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/127898 | 5 | It is standard that every Borel probability measure on a polish space $X$ can be obtained as pushforward of the uniform measure $\lambda$ on $[0,1]$ along an almost-everywhere-defined Borel-measurable function $d: [0,1] \to X$ . (In fact, $d$ can always be taken to be continuous on a measure-$1$ $G\_\delta$ subset. But... | https://mathoverflow.net/users/13506 | Obtaining conditional probabilities as pushforwards of [0,1] | It follows from a classification of morphisms in this category due to Rokhlin. If both the target space and all the conditional measures are purely non-atomic, then this map is (mod 0) just the coordinate projection of the unit square (endowed with the Lebesgue measure). If any of these measures has atoms, then it is e... | 6 | https://mathoverflow.net/users/8588 | 127900 | 71,237 |
https://mathoverflow.net/questions/127427 | 0 | Hi, maybe this is a stupid question, however none of my mathematicians colleagues could answer it properly. It's know that a finitely generated projective $A$-module $M$ is free if $A$ is a local ring. Now, consider an arbitrary finitely generated $A$-module $M$. If you pick all the possibles localizations of $M$, you ... | https://mathoverflow.net/users/17868 | Gluing free modules to get a finitely generated free module | I am assuming your ring is commutative. First of all, there is currently an error in your question, because you neither assume $M$ to be locally free, not projective. Your questions reads: "Now, consider an arbitrary finitely generated $A$-module $M$." My guess is you inadvertedly deleted this assumption when you edite... | 1 | https://mathoverflow.net/users/16046 | 127901 | 71,238 |
https://mathoverflow.net/questions/127918 | 3 | Given a compact subset $A$ of a Banach lattice $E$, is the following true?
There exist $u,v\in E$ so that $u\leq a\leq v$ for all $a\in A$.
This is true in case $E=C(X)$, $X$ compact, with the uniform norm. One way of disproving this conjecture is to construct a norm convergent sequence in $E$ which is not order b... | https://mathoverflow.net/users/33212 | Are compact sets in a Banach lattice order bounded? |
>
> One way of disproving this conjecture is to construct a norm convergent sequence in E which is not order bounded.
>
>
>
E.g. the sequence $(\frac{1}{n}e\_n)$ in $l^1$.
| 8 | https://mathoverflow.net/users/23141 | 127921 | 71,244 |
https://mathoverflow.net/questions/127877 | 10 | A metric space is *doubling* if any ball of radius $2R$ can be covered by $N$ balls of radius $R$ and $N$ is fixed once forever.
>
> Is there an example of complete length-metric space which is doubling, but the [Besicovitch covering theorem](http://en.wikipedia.org/wiki/Besicovitch_covering_theorem) does not hold?... | https://mathoverflow.net/users/26648 | Doubling space without Besicovitch covering theorem? | The Besicovitch covering theorem fails for example in the Heisenberg group, see
[ E. Sawyer and R. L. Wheeden,
*Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces*,
Amer. J. Math. **114** (1992), no. 4, 813–874. <http://www.jstor.org/stable/2374799> ]
*The following was more of a co... | 10 | https://mathoverflow.net/users/11716 | 127928 | 71,246 |
https://mathoverflow.net/questions/127720 | 4 | Let $C(\mathbb{R};{U}(n))$ denote the topological group of continuous functions $\mathbb{R}\to {U}(n)$ with pointwise multiplication and compact-open topology. My question is:
**Are these groups isomorphic for different values of $n$?**
I suspect the answer is no (it feels like it should be obvious), but proving th... | https://mathoverflow.net/users/10779 | Are the groups $C( \mathbb{R} ; U(n) )$ isomorphic? | How about this argument? If I remember correctly, the irreducible representations of $U(n)$ are either 1-dimensional or at least $n$-dimensional. Suppose that there was an isomorphism $\phi\colon C(\mathbb{R}; U(m)) \rightarrow C(\mathbb{R}; U(n))$ for $n < m$. We have the embedding $i\colon U(m) \rightarrow C(\mathbb{... | 6 | https://mathoverflow.net/users/9942 | 127929 | 71,247 |
https://mathoverflow.net/questions/127917 | 0 | **Dear All**, by the paper of Carter and Fong, we know the structure of 2-sylow subgroups of $GL(n,q)$. Let $W\_{r-1}=Z\_2\wr Z\_2\wr...Z\_2$ (r-1 times), and $W$ is a 2-sylow subgroup of $GL(2,q)$, then $W\_r=W \wr W\_{r-1}$ is a 2-sylow subgroup of $GL(2^r,q)$.If $n=2^{r\_1}+2^{r\_2}+...2^{r\_k}$, then $P=W\_{r\_1} \... | https://mathoverflow.net/users/33209 | 2-sylow subgroups | Derek's comment answers this question, however maybe I can add a little detail for the one aspect that is slightly tricky.
It should be pretty clear how to turn a Sylow 2-subgroup of $S\_{2^{r-1}}$ into a set of $2^r\times 2^r$-block matrices with blocks of size $2$. Which means that the only (potentially) tricky thi... | 3 | https://mathoverflow.net/users/801 | 127940 | 71,253 |
https://mathoverflow.net/questions/127939 | 7 | This is probably standard for group-theorists:
Let $G$ be a finite group. Is it true that the intersection of all normalizers of subgroups equals the center?
If so, where do I find a proof? What about the same question for infinite groups?
The original question can be reformulated as follows: Let $G$ be a finite grou... | https://mathoverflow.net/users/nan | Intersection of all normalizers | No. There are non-abelian groups $G$ for which all subgroups are normal, such as the quaternion group of order 8. So the intersection of all normalizers is just $G$.
| 15 | https://mathoverflow.net/users/22989 | 127943 | 71,254 |
https://mathoverflow.net/questions/127124 | 2 | I am trying to understand some basic facts about (almost) contact (metric) structure, especially on 3-manifolds 5-manifolds.
(1) I saw statement that "any compact oriented 3-manifold admit contact structure" by J. Martinet, and also statement by Geiges in his book that "the obstruction of almost contact structure is ... | https://mathoverflow.net/users/15884 | the existence of (almost) contact (metric) structure | I will respond under the impression that you're primarily interested in contact 3-manifolds. Much more is known in this low-dimensional case as techniques such as Dehn surgery etc. can be adapted (with some slightly non-trivial work) to work for contact manifolds in this dimension.
1) One easy way to construct contac... | 0 | https://mathoverflow.net/users/31530 | 127947 | 71,256 |
https://mathoverflow.net/questions/127920 | 0 | What is the proof for: 'An all pass transfer function/inner function can be represented by a Blaschke Product' ?
| https://mathoverflow.net/users/31955 | Representation of all pass transfer functions/inner functions as Blaschke product. | Atkinson, Discrete and continuous boundary problems, page 8.
| 1 | https://mathoverflow.net/users/25510 | 127951 | 71,258 |
https://mathoverflow.net/questions/127949 | 4 | The question is in the title, but let me specify what I mean by the category of graphs.
In the context of this question, the category of graphs is the category of symmetric *irreflexive* relations. That means, not the category of symmetric *reflexive* relations.
That means, no loops and at most one edge between verti... | https://mathoverflow.net/users/4814 | Is there any nontrivial monad on the category of graphs? | I take it morphisms $f: X \to Y$ are by definition functions that preserve the relation: if $x, x'$ are related in $X$, then $f(x), f(x')$ are related in $Y$.
It's easy to manufacture some silly examples of nontrivial monads, by finding suitable monoidal products on the category of graphs and then finding monoids wi... | 5 | https://mathoverflow.net/users/2926 | 127952 | 71,259 |
https://mathoverflow.net/questions/127961 | 5 | I have a question about primitive recursive functions. Maybe it's trivial, if it is I will move it into math.stackexchange.
Is there a primitive recursive function $f$ which is a bijection of $N$ onto $N$ such that $f^{-1}$ is **not** primitive recursive ?
| https://mathoverflow.net/users/32657 | A question about primitive recursive functions | The answer is yes. First, let $g$ be a total computable function whose rate of growth is too fast for it to be primitive recursive, such as the diagonal Ackermann function. Now, define $f(k)=2n$, if $k$ is the number coding up (in some canonical way) the computation of $g(n)$. That is, $k$ should encode a list of the e... | 10 | https://mathoverflow.net/users/1946 | 127964 | 71,265 |
https://mathoverflow.net/questions/127769 | 13 | I asked [this question on SE](https://math.stackexchange.com/questions/201705/applications-of-govorov-lazard-theorem) a long time ago, but never received an answer:
The Govorov-Lazard Theorem states that a (left) module over an unital ring is flat iff it is a direct limit of finitely generated free (left) modules.
... | https://mathoverflow.net/users/10194 | Applications of Govorov-Lazard Theorem? | One application which I find particularly beautiful is the following:
>
> Theorem: If $R$ is countable then any flat left $R$-module $M$ has projective dimension at most 1
>
>
>
This appeared first in
Jensen, 1966, On homological dimensions of rings with countably generated ideals
It has been used more re... | 6 | https://mathoverflow.net/users/11540 | 127978 | 71,272 |
https://mathoverflow.net/questions/127977 | 2 | We know that the grand Riemann hypothesis is a generalisation of the Riemann hypothesis and Generalized Riemann hypothesis. My question is about the existence of a similar generalisation of the Birch and Swinnerton-Dyer conjecture.
| https://mathoverflow.net/users/25947 | A generalisation of the Birch and Swinnerton-Dyer conjecture | The equivariant Tamagawa number conjecture generalizes the BSD conjecture. I am not sure if this is the most general conjecture available.
| 3 | https://mathoverflow.net/users/10400 | 127981 | 71,273 |
https://mathoverflow.net/questions/127821 | 6 | **Background and definitions**
Consider a random graph on $n$ vertices with a nicely behaved degree sequence. That is, letting $d\_i(n)$ denote the number of vertices of degree $i$, suppose that for all $i$, there exists a constant $\lambda\_i$ such that $d\_i(n)/n \to \lambda\_i$ as $n \to \infty$.
Let $G$ be a ra... | https://mathoverflow.net/users/22055 | The structure of small components in random graphs with a given degree sequence | I think the answer to your question is: It is "non-trivially" true that most small components are trees for all degree sequences where the giant component is not the whole graph.
The answer comes from the two papers you provided. Theorem 2 of the second paper gives what they call a "Discrete Duality Principle" which... | 3 | https://mathoverflow.net/users/839 | 127984 | 71,274 |
https://mathoverflow.net/questions/127982 | 1 | Good afternoon,
Take a submanifold $V$ of codimension $1$ of the sphere at infinity of $\mathbb{H}^n$ which is not the sphere at infinity of a totally geodesic hyperplane $\mathbb{H}^{n-1} \subset \mathbb{H}^n$. Now suppose that $f$ is an isometry of $\mathbb{H}^n$ fixing $V$ pointwise. Is true that $f$ must be the i... | https://mathoverflow.net/users/25511 | Fixed submanifolds of the sphere at infinity of $\mathbb{H}^n$ | Yes: you can easily see that $f$ must fix some point $x$ in the convex hull of the set $V$ (for exemple the center of an ideal triangle with vertices in $V$), and thus it fixes pointwise every geodesic from $x$ to a point of $V$; by your hypothesis on $V$ the union of all these geodesics is not contained in a proper to... | 4 | https://mathoverflow.net/users/32210 | 127990 | 71,277 |
https://mathoverflow.net/questions/127550 | 4 | Hejhal's algorithm [1] was a little gadget invented in the 90's for calculating the Hecke eigenvalues and Fourier coefficients of Maass wave forms. Later, Booker, Strombergsson, and Venkatesh (BSV) [2] took Hejhal's paper and made it more viable to perform high-precision calculations.
The basic idea is to assume the ... | https://mathoverflow.net/users/13151 | Hejhal's algorithm and computational methods for non-classical Maass wave forms | I don't think it is practical to directly find higher rank Maass forms along the lines of what Hejhal did, because they are functions of several variables and their Fourier expansions involve multiple sums. Not to mention the need to implement the appropriate special functions that appear. Even if you have the Fourier ... | 2 | https://mathoverflow.net/users/19964 | 127996 | 71,278 |
https://mathoverflow.net/questions/127906 | 7 | I think it is fair to say that the fields of Operator Algebras, Operator Theory, and Banach Algebras rely on **Gelfand representation and functional calculus** in a crucial way.
>
> I am curious about applications of these techniques beyond Functional Analysis. Do you know any?
>
>
>
I am aware of [this thread... | https://mathoverflow.net/users/35324 | Gelfand representation and functional calculus applications beyond Functional Analysis | There are some examples in ergodic theory.
Furstenberg's correspondence principle basically relays on Gelfand's representations.
Another example - the functional calculus gives a very easy proof of Von-Neumann's ergodic theorem.
Also, the spectral theorem gives you the existence of spectral measure for a measure ... | 5 | https://mathoverflow.net/users/8857 | 127999 | 71,281 |
https://mathoverflow.net/questions/128003 | 25 | Hi all. The question I have should be a rather simple one, but I just can't think it through.
So the Chern-Simons action reads
\begin{equation}
S = \int\_M {\rm tr} (A\wedge dA + \frac{2}{3} A\wedge A \wedge A)
\end{equation}
where $M$ is 3-fold, and similarly for higher dimensional manifold.
Now, my question is:
... | https://mathoverflow.net/users/15884 | How to understand Chern-Simons action | Often in the literature by "Chern-Simons theory" is meant by default $G$-[Chern-Simons theory](http://ncatlab.org/nlab/show/Chern-Simons%20theory) whose gauge group is a connected and simply connected semisimple compact group $G$, such as $G = SU$. In this case it so happens that all $G$-principal bundles on a 3-manifo... | 22 | https://mathoverflow.net/users/381 | 128007 | 71,283 |
https://mathoverflow.net/questions/128016 | 11 | Here is a basic, though very important, example:
Hilbert takes as primary the notion of “congruence” (or “equal”) between segments. His first axiom of congruence “requires the possibility of constructing a segment congruent to an assigned segment”. His second axiom reads: "if two segments are congruent to a third one t... | https://mathoverflow.net/users/29316 | Why do mathematicians prefer one definition over the other when they both define the same concept? | To give one answer to the question in the title: A reason to prefer one way of defining the same 'idea' over another is that it *generalizes better*. (Where of course what better means can depend.)
I have nothing to say about equivalence relations but since also other examples are asked for:
The notion 'prime numbe... | 19 | https://mathoverflow.net/users/nan | 128030 | 71,294 |
https://mathoverflow.net/questions/128029 | 1 | I'm trying to decompose the Kazhdan-Lusztig C' basis element associated to the longest word in $S\_n$, $C'\_{w\_0}$ into products and sums of elements $C'\_w$ where $w < w\_0$ in the Bruhat order. For example I know that $C'\_{s\_1s\_2s\_1}=C'\_{s\_1}C'\_{s\_2}C'\_{s\_1}-C'\_{s\_1}$.
The question I have is the follo... | https://mathoverflow.net/users/31116 | Decomposition of C' Kazhdan-Lusztig basis element associated to longest word in S_n | Is the expansion
$$C'\_w = C'\_{ws}C'\_s - \sum\_{z\leq ws;\, zs < z} \mu(z,ws) C'\_z$$
for $ws < w$ what you're looking for? ($s$ is a simple reflection.) Here, $\mu(z,ws)$ is the coefficient of $q^{(\ell(ws)-\ell(z)-1)/2}$ in the Kazhdan-Lusztig polynomial $P\_{z,ws}(q)$. Perhaps there is a simplification for the sp... | 1 | https://mathoverflow.net/users/32309 | 128032 | 71,295 |
https://mathoverflow.net/questions/128033 | 3 | Let $A, B$ be positive definite matrices. Then $A^r\circ B^r \le (A\circ B)^r$ for $0\le r\le 1$, where $\circ$ is Schur product. Here the inequality is in the sense of Loewner partial order.
How to prove this? Where can I find a reference?
| https://mathoverflow.net/users/24492 | Schur product, partial order | The proof follows from the following results:
>
> **Theorem.** (Thm. 1.6, in [2]) If $\Phi$ is a unital positive linear map from $\mathbb{M}\_m \to \mathbb{M}\_n$, and $f$ is an operator monotone function on $[0, \infty)$, then for every $A \ge 0$,
> \begin{equation\*}
> \Phi(f(A)) \le f(\Phi(A)).
> \end{equation... | 6 | https://mathoverflow.net/users/8430 | 128034 | 71,296 |
https://mathoverflow.net/questions/128028 | 13 | Let F be a quasi-coherent sheaf on a smooth projective variety X over an algebraically closed field and $D\_X(?)=RHom(?,\omega \_X[n])$ the dualizing functor. Is it the case that $D\_X(D\_X(F))$ is a sheaf? (i.e. $\mathcal H^n(D\_X(D\_X(F)))=0$ for $n\ne 0$). If so is there a convenient reference?
| https://mathoverflow.net/users/19369 | Is the derived double dual of a quasi-coherent sheaf a sheaf? | For coherent sheaves, by using resolutions, one checks that the assertion in question is true: for any perfect complex $K$, one even has $D\_X^2(K) \simeq K$. However, for larger quasicoherent sheaves $F$, the value $D\_X^2(F)$ fails to even be a sheaf; I give an example below in Corollary 5.
Fix a countable set $I$,... | 16 | https://mathoverflow.net/users/25792 | 128046 | 71,303 |
https://mathoverflow.net/questions/128039 | 6 | Does there exist a non trivial homomorphism from Thompson's group T to a linear group?
| https://mathoverflow.net/users/33071 | Thompson's group T | No: T is infinite, finitely presented and simple. Fg linear groups are residually finite, by Mal'cev's theorem. QED.
| 12 | https://mathoverflow.net/users/1463 | 128049 | 71,306 |
https://mathoverflow.net/questions/127937 | 4 | Please consider two processes:
Process 1 - I simulate random sequential adsorption of discs on the unit square in the continuum limit, randomly selecting real number coordinates and rejecting the point if placing a disc at this point leads to any disc-disc overlaps. I continue this process until I reach a disc surfac... | https://mathoverflow.net/users/33218 | Simulating random sequential adsorption in reverse | These distributions on sets of discs with density $P$ will differ.
Here is a big simplification: Instead of a surface, let's pack line segments on $\mathbb{R/Z}$. Let's place $3$ segments of length $1/10$, and then delete one, and consider the distribution on the distances between the intervals. Without loss of gener... | 1 | https://mathoverflow.net/users/2954 | 128052 | 71,308 |
https://mathoverflow.net/questions/128044 | 3 | M. Atiyah in "VECTOR BUNDLES OVER AN ELLIPTIC CURVE" defined ample line bundle $E$ on $X$ as satisfying the following conditions:
1. Canonical map $H^0(X, E)\to E\_x$ is surjective for any $x\in X$.
2. $H^q(X,E)=0$ for $q>0$.
But in standard textbooks like Hartshorne there is another definition:
$E^{\otimes n}$ i... | https://mathoverflow.net/users/11072 | Equivalent definitions of ample bundles | Even a very ample line bundle does not need to be ample in the sense of the first definition (consider $\mathcal O(1)$ on a smooth plane curve of degree $>4$ or, more generally, on any smooth hypersurface of high enough degree). I do not know whether ampleness in the sense of the 1st definition implies ampleness in the... | 2 | https://mathoverflow.net/users/29992 | 128055 | 71,309 |
https://mathoverflow.net/questions/128061 | 0 | Say that the variance of a constant mean scalar stochastic process can take finite number of values. The problem is to detect the the point of break in variance as observation data comes in.
I tried the following: (1)Compute sample variance going forward (Call it Vf(n)), starting from the first data point, (2)Compute... | https://mathoverflow.net/users/31955 | On Variance Break detection | This type of problem is called change-point analysis. A lot of work has been done on it (and it's more common to look for changes in the mean, rather than changes in the variance), and there are many packages you can use for it in R, [Matlab](http://www.mathworks.com/help/wavelet/ref/wvarchg.html), etc.
| 0 | https://mathoverflow.net/users/2954 | 128063 | 71,314 |
https://mathoverflow.net/questions/127316 | 6 | Let $A/K$ be an abelian variety defined over a nonarchimedean local field $K$ of characteristic $0$ and let $L$ be a finite extension of $K$. Consider the norm map $$A(L)\xrightarrow{N\_{L/K}}A(K)$$ I want to know if the following statement is true -
If $A\_{0}(K)$ is the subgroup of $A(K)$ specializing to the conne... | https://mathoverflow.net/users/13628 | Local Norm Mapping for Abelian Varieties | One doesn't need Tate local duality to analyze the good reduction case, and in general the answer is affirmative.
First, let's review the general nonsense for norm maps with commutative group functors (to sidestep representability issues). For any finite etale map of schemes $f:S' \rightarrow S$ and any commutative f... | 2 | https://mathoverflow.net/users/30180 | 128066 | 71,317 |
https://mathoverflow.net/questions/128056 | 6 | Let $A$ be a finite set of integers with $|A \hat{+} A| \leq K|A|$, where the $\hat{+}$ denotes restricted sumset: the set of all $a\_1 + a\_2$ with $a\_1, a\_2 \in A$ and $a\_1 \neq a\_2$.
Claim: $|A + A| \leq (K + o(1))|A|$, where $o(1)$ denotes a quantity tending to 0 as $|A|$ tends to $\infty$.
Sketch proof: L... | https://mathoverflow.net/users/5575 | Additive Combinatorics - reference request | For a somewhat similar argument, see Proposition 2.5 from Alon's 1987 paper ``Subset sums", available [here](http://www.tau.ac.il/~nogaa/PDFS/Publications/Subset%20sums.pdf).
| 7 | https://mathoverflow.net/users/9924 | 128068 | 71,319 |
https://mathoverflow.net/questions/126419 | 11 | Hebbey defines the Sobolev space of functions on a Riemannian manifold (M,g) as the completion of smooth functions under the Sobolev norm. However, I have seen (elsewhere) that Sobolev spaces have been defined as the collection of locally integrable functions whose weak derivatives (w.r.t to the Levi-Civita connection)... | https://mathoverflow.net/users/3709 | Density of smooth functions in Sobolev spaces on manifolds | According to pages 14 and 15 of:
* MR2343536 Reviewed Eichhorn, Jürgen Global analysis on open manifolds. Nova Science Publishers, Inc., New York, 2007. x+644 pp. ISBN: 978-1-60021-563-6; 1-60021-563-7 (Reviewer: Yuri A. Kordyukov)
The poof is given in:
* MR1066741 Reviewed Eichhorn, Jürgen Elliptic differential ... | 7 | https://mathoverflow.net/users/26935 | 128081 | 71,324 |
https://mathoverflow.net/questions/127699 | 10 | Suppose $G$ has a finite index subgroup $N$ such that $N$ acts properly and cocompactly on a CAT(0)-cube complex. Does $G$ also act properly and cocompactly on a CAT(0)-cube complex?
Edit: After searching the web a little I found that the answer to this question is no, in general. There exists for example a 3-dimensi... | https://mathoverflow.net/users/31670 | Are virtual cubulated groups cubulated? | The answer is 'yes' when your group $G$ is word-hyperbolic. This can be deduced from Sageev's theorem. I'll explain this here, but a good reference is Hruska--Wise's paper 'Finiteness properties of cubulated groups' (arXiv:1209.1074v2).
The first step is to notice that $G$ admits a proper, but not cocompact, action o... | 9 | https://mathoverflow.net/users/1463 | 128084 | 71,326 |
https://mathoverflow.net/questions/127866 | 4 | Let $\Omega$ be a bounded domain in $\mathbb{C}$. Let $X$ be a discrete set of points whose boundary is in the boundary of $\Omega$. Can I find an $L^2$ holomorphic function which vanishes on $X$? Can I solve the problem in weighted $L^2$ spaces?
If there are counterexamples, are precise conditions on the set $X$ kno... | https://mathoverflow.net/users/27968 | Weierstrass factorization with $L^2$ estimates? | The right place to start is the seminal work of Seip,
Kristian Seip, Beurling type density theorems in the unit disk, Invent. math.
1993, Vol 113, 1, pp 21-39 (look at the last sections). You will see that much is known in the case of the disc, but I fairly doubt that a complete characterization is known for arbitrary... | 1 | https://mathoverflow.net/users/31537 | 128087 | 71,328 |
https://mathoverflow.net/questions/128014 | 6 | Is there an elementary proof of this identity?
$$n + 1 - \sum\_{k=1}^{n} k^{k-1} \binom{n}{k} \frac{(n-k)^{n+1-k}}{n^{n}} =1 + \sum\_{k=1}^n \frac{n!}{(n-k)!n^k}\;?$$
The term on the right is the average time to find a duplicate birthday from the classic Birthday Problem and the sum on the left is a special case of... | https://mathoverflow.net/users/45564 | Elementary proof for identity involving sums of binomials | After canceling $1$'s and clearing denominators, the identity can be rearranged to this one:
$$n^{n+1} = \sum\_{k=1}^{n} \binom{n}{k} k^{k-1} (n-k)^{n-k+1} + \sum\_{k=1}^n \binom{n}{k} n^{n-k} k!$$
and now we proceed to give a bijective proof. The left side counts data of the form
* Endofunction $f: S \to S$ o... | 16 | https://mathoverflow.net/users/2926 | 128094 | 71,333 |
https://mathoverflow.net/questions/128099 | 0 | For a semigroup $S$ and a congruence $\rho$ on $S$, let's say that $\rho$ is good when for all $a,b\in S$ we have that $[ab]=[a][b],$ where $[x]$ denotes the congruence class of $x$ modulo $\rho$ and the product on the right-hand side is the product of sets: $AB=\lbrace ab\,|\,a\in A,\ b\in B\rbrace$.
What are the se... | https://mathoverflow.net/users/20803 | What are the semigroups in which congruence classes can be multplied like sets? | These semigroups are called perfect. They were studied in:
Fortunatov, V.A.
Perfect semigroups. (Russian)
Izv. Vyssh. Uchebn. Zaved., Mat. 1972, No.3(118), 80-89.
The class of perfect semigroups is closed under homomorphisms and includes all completely (0)-simple semigroups.
| 4 | https://mathoverflow.net/users/18814 | 128101 | 71,336 |
https://mathoverflow.net/questions/128004 | 0 | Let a constant $B \ge 1$, and let $l\_1 = 0$, $b\_1 = 0$ be the values of $l$ and $b$ (respectively) at time $t = 1$.
Let $l\_{t+1} = l\_t + 1$ if $b\_i < B$, and $l\_{t+1} = l\_t$ otherwise
Let $b\_{t+1} = b\_t + \frac{l\_{t+1}}{t}$
So here I can intuitively see that $\forall t \le B: l\_t = b\_t = t$, and that ... | https://mathoverflow.net/users/21685 | Giving a general term of a recursive function, and upper bound for it | There's no general upper bound.
Suppose $p\_t<1$ for every $t$ and $\sum\_{t=1}^\infty p\_t = \infty$.
For every $N$ there's a positive probability that $l\_N = 0$, then $l\_t$ will be larger than $NB$ eventually.
You can get a very basic probabalistic bound using Markov's inequality
First let $\tilde l\_t$ and... | 1 | https://mathoverflow.net/users/32372 | 128104 | 71,338 |
https://mathoverflow.net/questions/128100 | 3 | I am looking for classical and elementary reason that why non-singular cubics are not rational?
| https://mathoverflow.net/users/nan | non-singular cubics are not rational | As there is no precision about the dimension or about the field, I will try to give a description in some cases. There are plenty of other nice results about cubics.
1) Dimension 1: A smooth non-singular projective cubic curve is not rational. This is because the arithmetic genus, which is $g=(d-1)(d-2)/2=1$ is not e... | 20 | https://mathoverflow.net/users/23758 | 128105 | 71,339 |
https://mathoverflow.net/questions/127979 | 3 | Let $n\geq2$. We assume $0<\alpha\_n<\cdots<\alpha\_2<\alpha\_1<1$ and $0<\beta\_n<\cdots<\beta\_2<\beta\_1<1$
, $\alpha\_n=\beta\_n$, and there exists $1\leq j\_0\leq n$ such that $\alpha\_{j\_0}\neq \beta\_{j\_0}$.
My question: is there a complex number $s$ such that
$\sum\_{j=1}^n s^{\alpha\_j}=0$ and $\sum\_{j=... | https://mathoverflow.net/users/33232 | A question from complex analysis | The answer is no. Take the polynomials
\begin{align}
f(x) &= x^6 + x^5 + x^4 + x^3 + x^2 + x\\\
g(x) &= x^8 + x^6 + x^5 + x^4 + x^3 + x.
\end{align}
From
\begin{align}
f(x) &= x(x+1)(x^2+x+1)(x^2-x+1)\\\
g(x) &= x(x+1)(x^2+x+1)(x^2-x+1)^2
\end{align}
wee see that $f(x)$ and $g(x)$ have the same complex roots. Upon sett... | 4 | https://mathoverflow.net/users/18739 | 128106 | 71,340 |
https://mathoverflow.net/questions/128110 | 5 | I would like to ask for possible references for the following very general situation, a categorified version of Mackey functors.
The question is if there are other known constructions to associate to any subgroup $H$ of $G$ a category $C(H)$ and for any $H\leq K \leq G$ pairs of adjoint functors $Ind\_H^K:C(H)\righta... | https://mathoverflow.net/users/33263 | Categorified versions of Mackey's functor | Have a look at page 4 of <http://www.math.umn.edu/~webb/Publications/GuideToMF.ps> for a few examples of Mackey functors in different categories.
| 4 | https://mathoverflow.net/users/10400 | 128111 | 71,343 |
https://mathoverflow.net/questions/128102 | 2 | Let:
-- $x\_1,\ldots,x\_n$ be $n$ distinct points on the complex plane $\mathbb C$.
-- $r\_1,\ldots,r\_n$ be $n$ real numbers.
Consider the map
$$ z\mapsto u(z)=\int^z \frac{1}{(x-x\_1)^{r\_1}\cdots (x-x\_n)^{r\_n}} d\xi $$
It defines a multivalued holomorphic function on $\mathbb C\setminus (x\_1,\ldots,x\... | https://mathoverflow.net/users/33261 | Monodromy of "complex Schwarz-Christoffel maps | Monodromy is affine in the general case. It is generated by finitely many elliptic transformations of the form $az+b$. Proof: your function satisfies the differential
equation $u^{\prime\prime}=Ru',$ where $R$ is rational. The general solution of this equation
is obtained from a particular solution by the formula $au+b... | 0 | https://mathoverflow.net/users/25510 | 128118 | 71,346 |
https://mathoverflow.net/questions/128117 | 2 | I have a somewhat technical question about conjugacy in
quasi-reductive groups.
Let $k$ be a field (in my main case interest, $k$ is finite), $G$ be a connected
quasi-split reductive group over $k$. As is customary, let $S$ be a maximal $k$-split torus in $G$, $T$
its centralizer in $G$ (which is a maximal $k$-torus... | https://mathoverflow.net/users/9317 | a conjugacy question in quasi-split reductive groups | Yes. Consider the map sending $b \in B$ to $bdb^{-1}d^{-1}$. The fibers are cosets of $T$, the centralized of $d$, so the image is isomorphic to $U$. The image is also clearly contained in $U$, so by Ax-Grothendieck, or, in the finite field case, counting, the image is all of $U$.
| 4 | https://mathoverflow.net/users/18060 | 128122 | 71,348 |
https://mathoverflow.net/questions/128079 | 4 | Is there anything known/proved/conjectured about the distribution of:
$$B(n) = \frac{(p\_n-1)}{2} \bmod 2, \qquad p\_n \mbox{ is the } n\mbox{-th prime}$$
i.e. the bit 1 of the binary representation of the $n$-th prime number?
| https://mathoverflow.net/users/12875 | A "bit" of primes | First, I strongly second the recommendation of Gjergji Zaimi to read the paper by Granville and Martin; here is a [link to the arXiv version](http://arxiv.org/abs/math/0408319) in addition.
Some initial and partial information:
On a very rough scale the frequency counts of primes with "bit 1" equal to $0$ and $1$,... | 9 | https://mathoverflow.net/users/nan | 128125 | 71,349 |
https://mathoverflow.net/questions/128113 | 9 | A unimodular matrix $M$ is a square integer matrix having determinant $+1$ or $−1$.
A totally unimodular matrix (TU matrix) is a matrix for which every square non-singular submatrix is unimodular. A totally unimodular matrix need not be square itself. Obviously, any totally unimodular matrix has only $0$, $+1$ or $−1$ ... | https://mathoverflow.net/users/33264 | Inverse of a totally unimodular matrix | The answer is yes, because if $B=A^{-1}$, then we have an equality between minors:
$$B(I,J)=\pm\frac{A(J^c,I^c)}{\det A},$$
for every subsets $I,J\subset[[1,n]]$ of same cardinals. This formula generalizes that giving the entries of $A^{-1}$ in terms of minors of $A$. The $\pm$ sign is not essential to prove the stabil... | 14 | https://mathoverflow.net/users/8799 | 128139 | 71,359 |
https://mathoverflow.net/questions/128152 | 17 | A virtual currency called *bitcoins* has been in the news recently. It is said that in order to "mine" bitcoins, you have to solve hard mathematical problems.
Now, there are two kinds of mathematical problems. The difference is best explained by the following beautiful quotation from Langlands :
>
> [T]here is an... | https://mathoverflow.net/users/2821 | Which hard mathematical problems do you have to solve to earn bitcoins ? | Bitcoin mining is based on hash functions. Specifically the SHA-256 hash function, which maps arbitrary bit strings to 256-bit outputs in such a way that nobody knows how to find a collision (two inputs with the same output), although the pigeonhole principle implies collisions exist. Bitcoin mining doesn't involve fin... | 30 | https://mathoverflow.net/users/4720 | 128154 | 71,367 |
https://mathoverflow.net/questions/128162 | 1 | I was wondering if there are any results that studied the growth of $\left|\frac{1}{\Gamma(s)}\right|$ where $0 < \Re(s) < 1$ and as $\Im(s) \to \infty$? Any pointers to any results, papers, references will be highly appreciated.
Thanks.
| https://mathoverflow.net/users/2865 | Growth of the reciprocal gamma function in the critical strip | According to Gradshteyn-Ryzhik: Tables of integrals... 8.328.1, for fixed real $x$ and for $|y|\to\infty$ one has
$$
|\Gamma(x+iy)|\sim\sqrt{2\pi}e^{-\frac\pi 2|y|}|y|^{x-\frac12}.
$$
| 6 | https://mathoverflow.net/users/nan | 128164 | 71,372 |
https://mathoverflow.net/questions/128158 | 2 | In 2D perimeter(P) of a convex set around origin may be written as $P=1/2 \int m(\theta) d\theta$. Where $m(\theta)$ is the diameter of the set in the $\theta$ direction. This is related to Cauchy-Crofton formula. The question is if anything similar known at higher dimensions ?
| https://mathoverflow.net/users/18659 | Surface area of a convex set | A good reference is Klain and Rota's little book *Introduction to Geometric Probability*, especially Section 5.5.
Here's the formula. The surface area of a compact convex subset $K$ of $\mathbb{R}^n$ is
$$
\frac{1}{\omega\_{n - 1}} \int\_{S^{n-1}} Vol\_{n-1}(\pi\_{\theta^{\bot}} K) d\theta.
$$
Here $\omega\_{n -... | 2 | https://mathoverflow.net/users/586 | 128167 | 71,374 |
https://mathoverflow.net/questions/127932 | 1 | Let $T$ be a first order theory, $M$ a model of $T$ equipped with a topology with a definable basis (i.e. every basic open is definable with parameters). Let $F: M\rightarrow M$ be a partial function and $(f\_a)\_{a\in M^k}$ be a uniformly definable family of functions such that for any open set $U\subsetneq dom(F)$ th... | https://mathoverflow.net/users/8145 | Approximating a function via definable functions | With your clarification (in a comment) that $U$ should be from the definable basis, the answer to your question seems to be negative. Notice first that the discrete topology on any model has a definable (with parameters, as you wrote in the question) basis, consisting of the singletons. Now you can uniformly define a f... | 1 | https://mathoverflow.net/users/6794 | 128168 | 71,375 |
https://mathoverflow.net/questions/128175 | 1 | Is there a name for finite-dimensional associative $F$-algebras having the Jacobson radical of codimension 1. Of course they are particular local algebras and, indeed, the converse is true provided $F$ is algebraically closed.
| https://mathoverflow.net/users/23674 | Associative algebras with Jacobson radical of codimension 1 | Those are local *basic* algebras.
| 2 | https://mathoverflow.net/users/1409 | 128177 | 71,379 |
https://mathoverflow.net/questions/128176 | 14 | I apologize as this question is not really mathematical, and therefore perhaps not
well-suited for this site. Please feel free to close it if you think it is not. My reason
for asking it here is that I am not satisfied (that is not convinced in any sense)
by many discussions relative to that question I have seen on var... | https://mathoverflow.net/users/9317 | Will quantum computing kill cryptography ? | This question seems a bit vague, but one answer is that there are cryptosystems such as NTRU that are based on (special cases) of the closest vector problem (CVP). At present, quantum computers would not significantly speed up the solution of the CVP. If I understand correctly, they would require doubling the length of... | 16 | https://mathoverflow.net/users/11926 | 128179 | 71,380 |
https://mathoverflow.net/questions/128183 | 1 | Let $X$ be a separable metric space, $\mu\_{n}$ a sequence of Borel probability measures
and $\mathcal{C}$ be a family of sets that is closed under finite unions and
interections, and that contains all the balls. If $\mu\_{n}(A)$ converges for
every $A\in\mathcal{C}$, does there exists a Borel measure $\mu\_{\infty}$
s... | https://mathoverflow.net/users/18384 | Existence of limit measure | Why isn't the following a (locally compact) counterexample? Let $X$ be the set of natural numbers, with the metric where the distance between every two distinct points is 1. So the topology is discrete, and the only balls are the singletons and the whole space. Let $\mathcal C$ consists of the finite sets and the whole... | 4 | https://mathoverflow.net/users/6794 | 128186 | 71,382 |
https://mathoverflow.net/questions/128187 | 5 | A function is additive if $f(x+y) = f(x) + f(y)$. Intuitively, it might seem that an additive function from R to R must be linear, specifically of the form $f(x) = kx$. But assuming the axiom of choice, that is wrong, and the proof is rather simple: you just take a Hamel basis of $\mathbb{R}$ as a vector space over $\m... | https://mathoverflow.net/users/5017 | Does a nonlinear additive function on R imply a Hamel basis of R? | To my knowledge this is an open problem.
If one looks at Herrlich **The Axiom of Choice**, there is a diagram (7.23, p. 156) of implications related to non-measurable sets (which include discontinuous solution to the Cauchy functional equation problem), one can see that this is pretty far down below the existence of ... | 13 | https://mathoverflow.net/users/7206 | 128188 | 71,383 |
https://mathoverflow.net/questions/128194 | 11 | Fix a prime $p$ and integer $n>1$, along with the ring $R$ of integers in a finite extension of the field $\mathbb{Q}\_p$ (for example $R = \mathbb{Z}\_p$).
>
> Is there an upper bound $C(n,p)$ on the orders of finite subgroups of $\mathrm{GL}\_n(R)$? Or can finite subgroups be arbitrarily large?
>
>
>
Probab... | https://mathoverflow.net/users/4231 | Upper bound on order of finite subgroups of GL_n(Z_p)? | Yes, for any fixed $p$-adic field $K$ the supremum of orders of finite subgroups of $\operatorname{GL}\_n(K)$ is finite and can be explicitly bounded above. There is a beautiful discussion of this tucked away somewhere in Serre's *Lie Algebras and Lie Groups*. (What I say in the following is almost entirely derived fro... | 10 | https://mathoverflow.net/users/1149 | 128201 | 71,393 |
https://mathoverflow.net/questions/128204 | 1 | Greetings to members here.
The question is how to calculate the solution $S(k)$ of the following recursive equation
$$J(k)S(k+1)J^{T}(k)=A(k)S(k)A^{T}(k)+R(k)$$
where $J$ and $A$ are rectangular not square. $R$ is positive-definite. Furthermore, $J$ and $A$ are with full-row rank.
| https://mathoverflow.net/users/25957 | On solution of a recursion with rectangular matrices | If $J$ has more columns than rows, the map $S \to J S J^T$ is not one-to-one, so your equation does not determine $S(k+1)$.
| 2 | https://mathoverflow.net/users/13650 | 128211 | 71,397 |
https://mathoverflow.net/questions/128217 | 1 | Let $R\_1$ and $R\_2$ be two subrings of the ring $R$ which commute in $R$ so that we have a ring homomorphism $R\_1\otimes\_\mathbb{Z} R\_2\rightarrow R$. Assume that $R$ is flat over $R\_1$ and $R\_2$. Is then $R$ also flat over $R\_1\otimes\_\mathbb{Z} R\_2$? Is there an easy counterexample?
| https://mathoverflow.net/users/27923 | Flatness and tensor product of rings | Take $R\_1 = R\_2 = R = {\mathbb Z}[x]$. Then $R\_1\otimes\_{\mathbb Z} R\_2 = {\mathbb Z}[x\_1,x\_2]$ and $R = {\mathbb Z}[x]$ is not flat over it.
| 5 | https://mathoverflow.net/users/4428 | 128218 | 71,399 |
https://mathoverflow.net/questions/128213 | 3 | Let $X$ be a smooth complex manifold with finite fundamental group. Suppose that a finite group $G$ acts on $X$ and let $\widetilde{X/G}$ be a resolution of singularities. Is $\pi\_1(\widetilde{X/G})$ always finite?
I think this is true but don't know a way to prove or a reference.
| https://mathoverflow.net/users/50973 | Is $\pi_1(\widetilde{X/G})$ always finite if $\pi_1(X)$ is finite? | The answer is **yes** when $X$ is simply connected. This can be proven as follows.
$\underline{\textrm{Step 1.}}$ The fundamental group $\pi\_1(X/G)$ is finite. More precisely, $\pi\_1(X/G)= G/N$, where $N$ is the smallest normal subgroup generated by those elements in $G$ which have fixed points on $X$.
For a pro... | 5 | https://mathoverflow.net/users/7460 | 128220 | 71,400 |
https://mathoverflow.net/questions/128222 | 24 | This is migrated by [math.stackexchange](https://math.stackexchange.com/questions/363118/how-we-do-actually-compute-the-topological-index-in-atiyah-singer) as I did not receive an answer. I do not know if it is too naive for this site.
I am taking a lectured class in Atiyah-Singer this semester. While the class is m... | https://mathoverflow.net/users/18850 | How we do actually compute the topological index in Atiyah-Singer? | As Johannes Ebert said, it's best if at first you stay away from boundary value problems. For some elliptic operators there may not even exist *local* boundary conditions satisfying the conditions guaranteeing Fredholmness; the Dolbeault operator is such an example. Therefore often one has to deal with pseudo-local bou... | 10 | https://mathoverflow.net/users/20302 | 128242 | 71,408 |
https://mathoverflow.net/questions/128212 | 8 | How can I most quickly find a big prime, p, for which 4p+1 is also prime? For example, p=37 works. I wonder if these special primes have been researched and some characteristics are known. Are there infinitely many of these primes?
| https://mathoverflow.net/users/20757 | special primes with p'=4p+1 | There are very likely infinitely many primes of this form but this is open.
If one where to count the number of such primes up to $x$, one expects to find
$$
\frac{C x}{( \log x)^2}
$$
for some constant $C$ that one could compute, so on the one hand not too few but still only a set of relative density (in the prim... | 13 | https://mathoverflow.net/users/nan | 128246 | 71,409 |
https://mathoverflow.net/questions/128244 | 2 | How can I compute the [curvature](http://en.wikipedia.org/wiki/Curvature) of the contour lines (equipotential lines) $\phi (\vec{r})=c$ for the scalar field $\phi (\vec{r})$ ? I expect the direction of the curvature vector to be along the gradient of the field, in analogy to the electric field vectors which are orthogo... | https://mathoverflow.net/users/33298 | Curvature of contour lines of a scalar field | The formula for computing the curvature of a curve defined by an implicit equation can be found in my notes at <http://u.math.biu.ac.il/~katzmik/egreglong.pdf> on page 32. It is closely related to the Reiss relation in algebraic geometry. See references there.
| 4 | https://mathoverflow.net/users/28128 | 128248 | 71,411 |
https://mathoverflow.net/questions/128254 | 2 | I recently came across noncommutative geometry and found it rather interesting. I should mention that I'm a graduate student considering options for my research and if I were to name an area which I'm interested in, then it would be functional analysis including operator algebras etc., and that was how I got to know ab... | https://mathoverflow.net/users/33299 | Possible directions in noncommutative geometry | 1)You look into the book "Noncommutative Dynamics and E0 semigroups" by William Arveson
2) There is an approach to attack multivariate operator theory through algebraic geometry.You may look to "Operator Theory and Complex Geometry" by douglas for an introduction
3)You may look into brown,douglas,fillimore's paper in e... | 0 | https://mathoverflow.net/users/30081 | 128256 | 71,416 |
https://mathoverflow.net/questions/128266 | 1 | For a homogeneous space $G/H$, endowed with a $H$-equivariant metric $g$, let $\ast$ be the corresponding Hodge star map. It seems that $\ast$ must also be $\ast$-equivariant, but I can't see how one would prove it. I badly recall that the Hodge star map can constructed as contraction with the uniquely determined highe... | https://mathoverflow.net/users/3787 | Homogeneous Spaces and Equivariant Hodge Maps | A $G$-invariant metric $g$ on $G/H$ is uniquely determined by its $H$-invariant value $g\_o$ at $T\_o(G/H)$ for the base point $o\in G/H$. The Riemannian volume form $vol(g)$ is $G$-invariant, and $\star$ is given by
$\phi^k\wedge \psi^{n-k} = (\Lambda^{n-k}g^{-1})(\star\phi^k,\psi^{n-k}).vol(g)$ where $\Lambda^{n-k}g... | 2 | https://mathoverflow.net/users/26935 | 128273 | 71,421 |
https://mathoverflow.net/questions/108247 | 4 | [Wikipedia](http://en.wikipedia.org/wiki/Numerical_methods_for_ordinary_differential_equations) presents a timeline of important developments in Numerical Methods for ODEs, namely:
```
1768 - Leonhard Euler publishes his method.
1824 - Augustin Louis Cauchy proves convergence of the Euler method. In this proof, Cauc... | https://mathoverflow.net/users/22714 | Numerical Methods for ODEs - History | Here are the sources:
[Leonhard Euler: *Institutiones calculi integralis* (1768)](http://books.google.de/books/about/Institutiones_calculi_integralis.html?id=H-dccgAACAAJ&redir_esc=y)
[Augustin Louis Cauchy: Cours d'Analyse: *Equations différentielles ordinaires et aux dérivées partielles* (1824)](http://books.goog... | 3 | https://mathoverflow.net/users/nan | 128287 | 71,428 |
https://mathoverflow.net/questions/128281 | 0 | [The chaos game](http://en.wikipedia.org/wiki/Chaos_game) is a way to construct (an approximation) of Sierpinski triangle. It's clear (using Thales' theorem!) that if we begin with a point on the sierpinski triangle, then we will never leave it. However, the choice of the beginning point is not important! The final sha... | https://mathoverflow.net/users/33254 | Sierpinski Triangle and the Chaos Game | Because the iterated function system that defines the Sierpinski gasket is a contraction mapping in the metric space of non-empty compact subsets of $\mathbb{R}^{2}$ with the Sierpinski gasket as its only fixed point. So if you start with any non-empty compact set it will "get closer" to the Sierpinski triangle each ti... | 1 | https://mathoverflow.net/users/11332 | 128292 | 71,430 |
https://mathoverflow.net/questions/128272 | 1 | Let $\Gamma$ be a multiple edge free (di)graph (with or without loop). Let $A$ be its adjacency matrix. It is clear that if $\lambda^2$ is an eigenvalue of $A^2$, then $\lambda$
or $-\lambda$ is an eigenvalue of $A$. What can we say about sign of $\lambda$ in general? I mean that can we exactly determine sign of eigenv... | https://mathoverflow.net/users/27831 | signs of eigenvalues | Just to build on the answer which Chris gave, one can replace two disjoint copies $K\_3$ with two disjoint copies of *any* graph $G$ (which is not bipartite, just to keep things simple). Let the vertices of $G$ be numbered $0,1,\cdots,v-1.$ Now make two graphs $G\_1$ (not bipartite) and $G\_2$ (bipartite) with vertex s... | 0 | https://mathoverflow.net/users/8008 | 128303 | 71,438 |
https://mathoverflow.net/questions/128233 | 12 | On $n$ nodes, we have $2^{n(n-1)/2}$ graphs. Asymmetric graph is a graph that has only trivial automorphism. We known that asymptotically almost all finite graphs are asymmetric. Therefore, in the limit, the ratio of asymmetric graphs approaches 1.
However, I did not find any reference that provides lower bound on t... | https://mathoverflow.net/users/8784 | How dense is the set of asymmetric graphs? | Almost all non-asymmetric graphs have exactly one non-trivial automorphism, namely a transposition swapping two vertices. So, an accurate estimate of their number is
obtained by taking an arbitrary graph with one vertex less, choosing a vertex $v$, adding a new vertex $w$ with the same neighbours as $v$, then either jo... | 12 | https://mathoverflow.net/users/9025 | 128308 | 71,440 |
https://mathoverflow.net/questions/128311 | 0 | Is there any routine technique to find a set of permutations which generate a Sylow 2-subgroup of the symmetric group $S\_{2^{r−1}}$?
| https://mathoverflow.net/users/33209 | the symmetric group $S_{2^{r−1}}$ | The Sylow $2$-subgroup of $S\_n$, where $n$ is a power of $2$, is the automorphism group of a perfect binary tree with $n$ leaves. So, by induction, you can generate the Sylow $2$-subgroup by generators of $S\_{n/2}$, plus anything which swaps the children of the root:
$\langle (1~2), \\\ (1~3)(2~4),\\\ (1~5)(2~6)(3~... | 2 | https://mathoverflow.net/users/2954 | 128316 | 71,444 |
https://mathoverflow.net/questions/128330 | 2 | What are the conjugacy classes of $PSL (3,q)$ and $PSU(3,q)$?
| https://mathoverflow.net/users/33320 | Conjugacy classes in PSL(3,q) and PSU(3,q) | This is well-known, and there are a number of relevant references. Firstly, there are these by Wall (they are pretty hard to read though).
>
> Wall, G. E. Conjugacy classes in projective and special linear groups. Bull. Austral. Math. Soc. 22 (1980), no. 3, 339–364.
>
>
> Wall, G. E. On the conjugacy classes in t... | 3 | https://mathoverflow.net/users/801 | 128333 | 71,448 |
https://mathoverflow.net/questions/128328 | 2 | In a [previous post](https://mathoverflow.net/questions/127932/approximating-a-function-via-definable-functions/128168#128168) I asked about the definability of a function that can be "approximated" by a uniformly definable family of functions. Nevertheless, the notion of approximation I gave was too weak and a counter... | https://mathoverflow.net/users/8145 | Approximating a function via definable functions II | Let $M$ be the structure consisting of a countable universe with the following structure. There is a binary function $E$ such that, given any finitely many distinct elements $d\_1,\dots,d\_k\in M$ and any (not necessarily distinct) $a\_1,\dots,a\_k\in M$ there is some $q\in M$ with $E(q,d\_i)=a\_i$ for all $i=1,\dots,k... | 3 | https://mathoverflow.net/users/6794 | 128342 | 71,452 |
https://mathoverflow.net/questions/126506 | 2 | In my studies on the Ricci flow, I was faced with a problem. To prove the existence and uniqueness of solutions to the Ricci flow, it is proved that the Ricci flow is a Parabolic PDE type. Then one can find that it is weakly parabolic, so short-time existence does not follow from standard parabolic theory and use the D... | https://mathoverflow.net/users/32817 | Strongly parabolic PDE vs weakly parabolic PDE | Here is a toy model: Consider a function $u=u(t,x,y,z)$. Then the standard heat equation $\partial\_tu=(\partial\_x^2+\partial\_y^2+\partial\_z^2)u$ is strongly parabolic, while e.g. the equation $\partial\_tu=(\partial\_x^2+\partial\_y^2)u$ is only weakly parabolic (more commonly called degenerate parabolic).
Simila... | 4 | https://mathoverflow.net/users/22029 | 128348 | 71,456 |
https://mathoverflow.net/questions/128326 | 6 | Suppose $X$ is a normal projective variety over $\mathbb C$. In the case $X$ is smooth according to Hodge theory $h^1(X,O(X))$ is the dimension of the space of holomorphic $1$-forms on $X$ and this number is equal as well to the half of the first Betti number $b\_1(X)/2$ .
I would like to know what happen in the cas... | https://mathoverflow.net/users/13441 | $H^1(X,O_X)$, holomorphic $1$-forms, and $b_1(X)/2$ for normal $X$. | (Although I have pretty much "retired" from Mathoverflow, I will answer this, since the answer is nice but probably not all that well known.)
>
> Theorem. If $X$ is complex normal projective variety, then it is still true that $b\_1(X)=2h^1(\mathcal{O}\_X)$.
>
>
>
Proof. Let $\pi:\tilde X\to X$ be a desingular... | 12 | https://mathoverflow.net/users/4144 | 128351 | 71,458 |
https://mathoverflow.net/questions/128265 | 5 | It is well-known that a random graph a.e. has diameter 2. It is also well-known that the number of distinct eigenvalues of a graph is at least the diameter plus one.
But what is known about the expected number of distinct eigenvalues of a random graph?
| https://mathoverflow.net/users/22051 | How many distinct eigenvalues does a random graph have? | In [this recent paper](http://arxiv.org/abs/1103.3869) of Erdos, Knowles, Yau, and Yin, it is shown that in the bulk of the spectrum, the spacing between eigenvalues of an Erdos-Renyi graph on $n$ vertices obeys GOE statistics asymptotically. This implies that most of the eigenvalues are simple (i.e. $n-o(n)$ of the $n... | 13 | https://mathoverflow.net/users/766 | 128361 | 71,463 |
https://mathoverflow.net/questions/128313 | 1 | I see the following theorem in Lihe Wang's [A geometric approach to the Calderon--Zygmund estimates](http://www.math.uiowa.edu/~lwang/cccalderon.pdf)
>
> (Modified Vitali) Let $0<\varepsilon<1$ and let $C\subset D\subset B\_1$ be two measurable sets with $|C|<\varepsilon |B\_1|$ and satisfying the following proper... | https://mathoverflow.net/users/33314 | On the proof of Modified Vitali Lemma. | The function $f\_x \colon r \mapsto \frac{|C\cap B\_r(x)|}{|B\_r(x)|}$ is clearly continuous, so define
$$r\_x = \sup\{r < 2 ~:~ f\_x(r) = \epsilon\}.$$
| 0 | https://mathoverflow.net/users/11716 | 128369 | 71,467 |
https://mathoverflow.net/questions/128353 | 2 | What books one must read and in what sequence to learn a low-dimensional topology at the grad level? The goal is to read in about a year at least something about Geometrization conjecture of Thurston. The background of the OP is merely Basic Topology by MA Armstrong and undergrad level abstract algebra.
Please suggest ... | https://mathoverflow.net/users/27093 | Learning roadmap for geometric topology | One of the best introductions to the subject is certainly Thurston's [Three-dimensional Topology and Geometry, Vol.1](http://books.google.de/books?id=9kkuP3lsEFQC&printsec=frontcover&dq=Three-Dimensional+Geometry+and+Topology&hl=de&sa=X&ei=rml1UdjXH5SE0QHSs4GQBw&ved=0CDMQ6AEwAA#v=onepage&q=Three-Dimensional%2520Geometr... | 6 | https://mathoverflow.net/users/2039 | 128378 | 71,471 |
https://mathoverflow.net/questions/128318 | 12 | This question is inspired by some interesting comments on [this recent question](https://mathoverflow.net/questions/128194/upper-bound-on-order-of-finite-subgroups-of-gl-nz-p).
Fix an integer $n \geq 1$ and a finite subgroup $G$ of $\mathrm{GL}\_n(\mathbf{C})$. It is known that there are infinitely many primes $p$ su... | https://mathoverflow.net/users/6506 | Embeddings of finite groups into GL(n,Q_p) | $\def\Gal{\mathrm{Gal}}$
$\def\Res{\mathrm{Res}}$
$\def\GL{\mathrm{GL}}$
$\def\F{\mathbf{F}}$
$\def\Q{\mathbf{Q}}$
**Edited to include more details.**
Let $K$ be a field whose characteristic is prime to the order of $G$.
The algebra $K[G]$ is a product of matrix algebras over division rings.
Given any *absolutely i... | 9 | https://mathoverflow.net/users/33127 | 128389 | 71,474 |
https://mathoverflow.net/questions/128393 | 5 | Given categories $X$ and $Y$ and a strong functor
$$D:X^{op}\times Y\to Cat$$
we can of course build the oplax colimit
$$\mathrm{colim}^{oplax}\_{X^{op}\times Y}D$$
via the usual (covariant) grothendieck construction:
*Objects* are triples $(x,d,y)$ with $d\in D(x,y)$.
*Morphisms* $(x,d,y)\to (x',d',y')$ ar... | https://mathoverflow.net/users/1261 | grothendieck construction for profunctors | [I guess that by $x\_\*g$ you mean $D(\mathit{id}, g)(x)$ and by $f^\*y$ you mean $D(f, \mathit{id})(y)$.]
Actually, your second construction is the usual Grothendieck construction for "$\mathbf{Cat}$-valued distributors" (BTW, this term may be misleading a bit, because in a $\mathbf{Cat}$-valued distributor $\mathbb... | 6 | https://mathoverflow.net/users/13480 | 128396 | 71,475 |
https://mathoverflow.net/questions/128382 | 1 | Using elementary matrix row and column operations on the system of two diophantine equations, namely, $N=an+b$ and $N=cn+d$, where $n\in\mathbb{N}^0$, it can be shown that the intersection of these two arithmetic progressions is another arithmetic progression $N=(ac)n+c\delta+d$ where $\delta\in\mathbb{N}:a|\left(c\del... | https://mathoverflow.net/users/33333 | Intersection of two arithmetic progressions | You do not need a divisibility criterion, the intersection of two such arithmetic progressions can be found using the chinese remainder theorem.
In your example notice that any such $N$ in the intersection satisfies:
$N \equiv 3 \bmod 5$
and
$N \equiv 5 \bmod 7$
Solving gives $N \equiv 33 \bmod 35$.
| 3 | https://mathoverflow.net/users/21698 | 128397 | 71,476 |
https://mathoverflow.net/questions/128278 | 3 | Let $X, Y$ be irreducible projective varieties and $Y$ be smooth. Let $f:X \to Y$ be a flat projective morphism. Assume that a special fiber of $f$ is non-reduced i.e., there exists an irreducible component of the fiber which is of multiplicity greater than $1$. When can we say that the generic fiber or $X$ is non-redu... | https://mathoverflow.net/users/32151 | Morphism with non-reduced special fibre | I think there is some confusion here. Either on your part or on mine. I don't think being non-reduced is equivalent to having a non-reduced component. A scheme may have a fat point, but be irreducible and generically reduced. Similarly, I don't quite understand what you want. Do you want the generic fiber be non-reduce... | 3 | https://mathoverflow.net/users/10076 | 128403 | 71,478 |
https://mathoverflow.net/questions/128405 | 2 | I would like to know if the following statement is correct.
**Statement**. Let $X$ be a normal projective variety with $Pic(X)=\mathbb Z+torsion$. Let $L$ be an ample line bundle on $X$ and let $D$ be an effective $\mathbb Q$-Cartier divisor in $X$. Consider the projective cone $C$ over $X$ corresponding to $L$. Then... | https://mathoverflow.net/users/13441 | Conical divisor over a $\mathbb Q$-Cartier divisor. | I'm going to assume that $L$ induces a projectively normal embedding (so I feel comfortable talking above divisors, or you can take Spectrum of section rings instead of cones). Then this is certainly true for the affine cone which should be all you need, since away from the cone point there is nothing to check. In othe... | 5 | https://mathoverflow.net/users/3521 | 128409 | 71,481 |
https://mathoverflow.net/questions/128380 | 3 | Let $\sigma :\mathbb{N}\rightarrow\mathbb{R}$ an injective sequence of real numbers.
There exists an infinite set $A=$ { ${a\_{1},a\_2,\ldots ,a\_n,\}\ldots$ } $ \subset{N}$ such that
i) $\sigma\_{|A}$ is monotone
ii) $a\_n=O(n^2)$ ?
| https://mathoverflow.net/users/33332 | Chains or Antichains slowly increasing | For $0 \leq k \lt 2^j$ , let $\sigma(k+2^j)=(2k+1)/2^{j+1}$ . Let A be a subset of integers such
that $\sigma\mid\_A$ is monotonic. Then $a\_{n+1} - a\_n$ is greater than $a\_n/4$ infinitely often,
which cannot hold if $a\_n$ is $O(n^d)$ for any positive integer $d$.
Gerhard "Can't Make It Much Simpler" Paseman, 2013... | 3 | https://mathoverflow.net/users/3528 | 128411 | 71,482 |
https://mathoverflow.net/questions/128249 | 4 | I am interested in approximating the sum of the squares of the multinomial coefficients, i.e.
$a\_\ell^p := \sum\_{k\_0+\ldots+k\_p = \ell} (\frac{\ell!}{k\_0! \ldots k\_p!})^2$
or more general,
$a\_\ell^{\alpha\_0,\ldots, \alpha\_p} := \sum\_{k\_0+\ldots+k\_p = \ell} (\prod\_{i=0}^p \alpha\_i^{k\_i})^2(\frac{\... | https://mathoverflow.net/users/32922 | Estimate on sum of squares of multinomial coefficients | Douglas already commented that the asymptotics for fixed $p$ and $l\to \infty$ shoudl follow from standard methods. One gets
$$a\_{\ell}^p\approx (p+1)^{2\ell+\frac{p+1}{2}}(4\pi \ell)^{-\frac{p}{2}}.$$
See theorem 4 in ["Counting Abelian squares"](http://arxiv.org/abs/0807.5028), by Richmond and Shallit. Notice that t... | 7 | https://mathoverflow.net/users/2384 | 128418 | 71,485 |
https://mathoverflow.net/questions/128416 | 4 | I've been reading about space filling curves, and been asking myself this question.
If $f: \mathbb{R}^{2} \rightarrow \mathbb{R}$ is a continuous open map, is it true that $\forall x \in$ range$(f)$ , $f^{-1}(x)$ is always uncountable?
| https://mathoverflow.net/users/33344 | Uncountable Pre-Image | I think so, yes. Let $x$ be in the range of $f$ and define $U = f^{-1}((-\infty, x))$, $V = f^{-1}((x,\infty))$. Since $f$ is open, $U$ and $V$ are both nonempty. So they are disjoint nonempty open sets, which means that the complement of $f^{-1}(x)$ is disconnected. But the complement of any countable subset of ${\bf ... | 14 | https://mathoverflow.net/users/23141 | 128421 | 71,486 |
https://mathoverflow.net/questions/128425 | 5 | I was wondering whether the set $\lbrace f\in H\_0^1(\Omega)|\|f\|\_{L^\infty(\Omega)}\leq 1\rbrace$ is compact in $H\_0^1(\Omega)$ or not. Here $\Omega$ is a convex domain in $\mathbb{R}^3$ with Lipschitz boundary.
Thanks.
| https://mathoverflow.net/users/33350 | Compactness in Sobolev spaces | No. As a general rule, in order to obtain compactness in some norm, one needs control of a higher regularity than what is associated to that norm, in order to shut down an "escape to frequency infinity". For instance, $H^1\_0$ has one degree of regularity, so one needs to control a norm involving more than one derivati... | 21 | https://mathoverflow.net/users/766 | 128426 | 71,488 |
https://mathoverflow.net/questions/128401 | 1 | I need to prove that $PGL\_2(\mathbb{R})\cong SO\_3(\mathbb{R})$. Abstract considerations show that both can be identified with the group of projective motions of a conic curve. But maybe there is more explicit isomorphism (in matrix form, for example)?
| https://mathoverflow.net/users/32920 | About isomorphism of $PGL(2)$ and $SO(3)$ | Put the bilinear form $\langle, \rangle$ on $2 \times 2$ real matrices by setting $\langle A,B \rangle = {\rm tr}(AB).$ The space of matrices breaks with respect to this form as the orthogonal direct sum of the space of scalar matrices and the $3$-dimensional subspace of matrices of trace zero. Now ${\rm GL}(2,\mathbb{... | 4 | https://mathoverflow.net/users/14450 | 128427 | 71,489 |
https://mathoverflow.net/questions/128434 | 2 | Let $A$ and $B$ be two given hermitian positive semi-definite matrices, then what is the solution for
$$\max\_{x\neq 0}\frac{x^HAx}{x^HBx+1}.$$
I am looking for closed form solutions.
If the denominator didn't have that $1$, this is standard generalized rayleigh quotient and would be unbounded.
I know how to so... | https://mathoverflow.net/users/27249 | An Interesting variant of Rayleigh Quotient | Replacing any nonzero $x$ by $tx$ with real $t$,
$ \dfrac{(tx)^H A (tx)}{(tx)^H B (tx) + 1}$ increases to $\dfrac{x^H A x}{x^H B x}$ as $t \to \infty $. Thus the supremum is your generalized Rayleigh quotient. If $B$ is positive definite, the supremum (not maximum, as it is not attained) is the largest eigenvalue of $B... | 7 | https://mathoverflow.net/users/13650 | 128435 | 71,491 |
https://mathoverflow.net/questions/102027 | 3 | Hi,
given a compact manifold M we can always alter a given Morse function f to a self-indexing one (i.e., one where every critical point c has $f(c) = \operatorname{index}(c)$) - a proof of this may be found in, e.g., "Lectures on the h-Cobordism Theorem".
But what about non-compact manifolds? Is it always possible... | https://mathoverflow.net/users/13356 | Self-indexing Morse functions on non-compact manifolds | Since Ryan has reawakened this question, let me add a few remarks. One way to get a handle structure on a manifold $M$ is to start with a smooth triangulation of $M$, so I will assume such a triangulation exists. A nice neighborhood of the 0-skeleton then gives a collection of 0-handles. Enlarge this to a nice neighbor... | 9 | https://mathoverflow.net/users/23571 | 128436 | 71,492 |
https://mathoverflow.net/questions/128385 | 7 | Let $A$ be a commutative ring. Let $f\in A\setminus\{0\}$ and $I\subseteq A$ any ideal. I would like to define the **multiplicity of $f$ at $I$** as
$$\mu\_f(I):= \max\{\, d\ge 0 \mid f\in I^d\,\},$$
where $I^0:= A$. In the case where $A$ is Noetherian and either local or an integral domain, the Krull Intersection The... | https://mathoverflow.net/users/9947 | The notion of multiplicity in algebraic geometry | In addition to two good answers, maybe one case the question has a positive answer is when $(R,m)$ is a regular local ring and $f$ is a nonzero element. For a Noetherian local ring $A$ and ideal $I$ which is primary to the maximal ideal, let $e(I, A)$ denote the Hilbert-Samuel multiplicity of $A$ with respect $I$. Then... | 5 | https://mathoverflow.net/users/22388 | 128437 | 71,493 |
https://mathoverflow.net/questions/128439 | 4 | Let $C$ be a topos with subobject classifier $\Omega$. Let $F$ be the endofunctor $x \mapsto \Omega^{\Omega^x}$ on $C$. Does there exist $C$ such that $F$ has an initial algebra? What if $\Omega$ is replaced with the coproduct $2 = 1 \sqcup 1$ of two copies of the terminal object? (What if it is replaced with any objec... | https://mathoverflow.net/users/290 | Does this kind of endofunctor ever have an initial algebra? | By Lambek's theorem, any initial algebra for an endofunctor $F$ has the property that its structural morphism $\alpha : F A \to A$ is an isomorphism. So we seek an object $A$ such that $P P A = \Omega^{\Omega^A} \cong A$.
However:
**Proposition.** If a topos $\mathcal{E}$ contains an object $A$ such that $P A$ is a... | 13 | https://mathoverflow.net/users/11640 | 128443 | 71,495 |
https://mathoverflow.net/questions/128440 | 0 | Let $\Lambda\_{n}$ be the set of all Lagrangian subspaces of $C^{n}$, and $P\in\Lambda\_{n}$. Put $U\_{P}= ( Q\in\Lambda\_{n} : Q\cap (iP)=0 )$. There is an assertion that the set $U\_{P}$ is homeomorphic to the real vector space of all symmetric endomorphisms of $P$. And then in the proof of it there is a fact that th... | https://mathoverflow.net/users/19582 | Lagrangian submanifolds | It is very elementary; the graph representation refers to the cartesian product $P\times iP$ coming from the real vector space direct sum decomposition $\mathbb{C}^n = P\oplus iP\sim P\times iP$.
Here the direct sum decomposition is possible because $\dim \_ \mathbb{R}(P)=n $ and $P\cap iP=0$, from the definition of ... | 2 | https://mathoverflow.net/users/6101 | 128448 | 71,498 |
https://mathoverflow.net/questions/128451 | 2 | I have a complete metric space $Y$, some non-metrizable(!) Hausdorff compactification $Z$ of it and a subspace $X \subset Y$.
Furthermore, I do have a uniformly continuous function $f$ on $X$. So there is a uniformly continuous extension of $f$ from $X$ to the closure of $X$ in $Y$.
>
> Can we extend f to a unifo... | https://mathoverflow.net/users/13356 | Extending uniformly continuous functions on subspaces to non-metrizable compactifications | The closure of $X$ in $Z$ is compact , so there is no hope if $f$ is not bounded. If it is bounded then so is its extension to the closure of $X$ in $Y$ and this gives a bounded uniformly continuous function on the closure which can be extended to a bounded continuous function on $Y$ by the Tietze extension theorem. Th... | 3 | https://mathoverflow.net/users/26013 | 128453 | 71,500 |
https://mathoverflow.net/questions/128456 | 1 | I've seen stated offhand in many sources that the cuspidal subgroup of the Jacobian of $X\_0(N)$ is finite.
Do they mean that the subgroup of the jacobian generated by **$\mathbb{Q}$-rational** cusps is finite, or do they mean the subgroup generated by **all** cusps is finite?
I know that Ogg (and also Mazur) does ... | https://mathoverflow.net/users/15242 | reference request for the finiteness of cuspidal subgroup of $X_0(N)$? | This is the Manin-Drinfeld theorem.See [related MO question](https://mathoverflow.net/questions/124502/does-the-manin-drinfeld-theorem-hold-over-number-fields)
| 2 | https://mathoverflow.net/users/11786 | 128465 | 71,503 |
https://mathoverflow.net/questions/128462 | 7 | Conjugate gradient was originally presented in the 50's before the modern understanding of Krylov subspaces (and the resulting iterative methods) was fully realized. As such, the method was derived using different tools and language. My question is, who was the first person to observe that Conjugate Gradient is a Krylo... | https://mathoverflow.net/users/17416 | Who first observed that Conjugate Gradient for Symmetric Positive Definite linear systems is a Krylov method? | [Krylov Subspace Methods for Solving Large Unsymmetric Linear Systems](http://www.ams.org/journals/mcom/1981-37-155/S0025-5718-1981-0616364-6/S0025-5718-1981-0616364-6.pdf), Y. Saad, *Mathematics of Computation* **37**, 105-126 (1981).
>
> The purpose of the present paper is to
> generalize the conjugate gradient
... | 4 | https://mathoverflow.net/users/11260 | 128466 | 71,504 |
https://mathoverflow.net/questions/128460 | 3 | Let $M$ be a closed orientable Riemannian manifold. Recall that a plane field on a Riemannian manifold is said to be geodesic if any geodesic tangent to the plane field at one point is tangent to it at every point.
Is it true that if $E\subset TM$ a one-dimensional geodesic plane field, then there exists $X\in \Gamma(... | https://mathoverflow.net/users/30176 | Is geodesic plane field a Killing field? | (I assume by a plane field you mean a [distribution](http://en.wikipedia.org/wiki/Distribution_%28differential_geometry%29).)
No, take $\mathbb{T}^3$ parametrized as $(x,y,z) \in [0,2\pi)^3$. The field $v = \partial\_x + \sin(z) \partial\_y$ is geodesic. But for any $\phi(x,y,z)$ the deformation tensor $\mathcal{L}\_... | 6 | https://mathoverflow.net/users/3948 | 128467 | 71,505 |
https://mathoverflow.net/questions/128332 | 2 | Good morning,
I'm trying to understand the following fact, that is stated in Gromov and Thurston's paper "Pinching constants for hyperbolic manifolds" :
Let $M$ be a (at least) 3-dimensional compact oriented hyperbolic manifold, with an action of $\mathbb{Z}\_i$
by diffeomorphism. Suppose the set of fixed points ... | https://mathoverflow.net/users/25511 | Fixed point set of an isometric group action on an hyperbolic manifold | Here is the setup with which you are dealing: You have a smooth closed manifold $M$ and a finite cyclic branched covering $p: M\to M'$ ramified over a codimension 2 totally geodesic submanifold $V'\subset M'$, where $M'$ is hyperbolic. Let $V=p^{-1}(V')$. Furthermore, you know (from the construction) that $p: V\to V'$ ... | 4 | https://mathoverflow.net/users/21684 | 128469 | 71,506 |
https://mathoverflow.net/questions/128472 | 0 | Let $X$ be a countably infinite (or larger) set with the cofinite topology. for every $x\in X$ is there exists a family $\xi\subset\tau$ such that $\lbrace x\rbrace=\bigcap\xi
$ ? If the answer is yes, then what is the cardinality of $\xi$ ?
| https://mathoverflow.net/users/33361 | A question on cofinite topology. | You should mean $\{x\}=\bigcap\xi$, and the answer is clearly yes, since we can take $\xi$ equal to the set of all open sets containing $x$. Any point $y$ other than $x$ is excluded in this intersection by the open set $X-\{y\}$. The cardinality of this $\xi$ is the same as the number of finite subsets of $X$, which is... | 1 | https://mathoverflow.net/users/1946 | 128475 | 71,508 |
https://mathoverflow.net/questions/128470 | 6 | I was reading through Jitsuro Nagura's [proof](https://projecteuclid.org/journals/proceedings-of-the-japan-academy-series-a-mathematical-sciences/volume-28/issue-4/On-the-interval-containing-at-least-one-prime-number/10.3792/pja/1195570997.full) that there is always a prime between $x$ and $\frac{6x}{5}$ when $x \ge 25... | https://mathoverflow.net/users/15915 | What are the best known lower and upper bounds for the second Chebyshev function $\psi(x)$ | The most recent results on bounds for $\psi(x)$ are from this year:
[Sharper estimates for Chebyshev's functions $\vartheta$ and $ψ$](http://arxiv.org/abs/1302.7208), February 2013.
>
> In this article we present some
> improved results for Chebyshev's
> functions $\vartheta$ and $\psi$ using
> the new zero-fr... | 5 | https://mathoverflow.net/users/11260 | 128481 | 71,510 |
https://mathoverflow.net/questions/128477 | 12 | Denote by $|A|$ the measure of $A$ (Can be Lebesgue measure) under what conditions on a function $f:\mathbb{R}^m \to \mathbb{R}$ the preimage of a null set is zero. i.e.
$|A|=0 \Rightarrow |f^{-1}(A)| =0$
A special interest for conditions on not necessarily smooth functions
| https://mathoverflow.net/users/32999 | Preimage of zero measure sets | For example, it is sufficient that $f\in C^1$ and the set $\lbrace x | \nabla f(x) = 0 \rbrace$ has measure zero.
To prove this, note that this is true locally, in a neighborhood of each point where $\nabla f \neq 0$, due to the implicit function theorem.
Now the claim follows from the fact that $f^{-1}[A] \subset Z ... | 14 | https://mathoverflow.net/users/22758 | 128482 | 71,511 |
https://mathoverflow.net/questions/128479 | 5 | I am sorry if this question is not for mathoverflow. I asked the same question in tex stackexchange and got an answer that this question is out of topic. If it's the same here, please let me know.
There are two setbuilder notations, the vertical bar and the colon. In some cases, it is better to make a choice. For exa... | https://mathoverflow.net/users/11846 | Can I use both of setbuilder notations in one article? | As the author, such things are generally up to you. It would not be the first time an author chose clarity over consistency of notation. I think many readers appreciate such tradeoffs, although I am sure there is someone who doesn't. If you are going to use inconsistent notation, it is polite to make a footnote or some... | 6 | https://mathoverflow.net/users/5963 | 128483 | 71,512 |
https://mathoverflow.net/questions/128484 | 1 |
>
> Let $R$ be a commutative finitely generated $\mathbb{Z}$-algebra. Then the nilradical is equal to the Jacobson radical.
>
>
>
I am not able to make much traction on this, nor can I find this result in any book I've look at. So far I've reasoned that since $R \cong \mathbb{Z}[x\_1,\dots,x\_n]/I$ for some idea... | https://mathoverflow.net/users/33365 | Finitely Generated Commutative Z-algebra | This follows from the Nullstellensatz (or the version of it), which says that a finitely generated algebra over a Jacobson ring is Jacobson. This can be found in, for example, Eisenbud's book on commutative algebra (I don't have it at hand so I can't be more specific right now, but it should be easy to find in the tabl... | 3 | https://mathoverflow.net/users/4351 | 128486 | 71,513 |
https://mathoverflow.net/questions/128468 | 2 | Resolvable designs are block designs with the additional property that the blocks can be partitioned into partitions of the points. It is easy to see that lines in affine space form a resolvable design since parallel lines partition the space, and so we can partition the lines according to direction.
Can lines in pro... | https://mathoverflow.net/users/29873 | Resolvable designs from projective space | The fact that every odd dimensional projective geometry ${\rm PG}(n,2)$ over $\mathbb{F}\_2$ admits a line parallelism (i.e., the Steiner $2$-design formed by the points and lines of ${\rm PG}(n,2)$ with $n$ odd is resolvable) is a corollary of the classical result proven here:
[R. D. Baker, *Partitioning the planes ... | 3 | https://mathoverflow.net/users/27829 | 128493 | 71,516 |
https://mathoverflow.net/questions/128490 | 3 | One can construct topological spaces with prescribed homotopy groups or, say, homology groups.
But is it possible to construct a ring with any given $K\_0$ group? What about $K\_1$ group et.c.?
I know very little about K-theory, so this question might be silly.
Thanks!
| https://mathoverflow.net/users/32741 | ring with prescribed K group | Every abelian group $G$ is the class group of some Dedekind domain $R$ (theorem of Luther Claborn), so we have $K\_0^{red}(R)= G$.
| 3 | https://mathoverflow.net/users/10503 | 128495 | 71,518 |
https://mathoverflow.net/questions/128344 | 6 | I am novice in the algebraic K- theory and don' t know if this is the right place for the following questions. So some people might consider them as basic questions.
Consider an exact monoidal category and its K- groups as introduced by Quillen.
Do $K\_n(\mathcal C)$ have a ring structure? Clearly for n=0 the Groth... | https://mathoverflow.net/users/33326 | K-theory of monoidal categories | (I'm not sure the term *exact monoidal category* is a standard one in the literature, so I'll just assume I know what you mean by it.)
Yes, $K\_n$ commutes with products, as some have mentioned, but the tensor monoidal functor $\otimes : \mathcal C \times \mathcal C \to \mathcal C$ is not exact, so it doesn't induce ... | 5 | https://mathoverflow.net/users/15247 | 128502 | 71,521 |
https://mathoverflow.net/questions/127691 | 25 | Let $\boldsymbol{G}$ be a reductive group over a finite field $\mathbb{F}\_q$, $G = \boldsymbol{G}(\mathbb{F}\_q)$, $W = \mathrm{W}(\mathbb{F}\_q)$ the Witt vectors over $\mathbb{F}\_q$, and $K = \mathrm{Frac}(W)$ its fraction field. I'll abuse notation by also writing $\boldsymbol{G}$ for the corresponding (unramified... | https://mathoverflow.net/users/19801 | Reconciling Lusztig's results with the Langlands philosophy | The way I like to think about this is that a Langlands parameter for the group $G({\mathbb F}\_q)$ should be the "restriction to inertia" of a tame Langlands parameter for the group $G(K)$.
That is, a tame Langlands parameter (say, over ${\mathbb C}$) for $G(K)$ should be a
pair $(\rho,N)$, where $\rho$ is a map $W\_... | 12 | https://mathoverflow.net/users/14202 | 128528 | 71,530 |
https://mathoverflow.net/questions/128525 | 2 | Given a $N \times M$ matrix $X$ comprised of standard normal entries ($M > N$), I'm interested in approximating $E[trace((XX^T\frac{\gamma}{M} + I)^{-1}]$ in terms of $N, M$ and $\gamma$. Unfortunately, I can't necessarily assume $\gamma$ is small. I've had no luck in coming up with any kind of approximation. Thanks!
... | https://mathoverflow.net/users/33374 | Expectation of the trace of an inverse of a random matrix | let $\lambda\_{1},\lambda\_2,\ldots\lambda\_N$ be the eigenvalues of $M^{-1}XX^{T}$; including for convenience a factor $1/N$, the quantity you seek is
$$N^{-1}E[{\rm Tr}(XX^{T}\gamma/M)+I)^{-1}]=\int d\lambda \rho(\lambda)(\lambda\gamma+1)^{-1}$$
where $\rho(\lambda)=E[N^{-1}\sum\_n\delta(\lambda-\lambda\_n)]$ is ... | 4 | https://mathoverflow.net/users/11260 | 128532 | 71,533 |
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