93787 | 0 | We have two random variables $X$ and $Y$. Suppose $P\_1$ is the probability that $Pr[X > Y]$. $Z\_1$ and $Z\_2$ are two i.i.d. (identical and independent) random variables, and let $P\_2$ be the probability that $Pr[X+Z\_1 > Y+Z\_2]$.
Can $P\_2$ be greater than $P\_1$?
If we think $Z\_1$ and $Z\_2$ as noise, then ... | https://mathoverflow.net/users/7238 | Affect of noise on Random variable separation | It seems that you need more assumptions on $X$ and $Y$. Otherwise, take both to be constant and $X = x < Y = y$, so that $P\_1 = 0$. Then it is very easy to find examples such that $P\_2 > 0$.
However, this does not mean that the two random variables are made more separable...
| 2 | https://mathoverflow.net/users/12105 | 93788 | 55,067 |
https://mathoverflow.net/questions/93785 | 6 | Let $\ell\_1,\dots,\ell\_n$ be $d+1$-variate linear forms over complex numbers in variables $X=(X\_0,\dots,X\_d)$. Consider the $(n-d)$-fold products
$$\ell\_{i\_1}(X)\ell\_{i\_2}(X)\dots\ell\_{i\_{n-d}}(X)=\sum\_{|I|=n-d}a\_{I,J}X^I,\ J=(i\_1,\dots,i\_{n-d}),\\ 1\leq i\_1<i\_2<\dots < i\_{n-d}\leq n.$$
Define $\bi... | https://mathoverflow.net/users/11100 | do you know this determinant (basic commutative algebra)? | Here's a proof; I don't know any references. I will show that, if $L\_K=0$, then $\det A=0$. Since the $L\_K$ are distinct irreducible polynomials, this shows that $\prod L\_K$ divides $\det A$, and the two sides have the same degree.
Suppose that $L\_K=0$. Then the linear forms $\ell\_{k\_1}$, ..., $\ell\_{k\_{d+1}}... | 14 | https://mathoverflow.net/users/297 | 93793 | 55,071 |
https://mathoverflow.net/questions/93794 | 1 | I consider a linear map $T\colon X^\*\to Y^\*$, where $X^\*$ and $Y^\*$ are duals of Banach spaces. I would like to know if I can deduce that $T$ is weak\* continuous (I consider the weak\* topologies on both $X^\*$ and $Y^\*$) if I know either of the following:
1. $T$ is weak\* continuous when restricted to the unit... | https://mathoverflow.net/users/22832 | Weak* continuity of linear maps | Yes. From the Krein Smulian theorem (use Google) you get that $T^\*$ maps $Y$ into $X$.
| 6 | https://mathoverflow.net/users/2554 | 93800 | 55,073 |
https://mathoverflow.net/questions/93803 | 3 | Good evening,
I have a question concerning non-invertible operators.
*Let $H$ be a Hilbert space and $T$ a non-invertible bounded operator on $H.$ Is it true that $T$ is the limit of some sequence of invertible bounded operators on $H$?*
We can see that is true when $H$ is finitely dimensional. So the interestin... | https://mathoverflow.net/users/11376 | is a non-invertible operator a boundary point of the group of invertible operators? | This is true if $0$ belongs to the boundary of the spectrum of $T$ (more or less from the definition of the spectrum) but fails in general. The unilateral left shift $S: \ell^2({\mathbb N})\to\ell^2({\mathbb N})$ satisfies $S^\ast S=I\neq SS^\ast$.
If $(T\_n)$ were a sequence of invertible operators with $T\_n\to S$ ... | 7 | https://mathoverflow.net/users/763 | 93804 | 55,074 |
https://mathoverflow.net/questions/93731 | 2 | Hi,
I face a problem some time now (not a homework problem) and I believe it is related to matrix perturbations and how the null space behaves in these cases.
The distilled version of the problem is the following:
Let $\mathbf{A} \in \mathbb{C}^{n \times m}$ where $n < m$. Define as $\mathbf{N}$ an orthonormal ... | https://mathoverflow.net/users/17246 | Null Space Perturbations | What follows is just a trivial manipulation, but I do not think you can get better bounds than that for a generic perturbation.
If you write $\tilde{A}=A+\Delta A$, then you get $\tilde AN=(\Delta A )N$, which has norm smaller than $\left\Vert\Delta A \right\Vert$. Your denominator is essentially $\left\Vert A \right... | 2 | https://mathoverflow.net/users/1898 | 93806 | 55,075 |
https://mathoverflow.net/questions/93811 | 2 | Suppose $\varphi:\mathbb{N}\rightarrow\mathbb{N}$ is a finite-to-one map. We can then define a function $\varphi\_1:a\mapsto |\varphi^{-1}(a)|$. If this function is finite-to-one, we can similarly define $\varphi\_2=(\varphi\_1)\_1: a \mapsto |\varphi\_1^{-1}(a)|$ and so on.
For example, the identity function on $\m... | https://mathoverflow.net/users/21857 | "Stably" finite-fiber maps of the integers | If $\mathbb N$ starts with $0$ then you can't have both $\phi\_n$ and $\phi\_{n+1}$ surjective. That would mean there exists $a$ such that $\phi\_{n+1}(a)=0$, so that $a$ is not in the image of $\phi\_n$.
But if we use functions from positive integers to positive integers then we can even make a solution for which $\... | 2 | https://mathoverflow.net/users/6666 | 93814 | 55,080 |
https://mathoverflow.net/questions/93798 | 7 | Hi, first I would like to apologize for my English. It's not my native language and I feel my grasp of it is limited.
I've been reading [Burago's book on metric geometry](http://books.google.com.mx/books/about/A_course_in_metric_geometry.html?id=afnlx8sHmQIC&redir_esc=y) and I've that it mentions Finsler manifolds a... | https://mathoverflow.net/users/21762 | Alexandrov geometry techniques for Finsler manifolds. | As Anton mentioned in a comment, a non-Riemannian Finsler manifold cannot be an Alexandrov space. If you found an opposite statement in our book, I would appreciate the page number where it appears (I maintain a list of errors and misprints, there are way too many of them in the book).
Alexandrov spaces generalize Ri... | 10 | https://mathoverflow.net/users/4354 | 93816 | 55,081 |
https://mathoverflow.net/questions/93812 | 13 | The automorphism group of the algebra of $n$-dimensional matrices over a field $K$ is $PGL\_n(K)$. The automorphism group of $n-1$-dimensional projective space over $K$ is also $PGL\_n(K)$. Therefore, twists of them over $\bar{K}$ are both classified by the same Galois cohomology group, $H^1(PGL\_n(K))$.
Twists of $... | https://mathoverflow.net/users/18060 | Explicit Bijection between Central Simple Algebras and twists of $\mathbb P^n$ | References:
Artin, M.
*Brauer-Severi varieties*. Brauer groups in ring theory and algebraic geometry (Wilrijk, 1981), pp. 194–210,
Lecture Notes in Math., 917, Springer, Berlin-New York, 1982.
Knus, Max-Albert; Merkurjev, Alexander; Rost, Markus; Tignol, Jean-Pierre
**The Book of Involutions**. (English summary) ... | 13 | https://mathoverflow.net/users/339 | 93818 | 55,082 |
https://mathoverflow.net/questions/92645 | 7 | Grothendieck in SGA 1 introduces a proposition in expose 5 (proposition 3.1) which states:
Let $X$ be etale, separated of finite type over $Y$, locally noetherian, and let $G$ be a finite group which operates on $X$ by $Y$-automorphisms. Then $G$ operates admissibly and the quotient scheme $X/G$ is etale over $Y$.
... | https://mathoverflow.net/users/21523 | admissible group operation on etale, separated, finite type scheme | We can reduce to the case where $Y$ is affine, and in this case (as you observe), $X$ is quasi-projective over $Y$ by Zariski's main theorem. Consequently, the key step is to show that a finite group operating on a quasi-projective scheme does so admissibly (meaning that we can form the quotient nicely).
Admissibili... | 8 | https://mathoverflow.net/users/344 | 93824 | 55,086 |
https://mathoverflow.net/questions/93791 | 0 | I am curious. Is there a "slick" way of showing that given an arbitrary algebra $A$ with generating set $X$, an algebra endomorphism $\alpha : A\to A$ and a function (satisfying some conditions) $d : X\to A$, that $d$ extends uniquely to an $\alpha$-derivation $D : A\to A$?
| https://mathoverflow.net/users/22360 | alpha derivations | Suppose $V$ is a vector space, let $TV=\bigoplus\_{n\geq0}V^{\otimes n}$ be the tensor algebra on $V$ and let $I$ be an ideal of $TV$. Let $A=TV/I$ be the quotient algebra, let $p:TV\to A$ be the canonical projection and let $\alpha:A\to A$ be an endomorphism of algebras. Let, moreover, $\delta:V\to A$ be any linear ma... | 2 | https://mathoverflow.net/users/1409 | 93827 | 55,088 |
https://mathoverflow.net/questions/93829 | 4 | Here is an example from Ezra Miller's book: *Combinatorial Commutative Algebra*,p26-27
Let $f,g\in k[x\_1,x\_2,x\_3,x\_4]$ be a generic forms of degree $d$ and $e$, the *generic initial ideal* of $I=\langle f,g\rangle$ for both the *lexicographic order* and the *inverse lexicographic order*.
When $(d,e)=(2,2)$,the... | https://mathoverflow.net/users/9946 | How to find the generic initial ideal? | You should apply a generic linear coordinate transform to the ideal and then compute the initial ideal. The matrices for which the result is the generic initial ideal is a (Zariski) open subset (Lemma 2.6, in the book). In particular the complement is of lower dimension.
Leaving distribution issues aside, if you 'pi... | 3 | https://mathoverflow.net/users/5495 | 93843 | 55,092 |
https://mathoverflow.net/questions/93723 | 4 | Let $E$ and $F$ be $n$-dimensional representations of $S\_n$ of the form $V\oplus\mathbb{C}$, where $V$ is the standard representation. I just wonder if there is a formula to decompose $S^k(E\otimes F)\otimes\Lambda^m(E\otimes F)$ into irreducible modules of $S\_n\times S\_n$. Many thanks.
| https://mathoverflow.net/users/19282 | Symmetric group module decomposition | This is very far from a complete answer, but it should give enough results to enable the irreducible constituents to be computed for some small values of $k$, $m$ and $n$. I would be amazed if there was an easy formula for the decomposition. To clarify the notation, I'll write $\boxtimes$ for the outer tensor product, ... | 6 | https://mathoverflow.net/users/7709 | 93854 | 55,096 |
https://mathoverflow.net/questions/93849 | 2 | I'm studying Dehn surgery, and it says that the coefficient $(p,q)$ which says how the meridian curve on solid torus is attached will determine the entire resulting manifold. I'm wondering whether the coefficient $(p,q)$ of the meridian also determine the coefficient for longitude. If not, then can any $(r,s)$ with $ps... | https://mathoverflow.net/users/6569 | Does the coefficient of the meridian determine the coefficient of the longitude?(on Dehn surgery) | Yes, the longitude of the Dehn surgery solid torus can go to any $(r, s)$ such that the 2x2 determinant $ps-qr = \pm 1$. This is because the various choices for the longitude all differ by homeomorphisms of the torus which extend to homeomorphisms of the solid torus.
More generally, consider gluing together two mani... | 5 | https://mathoverflow.net/users/284 | 93857 | 55,099 |
https://mathoverflow.net/questions/93862 | 2 | Let $A$ be an algebra, say over $\mathbb{C}$ and finite-dimensional, but not necessary semisimple. I have the strong feeling, which I would like to prove and use, about the following rather natural approach.
Consider the set $S$ of central idempotents in $A$, then I want to turn the set of central idempotents $\Delta... | https://mathoverflow.net/users/22709 | Simplicial complex made of central idempotents of an algebra | The central idempototents of a finite dimensional algebra form a finite Boolean algebra with 1 as the max and the central primitive idempotents as the atoms. The decomposition of 1 into central idempotents is thus unique. So the central idempotents are the face lattice of a simplex. The order is $e\leq f$ if $e\in fA$.... | 7 | https://mathoverflow.net/users/15934 | 93863 | 55,102 |
https://mathoverflow.net/questions/93861 | 5 | <http://arxiv.org/abs/0908.4287>
I could not find any reviews for it, but if true its a major claim, because it says that $\Re(\rho) < 21/40$ where $\rho$ is a zeta zero.
My question:
>
> Are there any reviews of this paper, that reject or accept the claims made in this paper?
> Any references will be highly app... | https://mathoverflow.net/users/2865 | A note by N. A. Carella on zero-free regions | Well,
In this case it is a short elementary paper (6 pages) and it is easy to verify, although just the length of the paper and the importance of the result and some other aspects of the paper, such as the fact that it is in general mathematics part of arxiv and it does not seem as the autor uses latex is sufficient ... | 17 | https://mathoverflow.net/users/10811 | 93865 | 55,103 |
https://mathoverflow.net/questions/93869 | 2 | I have the following quadratic optimization problem: $\min\_{\vec{x}} |\vec{x}|^2$ subject to $\vec{x}^T G\_j \vec{x} \geq 1$, $j = 1 \ldots m$, where the $G\_j$ are positive semidefinite. $|\vec{x}|$ is the norm of the vector $\vec{x}$ and $T$ denotes transpose. Since the $G\_j$ are positive semidefinite, and we have ... | https://mathoverflow.net/users/18693 | Solving a non-convex quadratically-constrained quadratic program | Branch-and-Bound can be used to solve problems like this, but it's an exponential time algorithm that is not practical for large instances of the problem. What is $n$, the dimension of the $x$ vector? How big is your $m$?
| 2 | https://mathoverflow.net/users/9022 | 93876 | 55,107 |
https://mathoverflow.net/questions/93840 | 5 | Let $(M,g)$ be a Riemannian manifold and let $U\subset M$ be an open subset. Suppose that the tangent bundle over $U$ splits into two orthogonal distributions $TU=\mathcal{E}\oplus \mathcal{F}$.
Is it possible that the two $C^{\infty}(U)$-bilinear maps
\begin{align\*}I:\mathcal{E}\times\mathcal{E}&\to \mathcal{F}\... | https://mathoverflow.net/users/20823 | Local splitting of the tangent bundle with interesting properties | Yes, this can happen. A little experimentation with the structure equations shows that there is a $3$-parameter family of homogeneous examples in dimension $4$: Let $c\_1,\ldots,c\_4$ be nonzero constants satisfying $c\_1c\_2=c\_3c\_4$, and consider the simply-connected $4$-dimensional Lie group $G$ that has a basis of... | 5 | https://mathoverflow.net/users/13972 | 93890 | 55,113 |
https://mathoverflow.net/questions/93885 | 3 | I'm sure that many readers are already familiar with the well known Bonnesen inequality in the plane for a smooth, connected curve:
$(R\_{out} - R\_{in})^2 \leq \pi^2 (L^2 - 4\pi A),$
where $R\_{out}$ and $R\_{in}$ are the outer and inner radii respectively, $L$ is the length of the curve and $A$ is it's area.
... | https://mathoverflow.net/users/8755 | Different forms of Bonnesen's strong isoperimetric inequality in the plane. | I seriously doubt there is any such estimate (at least one that doesn't have the constant $C$ depend in a complicated and extremely unnatural way on the geometry of the curve).
My heuristic reasoning is as follows.
Let $\sigma$ be the unit circle and $f$ be a non-zero smooth function on $\sigma$ so that $\int\_{\sig... | 4 | https://mathoverflow.net/users/26801 | 93892 | 55,114 |
https://mathoverflow.net/questions/93887 | 5 | Can every μ-recursive function be defined using a single instance of the μ operator applied to a primitive recursive function?
According to Wikipedia, any μ-recursive function can be expressed as the μ-operator over a primitive recursive function ([source](https://en.wikipedia.org/wiki/%25CE%259C-recursive_function#N... | https://mathoverflow.net/users/22861 | Representation of μ-recursive functions | The answer is that you have to apply another primitive recursive function after the $\mu$ operator. Specifically, the [Kleene normal form](https://en.wikipedia.org/wiki/Kleene%27s_T_predicate) is that every recursive function $f$ has the form $f(n)=U(\mu x T(e,n,x))$, where both $U$ and $T$ are primitive recursive. The... | 9 | https://mathoverflow.net/users/1946 | 93895 | 55,116 |
https://mathoverflow.net/questions/93894 | 8 | What is the topology of $\mathbb{P}\_2 (\mathbb{C}) \setminus \mathbb{P}\_2 (\mathbb{R})$? For example what is the homology of this manifold with coefficients in $\mathbb{Z}$. I know that this is known but I can't find a good reference for it. Can anyone give me a reference?
| https://mathoverflow.net/users/13559 | topology of $\mathbb{P}_2 (\mathbb{C}) \setminus \mathbb{P}_2 (\mathbb{R})$? | $H\_i (CP^2 \setminus RP^2)\cong H^{4-i}(CP^2, RP^2)$ by Poincare-Alexander-Lefschetz duality (Bredon, Topology and Geometry, Theorem 8.3 on p. 351). The latter can be computed using the long exact sequence. $H^4 = Z$, $H^0=H^1=0$ is immediate. The piece
$$0 \to H^2 (CP^2,RP^2) \to H^2 (CP^2) \to H^2 (RP^2) \to H^3 ... | 11 | https://mathoverflow.net/users/9928 | 93900 | 55,118 |
https://mathoverflow.net/questions/93901 | 6 | Hi,
I am currently working through the paper by Bousfield and Gugenheim on rational homotopy theory, and have come to a point where they show why it is important to work over $\mathbb{Q}$, and not just any field (remark 9.7). They assert that if $Hom\_k(Hom\_{\mathbb{Z}}(G, k), k)=G$ for some non-trivial abelian grou... | https://mathoverflow.net/users/22870 | Mysterious property of $\mathbb{Q}$ | First, $k$ needs to be assumed of characteristic $0$. The mysterious property is that $\mathbb{Q}$ is a prime field:
If $Hom\_k(Hom\_Z(G, k), k)=G$, $G$ has to be a $k$-vector space, say $G=kT$ for a basis $T$.
$Hom\_Z (kT;k)$ is a $k$-vector space.
Let me first argue that $T$ has to be finite. Since $Hom\_k (kT;k... | 16 | https://mathoverflow.net/users/9928 | 93903 | 55,119 |
https://mathoverflow.net/questions/68866 | 4 | Let $p:X \to Y$ be a map of smooth algebraic varieties.
Let $C \subset T^\* X$ be a (locally closed) submanifold. Denote by $p\_\*(C) \subset T^\* Y$ the following set:
$$ \{(y,v) \in T^\*(Y)\mid\exists x \in p^{-1}(y) \text{ with } (x,(d\_x(p))^\*(v)) \in C \}.$$
This operation can describe (to some extent) what h... | https://mathoverflow.net/users/4690 | Direct image of Lagrangian subspaces of the co-tangent bundle | I've got a negative answer from Thomas Bitoun:
*Malgrange, Bernard*, [**Sur les images directes de ${\mathcal D}$-modules**](https://doi.org/10.1007/BF01168827), Manuscr. Math. 50, 49–71 (1985). [Zbl 0572.32014](https://zbmath.org/0572.32014).
It is at the very end of the paper, (6.3) page 23 of the electronic
vers... | 1 | https://mathoverflow.net/users/4690 | 93912 | 55,125 |
https://mathoverflow.net/questions/93909 | 2 | Good evening,
I'm reading some papers of Jim Agler and Nicholas Young, in which they prove a formula of integral representation with respect to a vector measure, but the integration is in the sense of Riemann, not of Lebesgue. And the proofs for Riemann integrals are often long.
Secondly, in the book of Rudin, Fun... | https://mathoverflow.net/users/11376 | Lebesgue integral with respect to vector measures? | Marc Rieffel has some notes that develop integration with respect to Banach-space valued measures from the ground up. The notes are very thorough. They are available here:
[http://math.berkeley.edu/~rieffel/measinteg.html](http://math.berkeley.edu/%7Erieffel/measinteg.html)
>
> Lectures notes from 1970 for the fi... | 3 | https://mathoverflow.net/users/703 | 93919 | 55,128 |
https://mathoverflow.net/questions/93917 | 1 | Assume $X$ is a measure space and $f : X \to [0,\infty]$ is an absolutely integrable function (that is $\int\_X f \; d \mu < \infty$). This question is about the asymptotic behaviour of the function $$E(\delta) = \int \_{f \leq \delta} f \; d \mu.$$
Since $f$ is absolutely integrable, it follows from the downward mon... | https://mathoverflow.net/users/4002 | A question about the tail of an absolutely integrable function | You can't say anything in general. For a simple example showing that you can't get any polynomial rate, consider the measure space $X = [1, \infty)$ with the Lebesgue measure and the function $f(x) = x^{-(1 + \epsilon)}$. Assuming I integrated it correctly, you get $E(\delta) = \frac{1}{\epsilon} \delta^{\epsilon/(1 + ... | 3 | https://mathoverflow.net/users/21652 | 93920 | 55,129 |
https://mathoverflow.net/questions/93911 | 1 | Let $n$ be a non-negative integer. $\;\;$ Let $\: f : \mathbb{R}^n \to \mathbb{R}^n \:$ and $\: g : \mathbb{R}^n \to \mathbb{R}^n \:$ be [Lipschitz](http://en.wikipedia.org/wiki/Lipschitz_continuity).
Define the relation $\stackrel{f}{\sim}$ on $\mathbb{R}^n$ by
$u \stackrel{f}{\sim} v \;\;$ if and only if ... | https://mathoverflow.net/users/nan | topological equivalence of ODEs | I assume that the dynamical systems in question are semigroups generated by $f$ and by $g$. Then the answer is negative: Take $n=2$, $f$, $g$ rotations of different orders. Then the quotient spaces are homeomorphic to ${\mathbb R}^2$ but the groups generated by $f$ and $g$ are not orbit-equivalent.
Or, maybe you mea... | 3 | https://mathoverflow.net/users/21684 | 93921 | 55,130 |
https://mathoverflow.net/questions/93102 | 0 | It is well known that pentagon+triangle identity of type (a1b) implies "all diagrams commute" [monoidal category](http://ncatlab.org/nlab/show/monoidal+category), in particular triangle identities of type (1ab) and (ab1). My question is that whether pentagon+triangle identity of type (1ab), or (ab1), implies triangle i... | https://mathoverflow.net/users/7341 | A question on triangle identities | The answer is: No for monoidal categories, but Yes for symmetric monoidal categories. Actually, you can get away with a lot less for the Yes. Intuitively, all you need is for the unit to commute with arbitrary elements, i.e. $\rho\_x = \lambda\_x \circ \tau: x\otimes 1 \rightarrow x$.
The No is a result of the obser... | 2 | https://mathoverflow.net/users/11540 | 93923 | 55,132 |
https://mathoverflow.net/questions/93916 | 15 | There is a literature dealing with
$$
\sum\_{k\le x}d(f(k))
$$
where $f$ is an irreducible polynomial and $d(n)$ is the number of divisors of $n$. Erdos 1952 shows that the sum $\asymp x\log x,$ which was improved to $Ax\log x+O(x\log\log x)$ by Bellman-Shapiro (cited in Scourfield). But these results only apply to irre... | https://mathoverflow.net/users/6043 | Sum of $\sum_{k=1}^nd(k^2)$ | In the case you are interested in there is a simple generating (Dirichlet) series:
$$ \sum\_{n=1}^\infty \frac{d(n^2)}{n^s} = \frac{\zeta^3(s)}{\zeta(2s)}.$$
From this you can either use a convolution argument or a Perron formula type argument to get an asymptotic formula. In particular, I believe it follows that
$$\s... | 29 | https://mathoverflow.net/users/3659 | 93924 | 55,133 |
https://mathoverflow.net/questions/93825 | 6 | Given a complex manifold, you can `weaken' its structure to give a smooth manifold. Is there an analogous construction that constructs a stack over the category of smooth manifolds from a stack over the category of complex manifolds?
Obviously, this is possible for a stack represented by the quotient of a complex ma... | https://mathoverflow.net/users/20919 | how to construct a $C^\infty$ stack from a holomorphic stack | Denote by $$u:CxMfd \to Mfd$$ the forgetful functor from complex manifolds to smooth manifolds. Let $$u\_!:St\left(CxMfd\right) \to St\left(Mfd\right)$$ denote its 2-categorical prolongation. Explicitly, this is given by the bicategorical Kan extension of $y\_{Mfd} \circ u$ along the Yoneda embedding $$y\_{CxMfd}:CxMfd... | 2 | https://mathoverflow.net/users/4528 | 93925 | 55,134 |
https://mathoverflow.net/questions/93462 | 3 | Let $S$ be a finite set and let $\mathcal{A} \subset\mathcal{P}(S)$ be a family of subsets of $S$. Consider the convex polytope spanned by the characteristic functions of members of $\mathcal{A}$ :
$$C=C\_ \mathcal{A}:=\operatorname{co}\{ \mathbf {1}\_A \, : \, A\in\mathcal{A} \}\, .$$
It's easy to see that every $\mat... | https://mathoverflow.net/users/6101 | The facial structure of the convex hull of a family of characteristic functions | In addition to the notes by Ziegler referenced in the comment above, there are a few general classes of 0/1 polytopes where one can say something about the facial structure.
One example is that of the independent set polytope, $P\_{I(M)}$ for a matroid $M$, where one uses the independent sets of the matroid to defin... | 2 | https://mathoverflow.net/users/10602 | 93928 | 55,136 |
https://mathoverflow.net/questions/88453 | 14 | Hello,
This question might be vague and not thought-through enough.
If we have a real positive number $x$, we can start to write it as a continued fraction:
$x = a\_0 + \frac{1}{x\_1} , \ldots , x\_n=a\_n + \frac{1}{x\_{n+1}}$ where $a\_i$ are non-negative integers, $x\_i$ non-nogative real numbers less than $1$.
S... | https://mathoverflow.net/users/2095 | Continued fractions and projective resolutions | Hi Sasha,
As I understand it your question have 2 parts:
1. Is there some relation between continued fractions and projective resolutions?
2. Is there some relation between rationality of a number and finite cohomological dimension of an object in an (abelian) category?
Now let me make those questions more concre... | 6 | https://mathoverflow.net/users/4690 | 93930 | 55,137 |
Subsets and Splits
No community queries yet
The top public SQL queries from the community will appear here once available.