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 |
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https://mathoverflow.net/questions/92638 | 8 | I have to do some messy calculations with weak 2-functors between bicategories, and I know the most efficient way to do it would be via some sort of string diagram methods. Also, it means that I can put pretty calculations in my paper, unlike some of this paper's 'ancestors' (written by others) where they simply can't ... | https://mathoverflow.net/users/4177 | Diagram calculus for weak 2-functors between bicatgories | I imagine you could easily apply Micah McCurdy's graphical calculus for monoidal functors (see [this paper](https://arxiv.org/abs/1110.5542) or [these slides](http://web.science.mq.edu.au/~mmccurdy/cms2010talk.pdf)) by just labeling the regions between strands as one usually does in the graphical calculus for bicategor... | 6 | https://mathoverflow.net/users/396 | 92643 | 54,484 |
https://mathoverflow.net/questions/92637 | 6 | I saw the following nice formula in an unpublished paper of H. Cohen. Let $L$ be a quartic field, whose Galois closure $\widetilde{L}$ has Galois group $S\_4$. Denote the cubic resolvent field of $L$ by $K\_3$, and let $K\_6$ be the Galois closure of $K\_3$.
Then, we have the following relation among Dedekind zeta fu... | https://mathoverflow.net/users/1050 | Generalizations/applications of a formula for the Dedekind zeta function? | As Keith says, such relations between permutation representations are often called Brauer relations, because Brauer was the first one to note that such isomorphisms of permutation representations give rise to relations between zeta functions (Kuroda noticed the same phenomenon at the same time). Any non-cyclic group ha... | 9 | https://mathoverflow.net/users/35416 | 92649 | 54,486 |
https://mathoverflow.net/questions/92652 | 4 | This question may be naive. Take an infinite set of distinct algebraic numbers (hence countable). List them out in a table (randomly) by picking a choice of ordering and change the diagonal numbers.
1) Is it possible to decide if this Cantor type diagonal number is algebraic or transcendental?
2) For some choice o... | https://mathoverflow.net/users/10035 | Are Cantor type numbers algebraic? | If one uses binary representation (so that changing the digit means swapping 0 and 1), then *every* real arises as a diagonal real. If you want the diagonal to be $d$, start with any list of algebraic numbers, and first change the diagonal digits on the list to be the dual to $d$. This is a finite change to each of the... | 18 | https://mathoverflow.net/users/1946 | 92654 | 54,489 |
https://mathoverflow.net/questions/92657 | 5 | I am looking for an example of a smooth irreducible quasiprojective variety $X$ over ${\mathbb C}$, such that when reduced over finite fields ${\mathbb F\_q}$, the number of its points is a polynomial $P(q)$ of $q$ with nonnegative (integer) coefficients, but $X$ has some odd cohomology.
Background: as discussed in ... | https://mathoverflow.net/users/6107 | Smooth variety with positive point-count polynomial and odd cohomology | The answer to your question is yes:
Let $X$ be the blowup of $\mathbb{A}^1 \times (\mathbb{A}^1 - \{0\})$ in a point. The number of points over a field of $q$ elements is $q(q-1) + q = q^2$ and $X$ has non-trivial $H^1$.
| 7 | https://mathoverflow.net/users/519 | 92661 | 54,492 |
https://mathoverflow.net/questions/92203 | 8 |
>
> We have a set $S$ with $k$ elements, a positive integer $n$, and subsets $S\_1, S\_2, \dots, S\_n,$ each with $n$ elements. For any two elements $a,b$ of $S$, there are at most two sets $S\_i$ containing both $a$ and $b$. Must $k$ be $\Omega(n^2)?$
>
>
>
If we require instead that, for any two $a,b$ there ar... | https://mathoverflow.net/users/21519 | Family of subsets such that there are at most two sets containing two given elements | I think I may have an answer. Let $p$ be a prime (for simplicity). Let $r=p^3$ and let $T\_1,\dots,T\_r$ be all possible graphs of quadratic functions defined on the integers mod $p$. These graphs live in a set of size $n=p^2$, so $r=n^{3/2}$. Note that no two quadratic functions agree in more than two places, so $|T\_... | 8 | https://mathoverflow.net/users/1459 | 92669 | 54,495 |
https://mathoverflow.net/questions/92618 | 2 | I can find many modification of the Jung-Abhyankar theorem. I can even find a new proof of the theorem (by K. Kiyek and J. L. Vicente). But I cannot find the original statement. Does any one know which paper/book is it in? Where can I access it?
| https://mathoverflow.net/users/21522 | What is the original statement of Jung-Abhyankar theorem? | The original papers are accessible online:
* H. W. E. Jung, [Darstellung der Funktionen eines algebraischen Körpers zweier unabhängigen Veränderlichen x, y in der Umgebung einer Stelle x = a, y = b.](http://www.digizeitschriften.de/index.php?id=resolveppn&PPN=GDZPPN002166585)
Journal für die reine und angewandte Math... | 3 | https://mathoverflow.net/users/1939 | 92671 | 54,497 |
https://mathoverflow.net/questions/92668 | 4 | I learned from MO [Subgroups of a finite abelian group](https://mathoverflow.net/questions/46115/subgroups-of-a-finite-abelian-group) that the problem of enumerating subgroups (not up to isomorphism) of finite abelian groups is a difficult one.
Are there simple formulas if one restricts to low rank for the subgroups?... | https://mathoverflow.net/users/22518 | Cyclic subgroups of finite abelian groups | I think that you can find the formulas that you are looking for in the paper "An arithmetic method of counting the subgroups of a finite abelian group" by Marius Tarnauceanu, Bull. Math. Soc. Sci. Math. Roumanie (N.S.) 53(101) (2010), no. 4, 373–386.
In particular, Theorem 4.3 seems relevant, but there are other resu... | 7 | https://mathoverflow.net/users/12858 | 92677 | 54,500 |
https://mathoverflow.net/questions/92482 | 4 | The collection of marked unimodular lattices in the euclidean $n$-space corresponds to the symmetric space $S\_n:=SO\_n(\mathbb{R}) \backslash SL\_n(\mathbb{R})$.
I have only recently been made aware that $S\_n$ is a so-called $linear$ symmetric space. Apparently this means the following. The collection of (marked) ... | https://mathoverflow.net/users/20516 | Linear symmetric spaces are spaces with ''orthogonal complements''? | Definition of "linear symmetric space" appears in the book of Borel and Ji "Compactifications of symmetric and locally symmetric spaces", p. 286. Their definition is not very precise, but can be restated as follows: A symmetric space $X$ is called linear if there exists a convex domain $D$ in the projective space such ... | 1 | https://mathoverflow.net/users/21684 | 92681 | 54,501 |
https://mathoverflow.net/questions/92571 | 4 | The Problem of the Mad King's Draft:
Suppose there is country which is ruled by a king who can be either 'mad' or 'normal.' The king rules a a large country with a continuum of citizens who have names $j$ from $[0,1]$. The citizens do not know whether the king is mad or not but believe both characters are equally lik... | https://mathoverflow.net/users/19774 | Conditional Probabilities - The Mad Kings' Draft | Let us formalize the story in the following way: Let $U\_1,U\_2,U\_3,U\_4$ be 4 independent random variables, uniformly distributed on $[0,1]$ (the possibly drafted citizens), and let $M$ be an independent Bernoulli random variable, with $\mathbb{P}(M=0) = \mathbb{P}(M=1) = 1/2$. ($M=1$ means that the king is mad.) The... | 6 | https://mathoverflow.net/users/19603 | 92686 | 54,503 |
https://mathoverflow.net/questions/92080 | 4 | I want to embark on a project about inverting a Moment-generating function of a probabilitiy distribution. That is given by
\begin{equation}
M\_X(t) = \text{E} \exp(tX)
\end{equation}
Since I have never done anything like this before, I am searching for some good references for this, especially references with worked ... | https://mathoverflow.net/users/6494 | Inversion of Moment-generating functions (aka Laplace transform of prob dist) | The inverse Laplace transform takes place in the complex domain and includes the residue theorem. You can probably do that analytically (as the above forwarded you to Wiki). But usually you can perhaps see how your usual transforms look like and try to invert by converting the result to a commonly used form which is ea... | 2 | https://mathoverflow.net/users/22524 | 92687 | 54,504 |
https://mathoverflow.net/questions/92666 | 10 | What is the etymology of **model**? The answer is of course pre-WWW, but the better part of an hour in the library searching both classic model theory and modal logic textbooks turned up nothing. Every book I touched, without exception, *uses* the word in the usual way - a structure consistent with some theory - but of... | https://mathoverflow.net/users/22517 | What is the etymology of model? | I'm sure the origin of the term is quite complex. Indeed, as Henry and David have pointed out, it is a very natural choice in this context.
I think the first *definition* of 'model' (not the first *use*) in the exact sense currently used in model theory is due to Tarski in *O pojȩciu wynikania logicznego* (*On the co... | 6 | https://mathoverflow.net/users/2000 | 92690 | 54,506 |
https://mathoverflow.net/questions/92700 | 5 | I'm trying to follow Hopkins' construction of the Serre Spectral Sequence, but some "obvious" things are not that obvious to me.
He starts with considering a double complex $C\_{\bullet,\bullet}$ with $C\_{p,q}$ to be a free $\mathbb{Z}$-module generated by the maps $\Delta[p]\times\Delta[q]\rightarrow E$ ($E$ is a t... | https://mathoverflow.net/users/22530 | Construction of Serre Spectral Sequence | The double complex has a horizontal differential $\partial'$ and a vertical differential $\partial''$ such that $\partial'\partial''=\partial''\partial'$. This gives rise to a total complex $TC\_n=\bigoplus\_{p+q=n}C\_{pq}$ with differential $\partial|C\_{pq}=\partial'+(-1)^p\partial''$. This can be filtered by $F\_p=\... | 4 | https://mathoverflow.net/users/12310 | 92705 | 54,513 |
https://mathoverflow.net/questions/92550 | 4 | Let $p\in [1,\infty)\setminus\{2\}$. Suppose $(e\_n)$ is a basic sequence in $\ell\_p$ (or $L\_p$) equivalent to the basis of $\ell\_p$ ($L\_p$). Is there a subsequence $(e\_{n\_k})$ such that $[e\_{n\_k}]$ is complemented?
| https://mathoverflow.net/users/22469 | Basic sequences in $\ell_p$ | The answer is yes also for $L\_p$, but I don't know a good book reference. For $2<p<\infty$, this is contained the paper of Kadec and Pelczynski--it is their second dichotomy theorem. Actually, they get that a normalized weakly null sequence has a subsequence that is either equivalent to an orthonormal sequence (in whi... | 3 | https://mathoverflow.net/users/2554 | 92708 | 54,514 |
https://mathoverflow.net/questions/92707 | 12 | Let $P(x;a,b) := \{an+b, 0\leq n \leq x \} $ denote an arithmetic progression. Further let $A(x;a,b)$ denote the number of elements of $P(x;a,b)$ that are squares. It's an old conjecture of Rudin that $A(x;a,b) \ll x^{1/2}$. Less ambitiously, Erdős posed the problem of showing that $A(x;a,b) = o(x)$. This was proven by... | https://mathoverflow.net/users/630 | Squares in an arithmetic progression | The conjecture of Rudin is about the maximum of $A(x; a,b)$, taken over all values of $a$ and $b$. Not with just keeping $a$ and $b$ fixed and letting $x$ get large. So, like you said, if $a$ is not $o(x)$, the proof of Uchida doesn't go through.
The latest news on this problem (as far as I know) can be found here: E... | 11 | https://mathoverflow.net/users/6698 | 92713 | 54,516 |
https://mathoverflow.net/questions/92696 | 19 | Can you make an example of a ***great*** proof by induction or construction by recursion?
Given that you already have your own idea of what "great" means, here it can also be taken to mean that the chosen technique :
* is vital to the argument;
* sheds new light on the result itself;
* yields an elegant way to fulf... | https://mathoverflow.net/users/13961 | Excellent uses of induction and recursion | A Classic: **Fix a positive integer $n$. Show that it is possible to tile any $2^n \times 2^n$ grid with exactly one square removed using 'L'-shaped tiles of three squares.**
It serves as a wonderful introductory example to proof by induction. Indeed, the proof can almost be represented with two appropriate figures. ... | 34 | https://mathoverflow.net/users/18131 | 92714 | 54,517 |
https://mathoverflow.net/questions/92703 | 2 | Hi,
Let $G\_1, G\_2$ be topological groups with $G\_1 \subset G\_2$ is closed. Let $\rho:G\_1 \to Aut(V)$ be a smooth irreducible representation. Can anyone tell me if there is a criterion/example/idea about when $\rho$ cannot be extended to $G\_2$, i.e., there does not exist smooth representation $\rho':G\_2 \to Aut(... | https://mathoverflow.net/users/9164 | Extending smooth irreducible representations | The above comments show that in general linear representations do not extend, so a better question is to ask when linear representations *do* extend! This leads rather quickly to Margulis' super-rigidity and the subsequent programme generated by this result.
| 3 | https://mathoverflow.net/users/14497 | 92717 | 54,519 |
https://mathoverflow.net/questions/92715 | 4 | Let $A=(a\_{ij})\_{i,j=1}^n$ be a symmetric real matrix, $M\_k:=det(a\_{ij})\_{1\leq i,j\leq k}$ be its minors and $M\_k\ne 0$ for all $k$. Then signs of eigenvalues of $A$ are equal (up to some permutation) to signs of $M\_1$, $M\_2/M\_1$, $\dots$, $M\_{n}/M\_{n-1}$. It is clear by induction, for example: when we repl... | https://mathoverflow.net/users/4312 | signs of eigenvalues of quadratic form | This is very closely related to Sylvester's law of inertia (see <http://en.wikipedia.org/wiki/Sylvester%27s_law_of_inertia>).
| 3 | https://mathoverflow.net/users/13650 | 92728 | 54,526 |
https://mathoverflow.net/questions/92616 | 3 | As is well known,
the normalized Ricci flow is defined for all $t>0$ on compact surfaces,
and every metric on a compact surfaces converges to a metric constant curvature if $X \neq S^2$ (at least I can't find a reference that asserts that same result for $X=S^2$; B. Chow's "Ricci flow on the 2-sphere" only shows that m... | https://mathoverflow.net/users/20557 | Does normalized Ricci flow on surfaces yield a bundle? | The formal reference for the result is C. Earle and J. Eells, "A fibre bundle description of Teichmüller theory", J. Differential Geom. Volume 3, Number 1-2 (1969), 19-43. The upshot is that there are two fibrations: One is the fibration of the space of Riemannian metrics $R(S)$ over the space $C(S)$ of conformal struc... | 13 | https://mathoverflow.net/users/21684 | 92745 | 54,531 |
https://mathoverflow.net/questions/92718 | 11 | Suppose that $k$ is an *algebraically closed* field. Let $F/k$ be a (possibly non-finitely generated) field extension. Is
$$
k[[x]] \otimes\_{k} F
$$
noetherian?
If not, is the natural map $k[[x]] \otimes\_{k} F \to F[[x]]$ injective?
| https://mathoverflow.net/users/5337 | Is the tensor product of a power series ring and a field noetherian? | The answer is no: for example, $k[[x]]\otimes\_k k((x))$ is not noetherian.
Indeed, if it were, so would be $ k((x))\otimes\_k k((x))$.
But this would contradict the following interesting general theorem of Vámos:
Given an extension of fields $K/F$ the tensor product $K\otimes\_F K$ is noetherian if and only i... | 17 | https://mathoverflow.net/users/450 | 92747 | 54,532 |
https://mathoverflow.net/questions/92755 | 49 | It is well known that most topological spaces can be studied via their algebra of continuous real-valued (or complex-valued) functions. For instance, in the setting of compact Hausdorff spaces, there is a complete dictionary between topological properties of the space $X$ and corresponding algebraic properties of the a... | https://mathoverflow.net/users/7392 | Is there an algebraic approach to metric spaces? | A very complete reference is the book "Lipschitz Algebras" by Nik Weaver. In there you will find various types of spaces of Lipschitz functions that can be associated to a metric space, and several results of the kind you are asking about.
From the book's introduction:
>
> Thus, there is a robust duality
> betwe... | 36 | https://mathoverflow.net/users/3536 | 92764 | 54,540 |
https://mathoverflow.net/questions/92740 | 0 | we have that the function (for suitable f)
$ F(x)= \sum\_{-\infty}^{\infty}f(x+n) $ is INVARIANT under any integer traslation
$ y=x+n$ for integer 'n'
however my question is can we find a lattice which is invariant under DILATIONS i mean under the transformation $ y=qx$ for integer (positive) or rational 'q' ??
... | https://mathoverflow.net/users/21933 | invariance under dilations | If you lattice $X$ contains at least a point $x$ it must contains all points $qx$ with $q$ rational. Hence $X$ is a dense set. As a consequence if your function $f$ is positive in some interval then function $F$ is infinite everywhere.
| 1 | https://mathoverflow.net/users/15120 | 92768 | 54,543 |
https://mathoverflow.net/questions/66886 | 16 | In the paper *Conformal Deformation of a Riemannian metric to a constant scalar curvature* of Richard Schoen (J. Differential Geom. 20(2) (1984) 479-495, doi:[10.4310/jdg/1214439291](https://doi.org/10.4310/jdg/1214439291)), in the first page, it says that
>
> "Note that the class of conformally flat manifolds of p... | https://mathoverflow.net/users/14579 | conformally flat manifold with positive scalar curvature | Products $M^m \times S^{n-m}$ will be conformally flat, where $M^m$ is a compact manifold of curvature $-1$ and $S^{n-m}$ has curvature $1$. If $n>2m$ then the scalar curvature of the product will be positive (positive curvature of the sphere dominates the negative curvature of the hyperbolic manifold, so the scalar cu... | 12 | https://mathoverflow.net/users/21684 | 92780 | 54,549 |
https://mathoverflow.net/questions/92684 | 10 | Hello,
this is a request for literature/a reference. I'm looking to do some calculations with the
symmetric group ($S\_6$ and higher) and would be interested in explicit expressions for
3-cocycles, i.e. elements of $H^3(S\_6, U(1))$.
Does anyone know whether these have already been calculated somewhere?
| https://mathoverflow.net/users/19421 | Explicit 3-cocycles for the symmetric group $S_6$ | I hope it's ok to advertize some GAP code on this site.
Let G be a finite group and let
$$R\_\*: \cdots \rightarrow R\_4 \rightarrow R\_3 \rightarrow R\_2 \rightarrow R\_1 \rightarrow R\_0$$
be a free $\mathbb ZG$-resolution of $\mathbb Z$.
A 3-cocycle with coefficients in $U(1)$ is a $\mathbb ZG$-linear homomo... | 11 | https://mathoverflow.net/users/22553 | 92781 | 54,550 |
https://mathoverflow.net/questions/92771 | 38 | Let $X$ be a smooth projective variety over $\mathbb{C}$. The following information is all equivalent (any of these numbers can be computed by a linear equation from any of the others):
1. the arithmetic genus of $X$
2. the constant coefficient of the Hilbert polynomial of $X$
3. $\chi(X, \mathscr{O}\_X)$
4. the "Tod... | https://mathoverflow.net/users/5094 | A geometric characterization for arithmetic genus | First let me note that there is an unfortunate clash in terminology: the arithmetic genus of a smooth complex projective variety $X$ of dimension $n$ can mean either
a) The number $\chi (X, \mathcal O\_X)$: the Hirzebruch arithmetic number in which you are interested .
b) The number $p\_a(X)=(-1)^n(\chi (X, \math... | 35 | https://mathoverflow.net/users/450 | 92807 | 54,563 |
https://mathoverflow.net/questions/92813 | 34 | It is well known that a (real) vector bundle $\pi : E\to B$ over a topological space (or manifold) $B$ is a fibre bundle whose fibres
$$F=\pi^{-1}(x), \ \ \ x\in B $$
over any $x\in B$, are diffeomorphic to a vector space $V$. On the other hand, a principal $G$-bundle is a fibre bundle $\pi : P\to B$ over $B$ wit... | https://mathoverflow.net/users/20783 | Vector bundles vs principal $G$-bundles | The difference is that, for a vector bundle, there is usually no natural Lie group action on the total space that acts transitively on the fibers. The fact that all of the fibers are, individually Lie groups, doesn't mean that there is a Lie group that acts on the whole space, restricting to each fiber to be a simply t... | 40 | https://mathoverflow.net/users/13972 | 92815 | 54,566 |
https://mathoverflow.net/questions/92794 | 3 | If we note $A\_{k}$ the category of affine algebraic groups defined over $k$ and $\mathcal{G}$ the category of finite groups, we have a functor $W:A\_{k}\longrightarrow \mathcal{G}$, where $W(G)$ is the Weyl group of an algebraic group $G$. Is that functor exact?
| https://mathoverflow.net/users/6849 | Exact sequence of Weyl groups | Yes, it is exact on the category of *connected reductive* groups and *normal* homomorphisms over an algebraically closed field $k$.
For an algebraically closed field $k$ and a connected reductive $k$-group $G$, one can construct a canonical *based root datum ${\rm BRD}\ G$*, so we obtain a canonically defined Weyl gr... | 5 | https://mathoverflow.net/users/4149 | 92822 | 54,570 |
https://mathoverflow.net/questions/88607 | 1 | Consider a rectangular $(m \times n)$ matrix $\underline E\_1$ with $m < n$ that has only $0$ or $1$ entries. It has exactly one $1$ entry in each row and not more than one $1$ entry in each column. Consider it being a selection of $m$ rows out of a $(n \times n)$ permutation matrix $\underline P$.
Given $\underline ... | https://mathoverflow.net/users/21444 | Complement to part of a permutation matrix | You get all possible $E\_2$ by starting with an $(n-m) \times (n-m)$ permutation matrix, and expand it to being $(n-m) \times n$ by inserting columns of zeros under each of the ones of $E\_1$. That is, there's an easy bijection between your set and the order $(n-m)$ permutation matrices. I think you knew this already. ... | 1 | https://mathoverflow.net/users/20281 | 92824 | 54,571 |
https://mathoverflow.net/questions/90818 | 2 | Let f(x) be a power series with complex coefficients, suppose f(0)=0. Is there a classification of equivalence classes of f(x) up to conjugation by another power series g(x) with g(0)=0? This question should be interesting in the field of complex analytic dynamical systems, since conjugate functions define similar dyna... | https://mathoverflow.net/users/9065 | infinitesimal classification of functions near a fixed point upto conjugation | This question is surely very interesting in holomorphic dynamics. A lot is already known, some of it going back to Koenigs (around 1880).
As you mentioned, a germ $f(x)= a\_{1}x+a\_{2}x^2 + \ldots $ with $\vert a\_{1} \vert \neq 0,1$ is holomorphically conjugated to its linear part $g(x)=a\_{1}z$.
When the germ is ta... | 2 | https://mathoverflow.net/users/15673 | 92829 | 54,573 |
https://mathoverflow.net/questions/92812 | 1 | Let $M$ and $N$ be $R$-modules with $R$ a commutative ring with identity. When we calculate $Tor\_i^R(M,N)$, usually first we choose projecive resolutions $P\_.$ and $Q\_.$ of $M$ and $N$, then we calculate the $i$-th homology group of the complex $P\_.\otimes Q\_.$. My question is: can we choose an injective resolutio... | https://mathoverflow.net/users/22562 | Calculation of $Tor_i^R(M,N)$ using an injective resolution of M and a projective resolution of N. | It is enough to replace one of the objects with its projective resolution. For example, you can replace $N$ with its projective resolution $Q$. Then the cohomology of the complex $M \otimes Q$ is equal to $Tor$'s. On the other hand, after that you can replace $M$ with ANY complex $C$ quasiisomorphic to it, for example ... | 3 | https://mathoverflow.net/users/4428 | 92835 | 54,578 |
https://mathoverflow.net/questions/92826 | 1 | Hello,
This probably just technical, but anyway:
In "Simplicial Homotopy Theory" by Goerss and Jardine, chap. III, par. 2, after cor. 2.12, they describe a model structure on $Ch^{+}$, the category of chain complexes in non-negative degress. Equivalences are quasi-isom., fibrations are those maps which are surjecti... | https://mathoverflow.net/users/2095 | About a statement in Jardine and Goerss "Simplicial Homotopy Theory" | The definition is correct. (That is, it correctly describes a model structure, and this is a commonly used model structure.)
Note that for nonnegatively graded chain complexes there is no chance of requiring all fibrations to be surjective in degree zero because (given the requirement that every map can be factored ... | 6 | https://mathoverflow.net/users/6666 | 92838 | 54,581 |
https://mathoverflow.net/questions/92825 | 6 | It is related to [this](https://mathoverflow.net/questions/92753/sums-of-powers-mod-p) question.
**Question.** Suppose that $p$ is a prime, $p-1$ is divisible by $q^2$ for some $q$. Is it true that every number modulo $p$ is a sum of two $q$-th powers.
If $q=2$, then (\*) is true. Indeed in that case $p\equiv ... | https://mathoverflow.net/users/nan | Sums of two same powers modulo $p$ | Suppose $p=q^2 + 1$. Then the set of $q$-th powers, including 0, has size $q+1$. The number of elements given by a sum of pairs of these is at most $(q+1) + \frac{(q+1)(q)}{2} = \frac{q^2 + 3q + 2}{2} < q^2 + 1$ whenever $q > 3$. So, for example, $6,7,10,11$ are not expressible as a sum of two fourth powers in $\mathbb... | 9 | https://mathoverflow.net/users/18086 | 92839 | 54,582 |
https://mathoverflow.net/questions/83019 | 0 | I have a differential operator defined by its Fourier transform:
$\left(\alpha k\_x^2 + \beta k\_y^2 + \gamma k\_x k\_y \right)^\delta \hspace{20pt} \alpha,\beta,\gamma,\delta \in \mathbb{R}$
I don't know how to do the inverse transform, but I know that it is impossible to compute in the most general case. However ... | https://mathoverflow.net/users/19830 | Fourier transform of a differential operator | Let me change slightly your notations and consider the quadratic form in $\mathbb R\_{\xi,\eta}^2$
$$
Q(\xi,\eta)=\alpha \xi ^2+2\gamma \xi \eta+\beta \eta^2,
$$
where $\alpha, \beta$ are real parameters.
This is a Fourier multiplier :
$
Fourier\bigl(Q(D\_x,D\_y)u\bigr)(\xi,\eta)=Q(\xi,\eta)\hat u(\xi,\eta).
$
Let us f... | 3 | https://mathoverflow.net/users/21907 | 92849 | 54,587 |
https://mathoverflow.net/questions/92847 | 7 | Does having vertex transitivity make the problem of calculating independence and chromatic numbers easier?
| https://mathoverflow.net/users/10035 | Vertex transitive graphs | Not particularly. There is a paper by Codenotti, Gerace, Vigna "Hardness results and spectral techniques for combinatorial problems on circulant graphs" Linear Algebra Appl. 285 (1998) 123-142 which shows that computing the chromatic number of a circulant graph is NP-hard.
(The pdf is available on Codenotti's web page.... | 11 | https://mathoverflow.net/users/1266 | 92850 | 54,588 |
https://mathoverflow.net/questions/85244 | 6 | Maybe this question be very simple, but I don't know why it is hard for me. Thanks for any guide and help.
We say a surface $S$ (2-dimensional metric(compact) Riemannian surface) is good (denote by $GS$), if every $2n$, $n\geq1$, points on surface can be separate by some geodesic to two distinct subsets $V\_1$ and $V... | https://mathoverflow.net/users/19885 | Good Surface,Bad Surface-Surface classification | Assuming that "geodesic" in this question means "simple closed geodesic", then every complete hyperbolic surface $S$ of finite area is "bad": You cannot even separate an arbitrary pair of points. The reason is that the union of simple closed geodesics on $S$ is nowhere dense (even more, its closure has Hausdorff dimens... | 11 | https://mathoverflow.net/users/21684 | 92853 | 54,589 |
https://mathoverflow.net/questions/92783 | 1 | Assume there are $m$ tasks, each task's working time conforms to some distribution, for instance an exponential distribution with mean $\lambda$.
So let the r.v. $X\_i$ is the working time of the $i$th task, and $\{X\_i\}$ are i.i.d. random variables.
There are $n$ machines to do these tasks, the scheduling police is... | https://mathoverflow.net/users/8379 | Random Task Scheduling Problem | The case where $n > m$ is "easy": if $F$ is the distribution function of each of the $m$ iid random variables, then the distribution function of the maximum is $F^m$. Of course, $F^m$ may be difficult to compute, so even bounding the expectation of the makespan can be tricky.
* Peter J. Downey, *Distribution-free bou... | 2 | https://mathoverflow.net/users/7252 | 92856 | 54,590 |
https://mathoverflow.net/questions/92852 | 6 | By default, let all algebras be complex and unital. I am concerned with the non-commutative algebras. I am wondering if the following might be true (at least for some classes of algebras, like semi-simple algebras).
Suppose $A$ is a complex, non-commutative algebra with a maximal right ideal which is not finitely gen... | https://mathoverflow.net/users/20746 | Left ideals vs right ideals | The answer is No. A ring is right (left) noetherian if all of its right (left) ideals are finitely generated. You can find an example of a ring R that is right noetherian but not left noetherian in:
Tsit-Yuen Lam, A first course in noncommutative rings.
| 8 | https://mathoverflow.net/users/22475 | 92857 | 54,591 |
https://mathoverflow.net/questions/86528 | 2 | Imagine we are making necklaces with $n$ beads, each bead is a different color from all others. Let's say we make one necklace. If we make another necklace with the same $n$ differently colored beads, choosing each bead at random, what is the chance that exactly $k$ of the $n$ nearest neighbor pairs of beads from the o... | https://mathoverflow.net/users/20840 | Counting Nearest Neighbors that Stay Nearest Neighbors after Random Rearrangements | An analytical solution to this question was given by David P Robbins in "The probability that neighbors remain neighbors after random rearrangements. Amer. Math. Monthly 87 (1980), 122-124." He uses an inclusion-exclusion argument to find an expression for $A(n,k)$, the number of random rearrangements that have exactly... | 4 | https://mathoverflow.net/users/20840 | 92859 | 54,592 |
https://mathoverflow.net/questions/92866 | 15 | There are various strict monoidal model categories of spectra (e.g. symmetric spectra) where the honestly commutative monoid objects model the "coherently commutative" ring spectra (which might otherwise be expressed using, say, operads). Is there an analog for spaces? That is, there a monoidal model category, Quillen ... | https://mathoverflow.net/users/344 | A model category of spaces where strict commutative monoids are $E_\infty$-spaces | Yes, such a model is developed in a paper of Blumberg, Cohen and Schlichtkrull about Thom spectra.
| 16 | https://mathoverflow.net/users/2353 | 92867 | 54,594 |
https://mathoverflow.net/questions/92889 | 2 | Perhaps my question is naive, but let me try.
Take a (real or complex) vector space $V$ and consider an ideal $\mathcal{I}$ of subsets of $V$ with the following property (call it (\*)): for each linear map $A\colon V\to V$ and each $S\in \mathcal{I}$ we have $A(S)\in \mathcal{I}$.
The ideal of finite sets enjoys th... | https://mathoverflow.net/users/20746 | Ultrafilters over vector spaces | Every ideal satisfying $(\\*)$ is included in an ideal maximal with respect to the property of satisfying $(\\*)$, by Zorn's lemma. However, such an ideal need not be maximal as an ideal.
No maximal ideal can satisfy $(\*)$, apart from trivial cases ($0$-dimensional vector spaces). Let $X$ be a subset of the base fie... | 4 | https://mathoverflow.net/users/12705 | 92894 | 54,604 |
https://mathoverflow.net/questions/92724 | 2 | Let $\ell\_{1,0}(S)=(c\_{0,0}(S),\Vert\cdot\Vert\_1)$ be a space of functions on a set $S$ with finite support, endowed with $\ell\_1$ norm. Could you answer the at least one of the following questions
1) which subspaces in $X$ are complementable?
2) which subspaces in $X$ are $C$-complementable, i.e. there exist ... | https://mathoverflow.net/users/19593 | Complementable subspaces of $(c_{00}(S),\Vert\cdot\Vert_1)$ | Just some simple remarks:
1. If $P$ is projection on $X$, then $P$ extends to a projection on the completion $\ell\_(S)$ with the same norm. Therefore, if $P$ has norm one, the closure of its range in $\ell\_(S)$ is the closed span of disjointly supported functions. The only way this can happens is for the range of $... | 4 | https://mathoverflow.net/users/2554 | 92895 | 54,605 |
https://mathoverflow.net/questions/92902 | 1 | Let $A\to B$ and $B\to C$ be epimorphisms of Abelian (infinitely generated) groups. Let $A\cong C$. Is then $A\cong B$?
| https://mathoverflow.net/users/18814 | non-Hopfian groups | Replacing $C$ by $A$, you ask the following: if there are epis $A \twoheadrightarrow B$ and $B \twoheadrightarrow A$, is there an iso $A \cong B$ ? The answer is no, take $A=\mathbb{Z}^{\oplus \mathbb{N}}$ and $B=A \times \mathbb{Q}$. There is an obvious projection $B \to A$. On the other hand, since $\mathbb{Q}$ is co... | 9 | https://mathoverflow.net/users/2841 | 92906 | 54,610 |
https://mathoverflow.net/questions/92891 | 1 | When working with group cosets in MAGMA is there a way of treating the cosets as subsets of the overlying group. Specifically I have a group $G$ and subgroups $H$ and $K$ . I wish to look at the intersection of a pair of cosets $Hh$ and $Kk$ for some $h,k\in G$ , but am unable to perform such operations in MAGMA when t... | https://mathoverflow.net/users/19783 | Working with group cosets in MAGMA | As far as I can see, the only way to do that directly with cosets $ C1$ and $C2$ of $G$ is
$\{ x : x\ {\rm in}\ G\ |\ x\ {\rm in}\ C1\ {\rm and}\ x\ {\rm in}\ C2 \}$
which looks very inefficient, because it is iterating over all of $G$.
I would suggest first find a right transversal $T$ of $H \cap K$ in $H$, and ... | 5 | https://mathoverflow.net/users/35840 | 92910 | 54,613 |
https://mathoverflow.net/questions/92899 | 11 | I recall reading that the forgetful functor $FinProdCat \to SymMonCat$ from categories with finite products and product preserving functors to symmetric monoidal categories and tensor preserving functors has a left adjoint.
To make this precise one has to insert lax, weak or strict in several places -- I am interest... | https://mathoverflow.net/users/733 | Left adjoint to the forgetful functor from finite product categories to symmetric monoidal categories | The 2-categories of finite-product categories and symmetric-monoidal categories are both 2-monadic over $\mathrm{Cat}$, in all possible senses. That is, there are strict 2-monads $P$ and $S$ on $\mathrm{Cat}$ such that $P$-algebras and strict, pseudo, lax, and colax $P$-morphisms coincide with finite-product categories... | 6 | https://mathoverflow.net/users/49 | 92913 | 54,616 |
https://mathoverflow.net/questions/92918 | 3 | $\DeclareMathOperator{\colim}{colim}
\DeclareMathOperator{\Spec}{Spec}$
[Edit1] I should point out that the colimits below are in the category of schemes, since the statements are trivially false for colimits in the category of affine schemes. [/Edit1]
In [this answer](https://mathoverflow.net/questions/23478/examp... | https://mathoverflow.net/users/21815 | Is lim R_i = O(colim Spec R_i) true for finite (co)limits? | We have an equivalence of categories $Aff\simeq Ring^{op}$ and a pair of adjoint functors $$\mathcal{O}:Sch\rightleftharpoons Ring^{op} : Spec$$ $$\mathcal{O} \dashv Spec$$
The category of affine schemes is clearly a full subcategory of $Sch$. This leads us to reformulation of the previous statement: the inclusion of f... | 8 | https://mathoverflow.net/users/10605 | 92923 | 54,619 |
https://mathoverflow.net/questions/92917 | 6 | Let $M$ be a smooth $n$-dimensional manifold, and let $FM = GL(M)$ indicate its [tangent frame bundle](http://en.wikipedia.org/wiki/Frame_bundle#Tangent_frame_bundle). Let $G$ be a fixed linear subgroup of $GL(n)$, and consider the space $\mathcal S$ of all [$G$-structures](http://en.wikipedia.org/wiki/G-structure) on ... | https://mathoverflow.net/users/238 | Random geometries | First of all, a canonical reference for special geometric structures is the book "Compact manifolds with special holonomy" by Dominic Joyce.
A1: As you observed, specifying an $O(n)$-structure is the same thing as picking a Riemannian metric, in other words a section of the bundle of positive symmetric 2-tensors. For... | 4 | https://mathoverflow.net/users/22029 | 92925 | 54,620 |
https://mathoverflow.net/questions/92865 | 9 | Is the condition that a module is reflexive an open condition?
That is, if $X$ is a smooth projective complex variety, $T$ a quasi-projective variety, and $F$ a finitely presented module on $X \times T$ that is $T$-flat, then we can form the locus $T' \subset T$ of points $t$ such that the restriction of $F$ to $X \... | https://mathoverflow.net/users/5337 | Is reflexivity an open condition? | This locus is indeed open. I will explain why using Kollar's "Hulls and Husks" (arXiv:0805.0576). More generally, this article studies in great detail when taking the double dual commutes with base change.
First, we may restrict to the open locus of $T$ where $F\_t$ is torsion-free (because reflexive sheaves are tors... | 9 | https://mathoverflow.net/users/2868 | 92926 | 54,621 |
https://mathoverflow.net/questions/92883 | 12 | A ***coin graph*** is a graph that can be represented by a set of disjoint, except possibly touching, *unit* disks in the plane (i.e. the disks are the vertices and the edges correspond to the pairs that touch each other). It's easy to show by induction that $\chi(G)\leq4$ for every coin graph $G$, as there's always a ... | https://mathoverflow.net/users/22580 | Small 4-chromatic coin graphs | Flo's example with 11 is best possible. Let $G$ be a minimal coin-graph which chromatic number four. Then $G$ does not have a cut vertex, nor a vertex of degree at most two. Moreover, there does not exist a separation $(A, B)$ of $G$ of order two with $G[A \cap B]$ an edge (as opposed to a pair of non-adjacent vertices... | 13 | https://mathoverflow.net/users/20940 | 92931 | 54,623 |
https://mathoverflow.net/questions/92908 | 0 | Hello, everyone!
As we know that by *Jensen's inequality*, for jointly convex function $f$ and $\sum\_ix\_i^2=1$, we have
$$f(\sum\_i{x\_i^2\lambda\_i},\sum\_i{x\_i^2\theta\_i)}\leq\sum\_i{x\_i^2f(\lambda\_i,\theta\_i)}\leq\max\_if(\lambda\_i,\theta\_i)\leq\sum\_if(\lambda\_i,\theta\_i),$$
and we get a bound of $f(\s... | https://mathoverflow.net/users/19399 | Is there relationship between $f\left({\sum_i(\mathbf{v}_i^\top\mathbf{x})^2\lambda_i},\sum_j{(\mathbf{u}_j^\top\mathbf{x})^2\theta_j}\right)$ and $\sum_i\sum_j{(\mathbf{v}_i^\top\mathbf{u}_j)^2f(\lambda_i,\theta_j)}$ if $f$ is jointly convex? | Consider the following case. Let $f(a,b) = a + b$, $\lambda\_1 = \theta\_1 = -1$, $\lambda\_2 = \theta\_2 = 1$, and $V = U = I\_2$. Then, $-2 \leq L\_1(\mathbf x) \leq 2$ and $L\_2 = 0$. Clearly, there exists no $\gamma\in\mathbb R$ that satisfies either $L\_1(\mathbf x) \leq \gamma L\_2$ or $L\_1(\mathbf x) \geq \gamm... | 2 | https://mathoverflow.net/users/22598 | 92937 | 54,627 |
https://mathoverflow.net/questions/92916 | 15 | Let $(C,W)$ be a category equipped with a subcategory of weak equivalences. Its "classification diagram" or "bisimplicial nerve" $N(C,W)$ is a bisimplicial set, for which $N(C,W)\_n$ is the nerve of the category of functors $[n]\to C$ and natural weak equivalences between them. If we fibrantly replace $N(C,W)$ in the c... | https://mathoverflow.net/users/49 | Does the classification diagram localize a category with weak equivalences? | It seems to me that the answer is yes. Here is a sketchy argument.
Let us fix some notations. I will write $i(X)$ for the maximal Kan subcomplex of a quasi-category $X$ and $Hom(A,B)$ for the internal Hom of simplicial sets $A$ and $B$. If $B$ is a quasi-category, then so is $Hom(A,B)$ for any simplicial set $A$, and... | 6 | https://mathoverflow.net/users/1017 | 92941 | 54,629 |
https://mathoverflow.net/questions/92929 | 11 | Let $C$ be a symmetric monoidal category, and $f : x \to y$ be a morphism in $C$. I would like to construct the localization $C\_f$ explicitly, which solves the universal property
$$\mathrm{Hom}\_{\otimes}(C\_f,D) = \{F \in \mathrm{Hom}\_{\otimes}(C,D) : F(f) \text{ iso}\}.$$
I am not interested in a general existenc... | https://mathoverflow.net/users/2841 | Localization of a symmetric monoidal category at a single morphism | I'll try to bend David White's answer towards the actual situation of your question. The outcome is somewhat clumsy and it totally looks like model structures can be eliminated from it, but anyway:
Assume your category C is closed monoidal and locally presentable. Then it is a monoidal model category with cofibration... | 5 | https://mathoverflow.net/users/733 | 92949 | 54,635 |
https://mathoverflow.net/questions/80803 | 25 | I've recently realized that there is a gap in my understanding of the Tannakian formalism for reconstructing an (algebraic) group from its category of (finite-dimensional) representations. To warm up, here is a version I do understand. Let $(\mathcal C,\otimes,\mathrm{flip})$ be a symmetric monoidal category $\mathbb C... | https://mathoverflow.net/users/78 | What does the Tannakian formalism reconstruct when fed the category of chain complexes? | Deligne does not do what you seem to want, which is give a theory internal to super vector spaces. To do so is probably an open question. The obvious version of Hopf algebras in SVect does not work: the category of comodules admits an action by the lines in SVect (commuting with the fiber functor), and thus cannot reco... | 8 | https://mathoverflow.net/users/4639 | 92956 | 54,638 |
https://mathoverflow.net/questions/92953 | 6 | Beurling's majorant is defined as the unique entire function $B(z)$ such that the Fourier transform of $B(x)$ is compactly supported in the interval $[-1;1]$, $B(x) \geq \text{sgn}(x)$ and $B(x)$ minimizes the integral,
$$
\int\_{-\infty}^{\infty} B(x) - \text{sgn}(x) \text{d} x=1
$$
Explicitly,
$$
B(z) = \left ( ... | https://mathoverflow.net/users/22603 | A question about the Beurling-Selberg majorant | Given such a $M$ note that $F(x)=M^2(x)$ will be (1) non-negative, (2) majorize $1\_{[a,b]}$, and (3) has $\hat{F}$ supported in $[-2\delta,2\delta]$. Thus it will be a (possibly not optimal) solution to the standard $2\delta$ Beurling-Selberg problem. This implies that
$$\int |M(x)|^2dx - (b-a) \geq \frac{1}{2\delta}... | 3 | https://mathoverflow.net/users/630 | 92960 | 54,641 |
https://mathoverflow.net/questions/92939 | 28 | Just a new guy in optimization. Is it true that all convex optimization problems can be solved in polynomial time using interior-point algorithms?
| https://mathoverflow.net/users/22601 | Can all convex optimization problems be solved in polynomial time using interior-point algorithms? | No, this is not true (unless P=NP). There are examples of convex optimization problems which are NP-hard.
Several NP-hard combinatorial optimization problems can be encoded as convex optimization problems over cones of co-positive (or completely positive) matrices. See e.g. "[Approximation of the stability number of a ... | 50 | https://mathoverflow.net/users/11100 | 92961 | 54,642 |
https://mathoverflow.net/questions/92914 | 2 | This question is inspired by the discussion at [this problem](https://mathoverflow.net/questions/92511/how-can-one-prescribe-the-pairwise-intersection-measuress-of-n-sets/92785).
Suppose I have a *design* consisting of a finite point set $U$ of size $|U|=m\_{\emptyset}$ and a family of $n$ subsets (sometimes called *... | https://mathoverflow.net/users/8008 | Design constraint systems over the reals | In spite of my comment, in fact a non-negative rational solution need not imply a non-negative integer solution. There are no projective planes of order $q=6$ (nor $q=10$) but there is a non-negative rational solution to the constraints for any $q$
Recall that the constraints for a projective plane of order $q$ are:
... | 0 | https://mathoverflow.net/users/8008 | 92965 | 54,644 |
https://mathoverflow.net/questions/50258 | 11 | What I would like to know is exactly what the title asks:
>
> What are the most general classes of
> simplicial complexes or posets for
> which the Charney-Davis conjecture is
> known, and what is the most general
> setting for which it might expected to be
> true?
>
>
>
I believe it conjectured, for exam... | https://mathoverflow.net/users/4558 | What are the most general classes of simplicial complexes or posets for which the Charney-Davis conjecture is known, and what is the most general setting for which it might expected to be true? | There is quite a lot that can be said about the Charney-Davis conjecture, so let me say a few things and perhaps add more later.
1) The conjecture is a discrete analogue of a well known conjecture by Hopf on the Euler characteristic of nonpositively curved manifolds in odd dimension. You consider cubical complexes an... | 18 | https://mathoverflow.net/users/1532 | 92976 | 54,648 |
https://mathoverflow.net/questions/87837 | 7 | The [Hirsch conjecture](http://en.wikipedia.org/wiki/Hirsch_conjecture) asserts that the graph (i.e. $1$-skeleton) of a $d$-dimensional [convex polytope](http://en.wikipedia.org/wiki/Polytope) with $n$ facets has [diameter](http://en.wikipedia.org/wiki/Distance_%28graph_theory%29) at most $n - d$.
After being open fo... | https://mathoverflow.net/users/4558 | Does the Hirsch conjecture hold for $n < 2d$? | The answer is no, as follows from the following Lemma of Klee and Walkup:
Lemma: If P is a d-polytope with n facets and we perform a "wedge" over any facet F we get a (d+1)-polytope P' with n+1 facets and with diameter(P') $\ge$ diameter(P).
Corollary: since there is a 43-polytope with 86 facets and diameter (at le... | 13 | https://mathoverflow.net/users/22608 | 92980 | 54,651 |
https://mathoverflow.net/questions/92940 | 1 | Let G be an affine connected algebraic group, and K[G] be its coordinate ring. Let Y be a subvariety of G defined as a zero set for some f in K[G]. For which f, Y is a closed subgroup of G
| https://mathoverflow.net/users/16049 | Let G be an affine connected algebraic group. When a subvariety of G with codimension one is a subgroup. | It is perhaps easiest to express this in terms of Hopf algebras. The coordinate ring $K[G]$ has the structure of a Hopf algebra; the subvariety $Y$ is a closed subgroup of $G$ if and only if the ideal $(f)$ is a Hopf ideal, i.e.
\begin{align\*}
\Delta(f) &\in (f) \otimes K[G] + K[G] \otimes (f); \\\\
\epsilon(f) &= 0; ... | 2 | https://mathoverflow.net/users/12858 | 92981 | 54,652 |
https://mathoverflow.net/questions/92979 | 3 | A convex polygon $P$ having an interior point $A$ in generic position (not on any line defined by two vertices) can be encoded combinatorially by associating to every vertex $v$ of $P$ the unique
edge $e(v)$ of $P$ such that $A$ is contained in the triangle defined as the convex hull of
$v$ and $e(v)$. (Equivalently, ... | https://mathoverflow.net/users/4556 | Realisation of convex polygons with an interior point from combinatorial data | Given $e$ so that it satisfies your condition of distinct triangles intersecting nontrivially we will prove that it comes from an associated polygon by induction on the number of vertices of the polygon.
First pick an $i$ so that our internal point is not contained in the triangle $(i-1,i,i+1)$, where indices are mod... | 4 | https://mathoverflow.net/users/2384 | 92990 | 54,655 |
https://mathoverflow.net/questions/92927 | 26 | Let $X$ be an integral proper normal curve over a (perfect) field $F$, of genus $\geq 2$. One variant of Grothendieck's "section conjecture" states that the sections $G\_F \rightarrow \pi\_1(X)$ of the exact sequence
\begin{equation}
1 \rightarrow \pi\_1(X\_{\bar{F}}) \rightarrow \pi\_1(X) \rightarrow G\_F \rightarrow ... | https://mathoverflow.net/users/19367 | Why does the Section Conjecture exclude curves of genus 1? | I think that Grothendieck had already observed that the map from rational points to sections is injective (for curves of genus at least 2 over a number field) and I believe that his proof works even for curves of genus $1$, so the thing that fails for curves of genus $1$ is surjectivity.
Consider any exact sequence o... | 28 | https://mathoverflow.net/users/519 | 92992 | 54,657 |
https://mathoverflow.net/questions/92974 | 9 | Wikipedia <http://en.wikipedia.org/wiki/Pettis_integral> defines the Pettis Integral for Banach space valued functions wrt to some measure space by duality.
It calls the Pettis & Bochner integral the weak & strong integral respectively, which implies some kind of relationship; also, apparently there is a Dunford int... | https://mathoverflow.net/users/22002 | What is the Dunford Integral and why is it useful? | Let $f \colon \Omega \to E$ be your function. $\mu$ is a measure on $\Omega$. Assume, for every $x^\* \in E^\*$, the composition $x^\* \circ f$ is $\mu$-integrable. The **Dunford integral** in general lies in $E^{\*\*}$, namely $\int\_A f d\mu$ is the element $u^{\*\*} \in E^{\*\*}$ defined by $u^{\*\*}(x^\*) = \int\_A... | 11 | https://mathoverflow.net/users/454 | 92994 | 54,658 |
https://mathoverflow.net/questions/92998 | 8 | Does anyone know what the Fourier transform (in the sense of distributions) of
$$
f(x) = (x^2 - 1)^{1/2}x, \quad |x|\ge 1,
$$
and $f(x) = 0$ otherwise, is?
| https://mathoverflow.net/users/19433 | Fourier transform | First of all observe that
$$ f(x)=\frac{1}{3}\frac{d}{dx} (x^2-1)^{\frac{3}{2}}\_+, $$
where for any real number $t$ we set $t\_+=\max(t,0)$. Thus it suffices to compute the Fourier transform of $(x^2-1)^{\frac{3}{2}}\_+$.
In Section 2.5 Chapter 2 of the book by Gelfand and Shilov, *Generalized Functions, vol.1, ... | 17 | https://mathoverflow.net/users/20302 | 93000 | 54,662 |
https://mathoverflow.net/questions/92683 | 2 | Consider a commutative group $G$ of finite type, a subgroup of finite index $H\subseteq G$, a noetherian commutative ring $A$, and a $G$-graded $A$-algebra $R=\bigoplus\_{g\in G}R\_g$ with no zero-divisors, and denote by $R\_H=\bigoplus\_{g\in H}R\_g$ the degree restriction of $R$ to $H$.
It is well-known that if $R$... | https://mathoverflow.net/users/11025 | Finiteness conditions and Veronese subrings | In case $G$ is finite, this cannot happen.
(This might extend to the general case of finitely generated groups, as Fred told me when we talked about this in my office :-) )
---
First, let me show that $R$ is of finite type over $R\_0$ in case $R\_0$ is noetherian and $G$ is finite. For that, it suffices to show... | 2 | https://mathoverflow.net/users/7001 | 93001 | 54,663 |
https://mathoverflow.net/questions/92736 | 3 | Let $V\_{\lambda}$ and $W\_{\lambda}$ be the irreducible representations of $S(n)$ and $\mathfrak{su}(N,\mathbb{C})$ associated to the partition $\lambda \in \mathbb{Y}$ of size $| \lambda |=n$ and length $l(\lambda) \leq N$. The following limit $$\frac{\dim V\_{\lambda}}{n!} = \lim\_{N \rightarrow \infty} \frac{\dim W... | https://mathoverflow.net/users/6862 | Two curious asymptotic results for dimensions of type A objects | This is an answer to Alexander's combinatorial reformulation of the question in comments to Bruce's answer.
dim $V\_\lambda$/$n$! is the chance that you will get a standard Young tableau if you assign the values 1 to $n$ to the boxes of a tableau of shape $\lambda$ according to a random permutation.
dim $W\_\lambd... | 4 | https://mathoverflow.net/users/468 | 93005 | 54,665 |
https://mathoverflow.net/questions/68006 | 14 | I'm interested in examples of theorems that employ the proof techniques that are utilized in the proof of the undecidability of Kolmogorov Complexity.
**Definition:(Sipser)** Let x be a binary string. We say that the minimal description of x, written as d(x), is the shortest string $\langle$M,w$\rangle$ where TM M o... | https://mathoverflow.net/users/15821 | Kolmogorov Complexity and Proof Techniques | Using the same technique, one can construct infinitely many statements which are true with probability arbitrarily close to 1, but are nonetheless unprovable. See lemma 4 in <http://theory.stanford.edu/~trevisan/cs172/notek.pdf>
| 4 | https://mathoverflow.net/users/6429 | 93015 | 54,671 |
https://mathoverflow.net/questions/93016 | 11 | **Question 1**: Given a co-commutative bialgebra, does there exist a sort of Grothendieck group type construction? Presumably this should take the form of a functor from co-commutative bialgebras to hopf algebras?
My motivation: the finite particle vectors in the symmetric Fock space $\mathbb{C}\Omega\oplus \bigoplus... | https://mathoverflow.net/users/10779 | A quantum Grothendieck group? | Connected graded bialgebras have an antipode (which is unique):
The following book gives two formulae:
[MR2724388](https://mathscinet.ams.org/mathscinet-getitem?mr=2724388)
Aguiar, Marcelo; Mahajan, Swapneel
Monoidal functors, species and Hopf algebras.
CRM Monograph Series, 29. American Mathematical Society, Provi... | 8 | https://mathoverflow.net/users/3992 | 93020 | 54,673 |
https://mathoverflow.net/questions/92525 | 3 | Suppose $n\in\mathbb{Z}^+$ is a nonsquare in $\mathbb{Z}$ but is a square mod $2^2,3^2,4^2,5^2,\ldots,k^2.$ How small can $n$ be?
On the ERH, there are no small pseudosquares: $L\_p>e^{\sqrt{p/2}}$. Heuristically, more is true: $\log L\_p\gg p/\log p.$ I am looking for an unconditional lower bound, even if very weak.... | https://mathoverflow.net/users/6043 | Bounds on pseudosquares | I will assume that $n$ is coprime to $2,3,\dots,k$. (In "EDIT" below I give a weaker bound for the case when this condition is not met.)
Let $n=m^2 d$, where $d>1$ is square-free. Then $r\mapsto (\frac{d}{r})$ is a nontrivial (quadratic) character mod $4d$. By assumption, $(\frac{d}{p})=(\frac{n}{p})=1$ for any prim... | 5 | https://mathoverflow.net/users/11919 | 93030 | 54,678 |
https://mathoverflow.net/questions/93034 | 8 | I am currently trying to understand the notion of criticality (as discussed, e.g., in Terence Tao's book on nonlinear dispersive equations) from a physical viewpoint. That's why i'm interested in the question which PDE from physics (apart from the Navier-Stokes equations) and geometry are supercritical with respect to ... | https://mathoverflow.net/users/22625 | Which PDE from physics (and geometry) are supercritical? | Generally speaking, supercriticality occurs when the dimension and/or the nonlinearity exponent is sufficiently large.
Sigma field models such as the harmonic map, wave map, or Schrodinger map equations become supercritical in three and higher spatial dimensions. (The critical two-dimensional case is probably the mos... | 22 | https://mathoverflow.net/users/766 | 93040 | 54,682 |
https://mathoverflow.net/questions/93026 | 6 | Let $X$ be a scheme over an algebraically closed field $k$ of positive characteristic $p$. Recall that the absolute Frobenius morphism $F : X \to X$ is the map which is the identity on points and the $p^{th}$ power morphism on functions. Recall also that we say that $X$ is Frobenius split if there is an $\mathcal O\_X$... | https://mathoverflow.net/users/1528 | Frobenius splitting over non-algebraically closed fields | There is of course the commutative algebra perspective on Frobenius splittings as well. Indeed, the general case there is *much* more general than what I think you are even considering. Let me make some comments on various generalizations of the notion of Frobenius splittings.
**The perfect field case:** Essentially ... | 12 | https://mathoverflow.net/users/3521 | 93045 | 54,686 |
https://mathoverflow.net/questions/93041 | -1 | Let $p(x)$ be a probability density function on the unbounded set $X \subseteq \mathbb{R}^n$, so that $\int\_X p(x) dx = 1$.
Let $F: X \rightarrow \mathbb{R}\_{\geq 0}$ a measurable but non-integrable function, i.e.
$$ \int\_X F(x) p(x) dx = \infty $$
I'm wondering if the following proposition is true:
$ \foral... | https://mathoverflow.net/users/22627 | Composed function made Lebesgue integrable? | Here is a way to start cooking such a function. From Fubini, we have for every non-negative function $g$ the equality $$\int g(x) p(x) dx = \int\_0^\infty du \int 1\_{g(x)>u} p(x) dx = \int\_0^\infty \mu(\{x:g(x)>u\}) du$$ (where $\mu$ is of density $p$).
So if you want $f\circ F$ to be integrable, pick any decreasin... | 1 | https://mathoverflow.net/users/9430 | 93046 | 54,687 |
https://mathoverflow.net/questions/92750 | 19 | The addition of $p$-typical Witt vectors ($p$ a prime number) is given by universal polynomials $S\_n=S\_n(X\_0,\dots,X\_n;Y\_0,\dots,Y\_n)\in\mathbb{Z}[X\_0,X\_1,\dots;Y\_0,Y\_1,\dots]$ determined by the equalities
$\Phi\_n(S\_0,\dots,S\_n)=\Phi\_n(X\_0,\dots,X\_n)+\Phi\_n(Y\_0,\dots,Y\_n)$ for all $n\ge 0$,
whe... | https://mathoverflow.net/users/17988 | Polynomials for addition in the Witt vectors | I found a formula for $S\_n$ in terms of the previous $S\_i$'s and the polynomials $R\_i$, which I'm quite happy about. In fact, I need the multivariate version of the $R\_i$, which I will construct all at the same time. Let us consider the ring of formal power series in countably many variables $X\_1,X\_2,\dots$ with ... | 10 | https://mathoverflow.net/users/17988 | 93047 | 54,688 |
https://mathoverflow.net/questions/93039 | 3 | Let g be a finite dimensional nilpotent lie algebra over a field k of characteristic zero. Let U(g) be the universal enveloping algebra and Z(g) be its center. Denote by Z\_1(g) the augmentation ideal of Z(g). If g is not abelian, is the extension of this ideal of infinite index in U(g)? In other words, is the quotient... | https://mathoverflow.net/users/6986 | Center of universal enveloping algebra of nilpotent lie algebra | Yes. Let $J(\mathfrak{g}) := (Z(\mathfrak{g}) \cap \mathfrak{g} U( \mathfrak{g} ) ) \cdot U(\mathfrak{g})$ denote the ideal of $U(\mathfrak{g})$ you're interested in.
First note that if $\mathfrak{g}$ is the $2n + 1$ dimensional Heisenberg Lie algebra $\mathfrak{h}\_n$ with basis vectors $x\_1,\ldots, x\_n, y\_1,\ldo... | 6 | https://mathoverflow.net/users/6827 | 93051 | 54,691 |
https://mathoverflow.net/questions/93037 | 3 | The spectrum $T(n)$ which is the telescope of a finite spectrum of type n along its self-map, has a unique Bousfield class $\langle T(n)\rangle$ which only depends on $n$. It is also known, from Ravenel's paper "Localization with respect to certain periodic homology theories" that $\langle S\rangle=\langle T(0)\rangle\... | https://mathoverflow.net/users/11546 | The Bousfield Class of the Infinite Wedge of Telescopes of Finite Spectra | The spectrum $H\mathbf{F}\_p$ is an acyclic for both $\bigvee \langle T(n) \rangle$ and $\bigvee \langle K(n) \rangle$. Therefore neither of these wedges equals $\langle S \rangle$.
| 9 | https://mathoverflow.net/users/4194 | 93053 | 54,693 |
https://mathoverflow.net/questions/93064 | 12 | For a subgroup $H$ of a given group $G$, I say $H$ is "big" if it has nonempty intersection with each conjugacy class of $G$.
I have known that, trivially, $G$ itself is "big". And if $H$ is a normal subgroup and it is "big", then $H=G$. I have also known that a finite group has no proper "big" subgroup. My question i... | https://mathoverflow.net/users/22635 | A subgroup intersects every conjugacy class | Yes, for example in [Osin's infinite group](https://arxiv.org/abs/math/0411039) with 2 conjugacy classes every proper subgroup is big. Of course if you do not care about the number of generators, you can consider the (much easier) infinitely generated group constructed by Higman-Neumann-Neumann where all non-identity e... | 13 | https://mathoverflow.net/users/nan | 93065 | 54,701 |
https://mathoverflow.net/questions/93069 | 0 | It is a standard fact from elementary complex analysis that a holomorphic function $f:\mathbb{C}\to \mathbb{C}$ is a conformal mapping. Now, suppose I have a map $f':\mathbb{R}^2\to \mathbb{R}^2$ which is a conformal mapping of the plane onto itself. Write $$f'(x,y) = (f\_1(x,y),f\_2(x,y)).$$ Is $f\_1 + if\_2$ holomorp... | https://mathoverflow.net/users/21254 | When are conformal maps holomorphic? | If we define **$f' : \mathbb{R}^2 \rightarrow \mathbb{R}^2$** by **$f'(x,y) = (x, -y)$**, then it is conformal, but the corresponding map $f\_1 + i f\_2$ is not holomorphic.
| 2 | https://mathoverflow.net/users/7455 | 93070 | 54,703 |
https://mathoverflow.net/questions/93068 | 2 | Suppose that for $n \geq 4$ we have $F(x\_1, \cdots, x\_n) \in \mathbb{Z}[x\_1, \cdots, x\_n]$ is a homogeneous polynomial. Consider a large prime $p$, and suppose that we consider points of the variety $F(x\_1, \cdots, x\_n) \equiv 0 \pmod{p}$. If we consider non-singular points, then it is easy to see that the number... | https://mathoverflow.net/users/10898 | Points on a projective variety modulo $p$ | If we identify $\mathbf{F}\_p$ with $X\_p=\{0,1,\ldots, (p-1)\}\subset [0,p]$, one can probably show that for many varieties $Y/\mathbf{Z}$ in $\mathbf{A}^n$ (e.g, many hypersurfaces), the intersection $Y\cap [0,B]^n$ has not much more than the expected number $B^{n-1}$ of points, for $B$ slightly larger than $\sqrt{p}... | 4 | https://mathoverflow.net/users/21428 | 93075 | 54,706 |
https://mathoverflow.net/questions/93086 | 5 | **EDIT.** (05-04-12) I have revised and improved the questions.
Let $A$ be a commutative $\mathbb{N}$-graded $R$-algebra, which is finitely generated by $A\_1$ as an $A\_0$-algebra. You may also assume that $A\_1$ is of finite type over $R$ and that $R$ is noetherian, if this is necessary. Let us denote by $\mathrm{g... | https://mathoverflow.net/users/2841 | Basics(?) about quasi-coherent modules on projective schemes | Question 1: The category $C$ is often described as the quotient
$$grMod(A)/TN(A)$$
where we denote by $TN(A)$ we denote The subcategory of $grMod(A)$ of graded modules $M$ such that $I^n m= 0$ for $n \gg 0$ for every $m \in M$, where $I := \oplus\_{n > 0} A\_n$, the *irrelevant* ideal. This subcategory constitutes th... | 3 | https://mathoverflow.net/users/6348 | 93090 | 54,713 |
https://mathoverflow.net/questions/93061 | 3 | I am looking for a simple degree conditon that ensures the existence of a k-factor in a graph. The k is supposed to be relatively high and I don't mind the condition being a bit strict. Ideally, something of the form $\delta(G) \geq f(k)$. Any suggestions? 10x!
To clarify a bit what I'm after: there is a theorem by N... | https://mathoverflow.net/users/22051 | Degree conditions for k-factor | A similar theorem is proved in ["Relating minimum degree and the existence of a k-factor"](http://www.math.unl.edu/~shartke2/math/papers/min-deg-k-factor.pdf) by Hartke, Martin and Seacrest.
They show that a graph $G$ on $n$ vertices with minimum degree $\delta\geq \frac{n}{2}$ contains a $k$-factor if $kn$ is even ... | 3 | https://mathoverflow.net/users/2384 | 93092 | 54,714 |
https://mathoverflow.net/questions/93088 | 3 | Let $X$ be locally compact Hausdorff space. Let $\mu$ be a Borel measure on it which is finite on compact and outer regular with respect to open sets and inner regular with respect to compact sets. Does an atom of such measure have to be a singleton (up to set of zero measure)?
| https://mathoverflow.net/users/19795 | Atoms of regular Borel measure | Without loss of generality your atom $A$ is compact (by inner regularity). Call a point $a\in A$ negligible if it has a neighborhood with zero measure inside $A$. Clearly the set $B$ of negligible points is open in $A$. If it is $A$ itself, then choose finite subcovering by those neighborhoods to get that measure of $A... | 7 | https://mathoverflow.net/users/4312 | 93094 | 54,715 |
https://mathoverflow.net/questions/90023 | 4 | Suppose $\mathfrak g$ is a finite dimensional Lie algebra over a field on characteristic zero and $G$ is a finite group of automorphisms of $\mathfrak g$.
>
> Does there necessarily exist a Levi subalgebra of $\mathfrak g$ which is $G$-invariant?
>
>
>
By Levi subalgebra I mean a semisimple complement of the s... | https://mathoverflow.net/users/1409 | Equivariant Levi subalgebras. | Yes. This results and its variations are contained in papers by E.J. Taft:
* Invariant Wedderburn factors, Illinois J. Math. 1 (1957), N4, 565-573 <http://projecteuclid.org/euclid.ijm/1255380679> .
* Invariant Levi factors, Michigan Math. J. 9 (1962), N1, 65-68 DOI:10.1307/mmj/1028998623
* Orthogonal conjugacies in a... | 3 | https://mathoverflow.net/users/1223 | 93095 | 54,716 |
https://mathoverflow.net/questions/93098 | 6 | Let $K$ denote a simplicial complex and $Y$ a path-connected topological space. Let us also denote by $K^n$ the $n$-skeleton of $K$. I would like to have an example for the following situation or a proof of its impossibility:
>
> A map $f^1:K^1\to Y$ that can be extended to $f^2:K^2\to Y$ and yet no such extension... | https://mathoverflow.net/users/14379 | A counter example in obstruction theory | Here is an example with CW complexes rather than simplicial complexes. I doubt that there is an important difference, although the simplicial case will require more bookkeeping.
Take $K=\mathbb{R}P^3$ and $Y=\mathbb{R}P^2$. We can give $K$ a CW structure with skeleta $\mathbb{R}P^k$ for $0\leq k\leq 3$. Let $f^1:\mat... | 15 | https://mathoverflow.net/users/10366 | 93103 | 54,720 |
https://mathoverflow.net/questions/93093 | 3 | I have a question about the definition of an elliptic surface. One defines an elliptic surface $S$ over a base curve $C$ (over some field $k$) as a surjective morphism $f: S \to C$ such that almost all fibres are smooth genus 1 curves, which is moreover relatively minimal with respect to the elliptic fibration - i.e. n... | https://mathoverflow.net/users/1107 | Singular fibres in the definition of an elliptic surface | In most papers on elliptic surfaces (at least among those I am aware of) the condition "there is at least one singular fiber" is to exclude elliptic surfaces that are a product or to enforce similar properties e.g., the Mordell-Weil group is finitely generated.
There are many examples of elliptic surfaces that are no... | 7 | https://mathoverflow.net/users/8621 | 93105 | 54,722 |
https://mathoverflow.net/questions/93089 | 4 | Let $\phi(x,t)$ be smooth function.
Let $\zeta= x-t$ and $\eta= x+t$. $u=\phi\_\zeta$, $v= \phi\_\eta$.
Let $u$, $v$ satisfies following equations:
1-
$$u\_\eta- v\_\zeta= 0$$
$$v^2u\_\zeta-(1+2uv)u\_\eta+ u^2v\_\eta=0$$
The roles of dependent and independent variables are then interchanged to give
2-$$\zeta... | https://mathoverflow.net/users/16031 | hodographic transformation | Another way to understand this 'transformation' is to think in terms of differential forms. Let $(u,v,\eta,\zeta)$ be coordinates on $\mathbb{R}^4$ and consider the pair of $2$-forms
$$
\Upsilon\_1 = du\wedge d\zeta + dv\wedge d\eta
\quad\text{and}\quad
\Upsilon\_2 = v^2 du\wedge d\eta + (1{+}2uv)\ du\wedge d\zeta - u^... | 7 | https://mathoverflow.net/users/13972 | 93107 | 54,724 |
https://mathoverflow.net/questions/93099 | 12 | Consider $n$ points generated randomly and uniformly on a unit square. What is the expected value of the area (as a function of $n$) enclosed by the convex hull of the set of points?
| https://mathoverflow.net/users/18011 | Area Enclosed by the Convex Hull of a Set of Random Points | **Update:** By a result of Buchta (Zufallspolygone in konvexen Vielecken, Crelle, 1984; available on digizeitschriften.de) there is a general formula for this expected value, it is
$$1 -\frac{8}{3(n+1)} \bigl( \sum\_{k=1}^{n+1} \frac{1}{k} (1 - \frac{1}{2^k}) - \frac{1}{(n+1)2^{n+1}} \bigr) $$
yielding (starting with ... | 21 | https://mathoverflow.net/users/nan | 93109 | 54,725 |
https://mathoverflow.net/questions/93119 | 13 | Let $\Sigma$ be a hermitian positive definite matrix and $L$ be its Cholesky decomposition so that $LL^\ast=\Sigma$. Furthermore, let's diagonalize $\Sigma$ as $\Sigma = P\Lambda P^\ast$. $\Lambda$ is a diagonal matrix containing the real, positive eigenvalues of $\Sigma$, let us denote as $\sqrt{\Lambda}$ the diagonal... | https://mathoverflow.net/users/8737 | Interesting relationships between Cholesky decomposition and diagonalization | First, let me rephrase your remark. Let $L=HU$ be the polar factorization of $L$ ($H$ hermitian positive definite, $U$ unitary). Then $\Sigma=LL^\ast=H^2$ tells you that the Hermitian part of $L$ is $\sqrt\Sigma$. Then $U=L\Sigma^{-1/2}$ is its unitary part.
On the other hand, you have $\sqrt\Sigma=LQ^\ast=QL^\ast$. ... | 6 | https://mathoverflow.net/users/8799 | 93129 | 54,734 |
https://mathoverflow.net/questions/93128 | 0 | For *semisimple* Lie groups it's just a lookup (I define "small" as dim(R)<5 and
put the dimensions of the Clebsch-Gordon series of RxR in parentheses):
A1(1\*1=1),A1(2\*2=1+3),A1(3\*3=1+3+5),A1(4\*4=1+3+5+7),SO2(2\*2=1+1+2),A2(3\*3=3+6),B2(4\*4=1+5+10),
and the products SO2#SO2,SO2#A1 and A1#A1. Hope I forgot none.
... | https://mathoverflow.net/users/11504 | List of small dimension Lie group irreps | These dimension formula only really make sense for reductive Lie groups or algebras. This means that every finite dimenional representation is completely reducible. If you want to do this for Lie groups or algebras which are not reductive then you need to make it clear what it is you are asking for. Reductive includes ... | 2 | https://mathoverflow.net/users/3992 | 93133 | 54,737 |
https://mathoverflow.net/questions/93145 | 0 | I'm trying to understand the formula presented for the sequence [A064532](https://oeis.org/A064532) from the OEIS, looks like a recurrence relation with complex numbers:
$a(10i+j) = a(i) + a(j), etc.$
Sorry if its a simple equation, but I wasn't able to understand it.
| https://mathoverflow.net/users/12033 | Understanding a sequence generation formula of the A064532 | Nothing to do with complex numbers.
Look at the number of holes in each digit
```
0 1 2 3 4 5 6 7 8 9
1 0 0 0 0 0 1 0 2 1
```
Now look at the total number of holes when you write the number in the usual way;
just take the weighted sum of the digits.
| 2 | https://mathoverflow.net/users/3992 | 93146 | 54,743 |
https://mathoverflow.net/questions/90766 | 3 | I have a zero mean multivariate normal probability distribution where WLOG each marginal variance is unity and all pairwise correlation coefficient are equal and positive. The number of elements in the random vector is arbitrary. I am interested in the probability distribution of the number of elements which exceed som... | https://mathoverflow.net/users/21868 | Exchangeable normal distribution mixing measure | The mixing distribution is the same as the long term average . Since your gaussians can be repesented as $Z + X\_i$ where $X\_i$ are i.i.d., and also independent of $Z$, the long term average for the events I think you are looking at is $\frac 1 n \sum^n 1\_{X\_i + Z > c}$. By conditioning on $Z$ this is seen to have d... | 3 | https://mathoverflow.net/users/22650 | 93155 | 54,746 |
https://mathoverflow.net/questions/93178 | 8 | Jacobson's theorem states that
If $R$ is a ring, and for every $x\in R$, there exists $n(x)\geq 2$ such that $x^{n(x)}=x$. Then
$R$ is commutative.
I wonder if the following stronger assertion(in case $R$ has unity) is true.
Let $R$ be a ring with unity. For every $x$ in $R$, there exists $n(x)\geq 2$ such that
... | https://mathoverflow.net/users/21090 | Jacobson's theorem and further | Yes, this is true.
By Jacobson's theorem, $R$ is commutative. Now the radical of $R$ is the intersection of all primes $P$ of $R$. Hence we have an embedding
$$\phi: R/rad(R) \to \prod\_P R/P.$$
For each $x \in R/P, x \neq 0$ there is $n \ge 1$ such that $x(x^n-1)=0$ and since $R/P$ is a domain, $x^n =1$, i.e. $x$ ... | 11 | https://mathoverflow.net/users/10194 | 93180 | 54,757 |
https://mathoverflow.net/questions/93183 | 4 | The Cantor Normal Form Theorem states that every ordinal $\alpha > 0$ can be uniquely expressed in the form $$\omega^{\beta\_1}k\_1 + \omega^{\beta\_2}k\_2 + \dots + \omega^{\beta\_n}k\_n$$ for some $n \ge 1$, positive integers $k\_1,k\_2,\dots,k\_n$ and ordinals $\alpha \ge \beta\_1 > \beta\_2 > \dots > \beta\_n$.
I... | https://mathoverflow.net/users/22663 | Cantor's Normal Form and Aleph_1 | As per the many comments, the issue is confusing ordinal and cardinal arithmetic. The number of such expressions that yield countable ordinals is uncountable, as it should be. This is because the $\beta$ can range over all countable ordinals (of which there $\aleph\_1$ many). Thank you to all commenters for replying so... | 6 | https://mathoverflow.net/users/22663 | 93186 | 54,760 |
https://mathoverflow.net/questions/93194 | 5 | Is there a complete characterization of those integer polynomials, that is $P\in{\mathbb Z}[X]$, such that $P(D)\subset D$, where $D$ is the unit disk ? At least, $P(X)=\pm X^k$ works, when $k\in\mathbb N$. But are there other ones, many other ones ?
| https://mathoverflow.net/users/8799 | Integer polynomials mapping the unit disk into itself | Since $P\in \mathbb{Z}[X]$, We have $P(0)\in \mathbb{Z}$. Suppose that $P(0)\neq 0$, then
$|P(0)|\geq 1$. In that case $P(D)\subset D$ is not satisfied unless $P$ is constant by open mapping theorem.
So, if $P$ is non-constant, then we must have $P(0)=0$.
Write $P(X)=XQ(X)$. Then $Q(X)=P(X)/X$. On a disk $D\_r$ of... | 19 | https://mathoverflow.net/users/21090 | 93195 | 54,764 |
https://mathoverflow.net/questions/91981 | 2 | The Monge-Ampere (whether real or complex, whether in a domain or on a manifold) equations usually studied are of the type $F(u, \nabla u, \mathrm{Hess}(u)) = 0$ where $F$ is a concave function of $\mathrm{Hess}(u)$. This is done for the purpose of $C^{2, \alpha}$ estimates. My question is: Has any work been done in th... | https://mathoverflow.net/users/3709 | Monge Ampere equations (concavity) | Do a google search for "Fully nonlinear uniformly elliptic equations" and you will turn up a lot of information. The best estimates for viscosity solutions are $C^{1,\alpha}$, in general, where $\alpha > 0$ is very tiny. This was work in the 1980s due to Caffarelli following the important work of Krylov-Safonov. There ... | 4 | https://mathoverflow.net/users/5678 | 93202 | 54,768 |
https://mathoverflow.net/questions/93207 | 4 | An old theorem of Pospisil asserts that for any infinite set $I$ the power-set algebra $\wp(I)$ has $\exp \exp |I|$ many maximal ideals containing the ideal of finite sets. This result is published in a rather obscure Czech journal but it seems it should be well-known and described in many textbooks/monographs. I would... | https://mathoverflow.net/users/15129 | Maximal ideals in Boolean algebras; reference request | The result on the number of maximal ideals can be found as Corollary 7.4. in the book Comfort & Negrepontis "The Theory of Ultrafilters", 1974. A proof can also be found at PlanetMath [here](https://planetmath.org/NumberOfUltrafilters).
| 6 | https://mathoverflow.net/users/35357 | 93209 | 54,772 |
https://mathoverflow.net/questions/93187 | 3 | Hello,
Let $Q$ be a finite quiver, let $M$ denote the arrow ideal and let $kQ$ denote the path algebra. Endow $kQ$ with the $M$-adic topology. Now let $\mathcal{A}$ be the set of all formal series ${\sum\_{\gamma} a\_{\gamma} \gamma , a\_{\gamma} \in K\}$ where $\gamma$ is a path. Then $kQ$ is naturally a $k$-algebra... | https://mathoverflow.net/users/22669 | Ring completion of $kQ$ | The metric you have given induces the M-adic uniformity on kQ. A ball in the metric around 0 contains all linear combinations of paths whose shortest term with non-zero coefficient is a path of length greater than or equal to n for a certain n which can be computed from the radius of the ball. So $M^n$ is contained in ... | 3 | https://mathoverflow.net/users/15934 | 93211 | 54,774 |
https://mathoverflow.net/questions/93217 | 11 | Does this concept is defined for every birational morphism？ What is the precise meaning？
Thank you for your comments.
| https://mathoverflow.net/users/3525 | What is the definition of exceptional divisor？ | The exceptional set is defined for every birational morphism $\pi : Y \to X$. This is defined as follows. Set $\Sigma \subset X$ to be the smallest closed subset of $X$ outside of which $\pi : (Y \setminus \pi^{-1}(\Sigma)) \to (X \setminus \Sigma)$ is an isomorphism.
In this case the *exceptional set* is defined to ... | 38 | https://mathoverflow.net/users/3521 | 93219 | 54,778 |
https://mathoverflow.net/questions/93218 | 1 | As we know, to prove the convergence of stochastic process, we could either show the convergence of finite dimensional distribution and tightness of the process, or use techniques of martingale problems. What about the following Markov process:
$L=\frac{1}{2}p(1-p)\frac{d^{2}}{dp^{2}}-\frac{\theta}{2}p\frac{d}{dp}+\l... | https://mathoverflow.net/users/18420 | Convergence of stochastic process | This drifted Wright--Fisher diffusion seems to converge to the rather degenerate process $X\_0 = p$, $X\_t = 1/2$ for $t>0$, which is why I would do it by hand: Show that for every $t>0$ and every $p\in(0,1)$, the process started from $p$ is with high probably near $1/2$ at time $t$, uniformly for $p\in[\epsilon,1-\eps... | 3 | https://mathoverflow.net/users/18032 | 93222 | 54,779 |
https://mathoverflow.net/questions/93002 | 19 | Let $S\_x$ be the set of finite sums of prime numbers $\geq x$. In other words, let $S\_x$ be the submonoid of $(\mathbf{Z}\_{\geq 0},+)$ generated by the set $\mathcal{P}\_{\geq x}$ of prime numbers $\geq x$.
It is easy to see that $S\_x$ contains every sufficiently large integer. This follows from the classical fac... | https://mathoverflow.net/users/6506 | Finite sums of prime numbers $\geq x$ | According to "The three primes theorem with almost equal summands" by Baker and Harman, every large odd $N$ is a sum of three primes each of size $\sim N/3$. (In fact, within $N^{4/7}$ of $N/3$-- this is much closer than we need, so we could use weaker results of earlier authors, if preferred.)
For odd $N$, this giv... | 4 | https://mathoverflow.net/users/22368 | 93235 | 54,784 |
https://mathoverflow.net/questions/93123 | 3 | Hello,
in G. Mikhalkin's Papaer "DECOMPOSITION INTO PAIRS-OF-PANTS FOR
COMPLEX ALGEBRAIC HYPERSURFACES":
<http://arxiv.org/pdf/math/0205011.pdf>
There is a lemma about the relation between intersection of a hypersurface with the boundary divisors in a toric variety and the truncation of the defining polynomial of a hyp... | https://mathoverflow.net/users/14105 | Hypersurfaces in Toric Varieties, Help understand a proof from Mikhalkin's paper | Let's consider an example. Triangle $(0,0),(0,d),(d,0)\ $ gives us $\mathbb CP^2$ and each integer point $\{(k,l) |k,l\geq 0, k+l\leq d\}\ $ corresponds to monomial $x^ky^l$ (or $x^ky^lz^{d-k-l}$ in projective coordinates). There is the same situation for any toric variety - ring of function is generated by monomials c... | 3 | https://mathoverflow.net/users/4298 | 93252 | 54,794 |
https://mathoverflow.net/questions/93257 | 7 | Is there a function $f$ such that for any presentation $$G=\langle x\_1,\ldots,x\_n \mid r\_1,\ldots,r\_k\rangle\quad \text{with}\quad |r\_i|\leq 3$$
$k\leq f(n)$ implies that $G$ has non-abelian free subgroups.
Of course $f=0$ works trivially, I am asking for bigger functions.
| https://mathoverflow.net/users/7307 | Number of relations and free subgroups | $f(n)=n-2$ works as I said in the comment above. On the other hand, $f(n) < n-1$ for $n\ge 4$. Indeed, the metabelian Baumslag-Solitar group can be given by relations $\langle x\_1,...,x\_{n-1},u,v,t\mid x\_1=x\_2, x\_2=x\_3, ..., x\_{n-2}=x\_{n-1}, tx\_1=u, ut^{-1}=v, x\_1^2=v\rangle $ with $n+2$ generators and $n+1$ ... | 10 | https://mathoverflow.net/users/nan | 93258 | 54,797 |
https://mathoverflow.net/questions/93243 | 4 | Does there exist a totally-ordered-without-endpoints proper class $L$ such that every closed interval in $L$ does have the order type of a closed interval in the Conway's surreal numbers, but $L$ as a whole does not have the order type of Conway's surreal numbers?
In case it helps, the thinking that lead up to my pos... | https://mathoverflow.net/users/10909 | Does there exist a surreal long line? | There are many such structures. For example, simply form L = N x [0,1) in the dictionary order, with the least element deleted, where N is the set of natural numbers including 0, and [0,1) is the semi-open interval of surreals. Since No (i.e. the ordered class of surreals) does not have a cofinal subset and L clearly d... | 7 | https://mathoverflow.net/users/18939 | 93260 | 54,798 |
https://mathoverflow.net/questions/93250 | 12 | Consider [Schur polynomials](http://en.wikipedia.org/wiki/Schur_polynomial) - symmetric polynoms of variables x\_1, ... x\_n, indexed by $d\_1 \ge d\_2 \ge ... \ge d\_n$.
I wonder what is known about the differential operators which have them as eigenfunctions ?
First, I am not sure such operators exist, but almost ... | https://mathoverflow.net/users/10446 | What differential operators have Schur polynoms as eigenfunctions ? Can this be deformed to trig. Calogero and Jack polynoms ? | Schur polynomials are defined to be the characters of irreducible representations of $G=GL\_n$. By the Weyl character formula,
$$s\_\lambda=\frac{1}{\delta}\sum\_{w\in W}(-1)^{l(w)}e^{w(\lambda+\rho)},$$
where $\delta$ is the Vandermonde determinant.
Clearly, the numerator is an eigenfunction of the Laplace operator ... | 13 | https://mathoverflow.net/users/18512 | 93270 | 54,803 |
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