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https://mathoverflow.net/questions/179711 | 3 | The analogous result to Cayley’s theorem or Yoneda lemma in semigroup theory represents semigroups as semigroups of functions from a set to itself. This suggests that a semigroup action consists of a semigroup $S$, a set $A$, and a mapping of the elements of the semigroup $S$ to functions from the set $A$ to itself. Bu... | https://mathoverflow.net/users/20781 | Is there a general notion of semigroup action? | Thanks to Andrej Bauer to drawing my attention to this question.
This is definitely related to restriction categories and I do recommend the articles by Steve Lack and myself ("Restriction categories I" TCS 2002). However, I though it may be useful to say a word or two about how the question relates to our developmen... | 3 | https://mathoverflow.net/users/57704 | 179869 | 90,358 |
https://mathoverflow.net/questions/179867 | 6 | I don't know much about singular cardinal combinatorics, so I apologize in advance if I write something that is wrong or looks funny. First let me recall some basic definitions.
Let $\lambda$ be a singular cardinal and let $(\kappa\_i : i < \text{cf}(\lambda))$ be a sequence of regular cardinals cofinal in $\lambda$.... | https://mathoverflow.net/users/1682 | A "good scale" that is not really a scale | Your pseudo-good scale is indeed equivalent to a good scale, although the good scale possibly lives on a different product. This follows easily from the construction of a scale given in Cummings' Notes on Singular Cardinal Combinatorics.
A basic theorem of Shelah: If a $<^\*$-increasing sequence of functions has stat... | 5 | https://mathoverflow.net/users/9377 | 179876 | 90,362 |
https://mathoverflow.net/questions/179881 | 6 | Let $\mathfrak{A}$ be a first-order structure over a relational language and let $\kappa$ be an infinite cardinal. Lets say that $\mathfrak{A}$ has the $\kappa$-*property* if for every structure $\mathfrak{B}$ of size $\kappa$ over the same language as $\mathfrak{A}$ we have: $$\mathfrak{B} \mbox{ embeds into }\mathfra... | https://mathoverflow.net/users/17836 | Does this property of a first-order structure imply categoricity? | The answer is no for uncountable cardinals $\kappa$. Let $\mathfrak{A}=\langle A,U\rangle$ be a set $A$ of size $\kappa$ with a unary predicate $U\subset A$, where $U$ and $A-U$ both have size $\kappa$. It follows that $\mathfrak{A}$ has your property, since every structure $\mathfrak{B}=\langle B,U^B\rangle$ of size a... | 6 | https://mathoverflow.net/users/1946 | 179884 | 90,366 |
https://mathoverflow.net/questions/149674 | 9 | Given a (connected) graph $G$ it is natural to want to rank its vertices, with the more "central" vertices ranked higher.
Two natural ways of doing it are:
1. By the degrees.
2. By the entries in a Perron eigenvector of the adjacency matrix.
These two methods coincide for regular graphs and for so-called harmonic... | https://mathoverflow.net/users/22051 | Coherence between different ranking methods of a graph's vertices | I'm not sure how common this is. It may depend on how you are determining random graphs.
There may be ties so let me separate the issues of the equivalence relation "same rank" from those of the linear order among equivalence classes. In the case of ranking by degrees there will have to be at least one tie. In the c... | 2 | https://mathoverflow.net/users/8008 | 179916 | 90,378 |
https://mathoverflow.net/questions/179919 | 6 | Given an aspherical 4-dimensional closed manifold $M$ with fundamental group $\mathbb{Z}^4$, it is homotopy-equivalent to $T^4 = S^1 \times \ldots S^1$, the 4-dimensional torus.
**Question 1:** Since I am no expert and could not dig out a reference I would be interested if it is open/known that under the circumstanc... | https://mathoverflow.net/users/40484 | (Smooth) Borel Conjecture for 4-dimensional torus | Freedman and Quinn's "Topology of $4$-manifolds", Chapter 11.5, contains the following statement:
>
> Let $f: M\to N$ be a homotopy equivalence of compact aspherical $4$-manifolds whose fundamental groups are poly-(finite or cyclic), restricting to a homeomorphism of boundaries. Then $f$ is homotopic rel $\partial ... | 10 | https://mathoverflow.net/users/8103 | 179921 | 90,380 |
https://mathoverflow.net/questions/179924 | 8 | If $(X,\tau)$ is a topological space, we can consider the product topology on $X\times X$ and take the closure of the diagonal $\Delta\_X = \{(x,x): x\in X\}$, which we denote by $\mathrm{cl}(\Delta\_X)$. Obviously, $\mathrm{cl}(\Delta\_X)$ is a symmetric binary relation.
Now we can take things upside down: Let $X$ b... | https://mathoverflow.net/users/8628 | Binary relations as the topological closure of the diagonal | The answer is no, not necessarily. For a counterexample, let $X=\mathbb{R}$ and let $aRb\iff a=b \text{ or } |a-b|\geq 1$, the "equal or differ by at least one" relation. This is symmetric and reflexive. Suppose $\tau$ is a topology on $\mathbb{R}$ with $\text{cl}(\Delta)=R$. For any real number $k$, the $k^{\rm th}$ s... | 8 | https://mathoverflow.net/users/1946 | 179928 | 90,382 |
https://mathoverflow.net/questions/179835 | 18 | The following was posted to math.stackexchange to no avail: <https://math.stackexchange.com/questions/908756/an-exercise-in-homology-computation-what-is-the-geometric-fixed-points-of-an-e>
The question I want to ask has a reasonably elementary formulation and I think there is a good chance it can be answered in this ... | https://mathoverflow.net/users/5181 | What is the geometric fixed points of an (equivariant) Eilenberg Maclane Spectrum? | Your identification with the geometric fixed points and the calculation for $C\_2$ is correct. As Akhil remarked, one obtains the same answer for $C\_{2^n}$ and this is calculated in Hill-Hopkins-Ravenel. In fact, for $G=C\_{p^n}$, the same method shows $\Phi^G H\mathbb{Z}=\mathbb{Z}/p[t]$ where $|t|=2$. The key ingred... | 19 | https://mathoverflow.net/users/8818 | 179933 | 90,385 |
https://mathoverflow.net/questions/179907 | 5 | Given a sequence $x\_1,x\_2,\dots$, let $D\_n$ be the $L^2$-norm of the function $f\_n$ whose value at $t \in [0,1)$ is $nt$ minus the number of $1 \leq i \leq n$ with $x\_i \leq t$. What can be said about the rate at which $D\_n$ must go to infinity, regardless of the choice of $x\_1,x\_2,\dots$?
That is, what theor... | https://mathoverflow.net/users/3621 | $L^2$ discrepancy bound for sequences in $[0,1)$ | This problem has been addressed in work of Roth and Davenport. Roth showed that for any sequence there must be $n$ with $D\_n$ larger than a constant times $\sqrt{\log n}$, and Davenport constructed sequences for which $D\_n$ grows like at most a constant times $\sqrt{\log n}$.
More precisely, for any set ${\mathcal... | 6 | https://mathoverflow.net/users/38624 | 179938 | 90,387 |
https://mathoverflow.net/questions/179908 | 5 | Consider the following result which I recently came across in a research paper in my area (Signal Processing)
>
> Let $X$ be a $N\times N$ positive semidefinite (psd) matrix whose rank
> is $r$. Let $A$ be any symmetric $N\times N$ matrix. Then, there exist
> a set of vectors $x\_1,\dots,x\_r$ such that \begin{al... | https://mathoverflow.net/users/27249 | Proof for a Rank-One Decomposition Theorem of Positive (semi) Definite Matrices | I don't know what's going on with the paper, but here is an argument regarding existence of such decompositions.
Given a rank one decomposition
$$X = \sum\_{i=1}^R x\_ix\_i^T$$
one has $\sum\_{i=1}^R x\_i^TAx\_i = {\rm Trace}(AX)$, so the question is how to make these pairings $x\_i^TAx\_i$ equal to each other. Cons... | 6 | https://mathoverflow.net/users/38468 | 179952 | 90,391 |
https://mathoverflow.net/questions/179853 | 13 | Let $f: \mathbb{R^2}\to \mathbb{R^2}$ be a Schwartz function. If the eigenvalues of $Df$ vanish everywhere, must $f$ be constant? Does an analogous result hold when we replace $2$ by $n$?
Any properties of $f$ would be of interest. For example, clearly $\nabla \cdot f = 0$.
| https://mathoverflow.net/users/57697 | Vanishing eigenvalues of Jacobian | **For some reason, the 'edit' button didn't appear for my earlier answer, maybe because it was already accepted. Thus, I'm adding the general $n$ argument as a separate answer.**
In fact, there is a stronger result: Suppose that $f:\mathbb{R}^n\to\mathbb{R}^n$ is smooth and has the properties that *(i)* the Jacobian ... | 11 | https://mathoverflow.net/users/13972 | 179956 | 90,393 |
https://mathoverflow.net/questions/179955 | 9 | In other words, given a segment in the hyperbolic plane is there a straightedge and compass construction of a segment incommensurable with it? In the Euclidean plane one can take the diagonal of the square built on the segment, as Pythagoreans discovered. However, equivalence of this incommensurability and irrationalit... | https://mathoverflow.net/users/51484 | Is there a straightedge and compass construction of incommensurables in the hyperbolic plane? | Yes. the fundamental theorem is that the constructible angles in the non-Euclidean plane are exactly the constructible angle in the Euclidean plane. Lengths come from the two laws of cosines and the law of sines. In particular, length 1 is not constructible. As i recall, positive length $x$ is constructible if and only... | 14 | https://mathoverflow.net/users/3324 | 179958 | 90,394 |
https://mathoverflow.net/questions/178528 | 15 | When constructing the Lie algebra $L(G)$ of a Lie group $G$, one usually uses the identification of the tangent space $T\_1 G$ with left invariant vector fields $\mathcal{V}^l(G)$ to construct the Lie bracket on $L(G)=T\_1 G$. However, in the literature on Lie groupoids and Lie algebroids some authors use left-invarian... | https://mathoverflow.net/users/5937 | Construction of the Lie functor: left vs. right invariant vector fields on Lie groups and Lie groupoids | My view is that one needs both, left and right invariant vector fields. Some reasons:
For $X\in\mathfrak g=T\_eG$ the left invariant vector field $L\_X$ has a flow consisting of right translations:
$$
\text{Fl}^{L\_X}\_t(x) = x.\exp(tX)
$$
and conversely. For a left action $\ell:G\times M\to M$ we need the right invar... | 13 | https://mathoverflow.net/users/26935 | 179959 | 90,395 |
https://mathoverflow.net/questions/179954 | 14 | It is well known that a finite 4-times transitive permutation group is Matthieu, symmetric, or alternating. Another way of stating this is that the set
$$
\Omega = \{(x\_1, x\_2, x\_3, x\_4), 1\leq x\_i\leq n, \mbox{$x\_1, \ldots, x\_4$ pairwise different}\}
$$
has the property that $S\_n$ acts transitively on $\Omega$... | https://mathoverflow.net/users/37555 | Which finite simple groups can be characterized by their action on a small set? | The answer is **NO** for $G=\mathrm{SL}\_n(p)$ with $p$ fixed and $n\to\infty$. The bound you give - $(\log|G|)^c$ - requires an action of $G$ on a set of a size that is polynomial in $n$. But all the actions of $G$ have order exponential in $n$ - see, for instance Section 5 of Kleidman and Liebeck's book. I have an e-... | 9 | https://mathoverflow.net/users/801 | 179968 | 90,400 |
https://mathoverflow.net/questions/179969 | 5 | Let $K$ be a global field, $\mathbb{A}\_K$ the ring of adeles, and $U$ a unipotent algebraic group over $K$. Why is $U(\mathbb{A}\_K)/U(K)$, when endowed with the quotient topology, compact?
| https://mathoverflow.net/users/53197 | Compactness of adelic quotients for unipotent groups over global fields | You don't mention if $U$ is assumed to be smooth or connected, but it doesn't matter. In general, if $H$ is any affine group scheme of finite type over a global field $K$ and if $H$ does not contain ${\rm{GL}}\_1$ as a $K$-subgroup scheme (as is the case when $H$ is a unipotent $K$-group scheme) then $H(K)\backslash H(... | 7 | https://mathoverflow.net/users/52824 | 179974 | 90,403 |
https://mathoverflow.net/questions/52393 | 22 | There was [this question](https://mathoverflow.net/questions/52318/) for which my response was unusally popular, so I dare to ask the following:
(1) Given a prime $p>2$, how many primes $\ell < p$ there exist which are quadratic residues mod $p$?
(2) Given a prime $p>2$, how many primes $\ell < p$ there exist which... | https://mathoverflow.net/users/11919 | Given a prime $p$ how many primes $\ell<p$ of a given quadratic character mod $p$? | GH from MO and Anonymous have commented above on (modest) lower bounds for the first problem. Let me mention here that a version of problem 2 (of producing many non-residues) appeared in work of [Bourgain and Lindenstrauss](http://www.ma.huji.ac.il/~elon/Publications/pos_entropy.pdf) in connection with the QUE conjectu... | 12 | https://mathoverflow.net/users/38624 | 179976 | 90,404 |
https://mathoverflow.net/questions/179927 | 3 | Given positive integers $m<n\in\mathbb{N}$ is there an algorithm to find integers $z\_1, z\_2\in \mathbb{Z}\setminus\{0\}$ such that $\frac{m}{n}$ is best approximated by $\frac{1}{z\_1}+\frac{1}{z\_2}$? (In other words, the goal is to minimize $|\frac{m}{n}- (\frac{1}{z\_1}+\frac{1}{z\_2})|$.)
| https://mathoverflow.net/users/8628 | Approximation of the form $\frac{1}{u}\pm\frac{1}{v}$ | The comments above give a good search algorithm. I wanted to note that the answer can depend very delicately on the exact value. If $\frac{m}{n}=\frac{1}{x}+\frac{1}{y}$ then also $\frac{m}{kn}=\frac{1}{kx}+\frac{1}{ky}$ this allows multiple solutions in some cases. For example all $11$ of these pairs give $\frac{5}{84... | 3 | https://mathoverflow.net/users/8008 | 179981 | 90,405 |
https://mathoverflow.net/questions/179979 | 3 | Consider a 2-dimensional smooth projective algebraic surface S over complex numbers. Could you recommend any exact references to the proofs of the following assertions (of course, if they are true):
1. If S is rationally chain connected then S is rational.
2. If through each point of S one can draw a rational curve t... | https://mathoverflow.net/users/48137 | Is each rationally chain connected surface rational? | By the comments of abx you only need an answer to question $3$. You get easily an answer if you assume the results in the classical book of Beauville on algebraic surfaces.
Namely, if $S$ is uniruled, then you can solve the indeterminacy (Theorem II.7) and get a dominant morphism $X \rightarrow S$ where $X$ is birati... | 2 | https://mathoverflow.net/users/46104 | 179996 | 90,409 |
https://mathoverflow.net/questions/179949 | 3 | After the uniform distribution (normalized Haar measure), the matrix Fisher distribution seems to be the most popular probability distribution on the Lie group SO(3). The density is proportional to exp tr(*K* *M*T *X*), where *M* is the mode and *K* (symmetric, often positive-definite) is the concentration. There are t... | https://mathoverflow.net/users/57736 | Two matrix Fisher distributions on SO(3)? | The question is discussed heavily by Sei, T., Shibata, H., Takemura, A., Ohara, K., & Takayama, N. (2013). Properties and applications of Fisher distribution on the rotation group. Journal of Multivariate Analysis, 116, 440-455. So I consider the question to be answered. But I will continue to check this page for a whi... | 1 | https://mathoverflow.net/users/57736 | 180009 | 90,414 |
https://mathoverflow.net/questions/180019 | 7 | I'm reading on Haar measure and we know that every locally compact group admits a Haar measure, is the same true for semigroups? if not, is there a class of semigroups that admits a Haar measure?
Thank you for the help.
| https://mathoverflow.net/users/48797 | Haar Measure on Locally Compact Semigroups | Not every locally compact semigroup admits a (locally finite) left-invariant measure. In fact, this has nothing to do with any sort of analytic technicalities and already fails for finite semigroups. For example, consider $S=\{a,b\}$ with $ab=a^2=a$ and $ba=b^2=b$. Then $aS=\{a\}$ and $bS=\{b\}$, so no (finite) measure... | 16 | https://mathoverflow.net/users/75 | 180020 | 90,418 |
https://mathoverflow.net/questions/180031 | -3 | I want to count the number of unique sequences of length n with the following constraints.
1. Each element of the sequence is an integer in $\lbrace 1,2,\dots,n\rbrace$.
2. Each two adjacent elements of the sequence differ at most by 1.
3. At least one element on the sequence is equal to 1.
The problem is to find ... | https://mathoverflow.net/users/57782 | How many sequences of length n satisfy these constraints? | I'm assuming "vector of length $n$" means the number of components is $n$. In this case, the answer is $3^{n-1}$. I will create a bijection between vectors ($n$-tuples) of the form described and words of length $n-1$ on an alphabet of three symbols.
Suppose given a vector $v = (a\_1, \ldots, a\_n)$ satisfying the thr... | 5 | https://mathoverflow.net/users/49066 | 180036 | 90,425 |
https://mathoverflow.net/questions/180037 | 1 | At a certain stage of my research work, I require the following fact to hold true.
A surface $S$ satisfies certain conditions so that it lies on a 4-sphere in $R^{21}$ (I have used the results proved in <http://www.sciencedirect.com/science/article/pii/S0926224508001125>). Hence, it also lies on a (20-dimensional) hy... | https://mathoverflow.net/users/25516 | Do lower dimensional spheres always lie on higher dimensional spheres? | This is an exercise in analytic geometry.
Let $x$ be the center of the $p$-sphere, which is contained in the affine $(p+1)$-plane $x+V$, and let $b$ be any point on the sphere.
Then we need to find a point $y$ in $x+V^\perp$ such that $\lvert y-a\rvert^2 = \lvert y-b\rvert^2$. This equation is equivalent to the fa... | 5 | https://mathoverflow.net/users/56885 | 180050 | 90,429 |
https://mathoverflow.net/questions/178728 | 4 | The following appears naturally in a certain context:
Let $P$ be a *graded* partially ordered set. Let $M$ be the subset of minimal elements of $P$. Define subsets $E\_i$ inductively as follows: First, let $E\_0:=M$. Then, if $|E\_i|\leq 1$, set $E\_{i+1}=\emptyset$. Otherwise, for each *incomparable* pair $x\neq y$ ... | https://mathoverflow.net/users/27923 | Directed subposet of a poset containing the minimal elements | After editing the above construction a little bit, the following holds: $E$ is the smallest subposet of $P$ which contains $M$ and satisfies the following property: If $x\in P\setminus E$ then there exists a greatest element $y\in E$ with the property $y<x$.
| 0 | https://mathoverflow.net/users/27923 | 180053 | 90,431 |
https://mathoverflow.net/questions/179104 | 1 | Let $F$ be a group which is strongly type $F\_\infty$ in the sense that every subgroup is of type $F\_\infty$. Here, type $F\_\infty$ means that the group admits a classifying space with compact skeleta.
Consider the semidirect product $B\_k\ltimes F^k$ where $B\_k$ is the braid group on $k$ strands. Think of its ele... | https://mathoverflow.net/users/27923 | Semidirect products with braid groups and type $F_\infty$ | In order to close this question, I cite YCor's answer in his comment above: One can replace $B\_k$ with the finite index subgroup $P\_k$ and in this case $P\_k\ltimes F^k=P\_k\times F^k$.
| 0 | https://mathoverflow.net/users/27923 | 180054 | 90,432 |
https://mathoverflow.net/questions/165746 | 26 | Set theory and model theory have many applications outside of logic, in particular in algebra, topology, analysis, ...
On the other hand model theory, in particular after Hrushovski, found many applications in algebraic geometry and Diophantine geometry.
**(A)** I wonder to know if there are any nontrivial applicat... | https://mathoverflow.net/users/11115 | Interactions between (set theory, model theory) and (algebraic geometry, algebraic number theory ,...) | **(C)**
Recently applied model theorists have touched many areas of algebra, algebraic geometry, number theory and even analysis structures.
**(1) Exponential fields:**
**Schanuel's conjecture** is a conjecture made by Stephen Schanuel in the **1960s**:
Given any $n$ complex numbers $z\_1,\dots,z\_n$ which are li... | 19 | https://mathoverflow.net/users/38966 | 180056 | 90,434 |
https://mathoverflow.net/questions/179012 | 8 | Let $\ M'\ M''\ $ be simply-connected Hausdorff compact manifolds (possibly with boundary for another variant of the question). Let $\ f:M'\rightarrow M''\ $ be a continuous function which induces an isomorphism of the cohomological rings. Does there exist a continuous function $\ g:M''\rightarrow M'\ $ which induces t... | https://mathoverflow.net/users/8385 | Inverse cohomological isomorphisms | The question has been answered in the comments. To justify writing this answer I'll sweeten it with some links and further details.
Compact manifolds (possibly with boundary) have the homotopy type of a CW-complex, see [the answers to this MO-question](https://mathoverflow.net/questions/156266/) which also provide l... | 5 | https://mathoverflow.net/users/50846 | 180057 | 90,435 |
https://mathoverflow.net/questions/180029 | 6 | Let $M$ be a compact 2-manifold of genus 2. Does there exist an orientation preserving homeomorphism $f:M\to M$, so that $f^n=id$ for some integer $n$, and $f$ doesn't have fixed points?
Using Lefschetz formula and the cohomology ring structure, I can prove that the order of such $f$ must be a multiple of 4 or 6. I ... | https://mathoverflow.net/users/37354 | Existence of orientation preserving, finite order self homeomorphism on a genus 2 surface without fixed point | The answer is yes, as it has been remarked in the previous answers. Here is a very explicit construction.
Let $T$ be the sphere $S^2 \subset \Bbb R^3$ with three disks removed. Take these three disks to be centered at the vertices of an equilateral triangle inscribed in the maximal circle $\{z = 0\} \cap S^2$ and wit... | 3 | https://mathoverflow.net/users/23193 | 180060 | 90,436 |
https://mathoverflow.net/questions/180058 | 13 | Let $n \geq 2$, $H \lneq (\mathbb{Z}/n\mathbb{Z})^\*$, $\zeta\_k$ a primitive $k$-th root of unity. Is it possible that $$\sum\_{h \in H} \zeta\_k^{h} \in \mathbb{Z}$$ for every $k$ dividing $n$ such that $n/k \leq |H|$?
I do not know the answer even if $k=n$. That is, can it be that $$\sum\_{h \in H} \zeta\_n^{h} \i... | https://mathoverflow.net/users/38889 | Can a sum of roots of unity be an integer? | **Below is the original and accepted proof, but see the by a factor of $100$ simpler later proof in the OP's answer!**
The answer is *yes* for the second question, with $n=8$, since $\zeta\_8+\zeta\_8^5=0$.
The answer is *no* for the first question. Let $\gamma\_k=\sum\_{h \in H} \zeta\_k^{h}$ be the sum from the... | 12 | https://mathoverflow.net/users/18739 | 180061 | 90,437 |
https://mathoverflow.net/questions/180065 | 8 | In a paper titled "RAMANUJAN’S UNPUBLISHED MANUSCRIPT ON THE PARTITION AND TAU FUNCTIONS WITH PROOFS AND COMMENTARY" by Bruce C. Berndt and Ken Ono, it is mentioned that Ramanujan derived the formula
\begin{align}\sum\_{n = 0}^{\infty}p(25n + 24)q^{n} &= 5^{2}\cdot 63\frac{(q^{5};q^{5})\_{\infty}^{6}}{(q;q)\_{\infty}^{... | https://mathoverflow.net/users/15540 | Generating function of $p(25n + 24)$ | Hirschhorn and Hunt have published a note [M. D. Hirschhorn; D. C. Hunt, A simple proof of the Ramanujan conjecture for powers of 5, J. Reine Angew. Math., 326 (1981), 1-17.](http://web.maths.unsw.edu.au/~mikeh/papers.html), where they give a proof (what they call "a simple proof") for the formula for the generating fu... | 12 | https://mathoverflow.net/users/32332 | 180067 | 90,440 |
https://mathoverflow.net/questions/180035 | 75 | In their 2001 paper defining periods, [Kontsevich and Zagier (pdf)](http://www.ihes.fr/~maxim/TEXTS/Periods.pdf) without further comment state that $e$ is conjecturally not a period while many other numbers showing up naturally (conjecturally) are. The former claim is repeated at many other internet sources including W... | https://mathoverflow.net/users/41139 | What are reasons to believe that e is not a period? | To my understanding, the reason is simple: in the almost 300 years since $e$ was discovered, no representation of it as a period has been found. I think this is quite a strong evidence.
Remark. To those who think that periods were introduced by Kontsevich and Zagier, I recommend the paper of Euler, On highly transcen... | 36 | https://mathoverflow.net/users/25510 | 180074 | 90,444 |
https://mathoverflow.net/questions/180052 | 1 | An associative $K$-algebra A is called reduced (or often basic) if $A/rad(A)$ has no nilpotent elements. It can be shown that this is equivalent to that $A/rad(A)$ is a isomorphic to a direct sum of division algebras. Here $rad(A)$ is the Jacobson radical of $A$.
In the article "Reduced group algebras" and a related ... | https://mathoverflow.net/users/57804 | Examples of reduced associative algebras | Let $Q$ be the quaternion group of order $8$ and let $P(Q)$ be the power set of $Q$ viewed as a monoid by $AB=\{ab\mid a\in A,b\in B\}$. Let $\mathbb QP(Q)$ be the contracted monoid algebra of $P(Q)$ over $\mathbb Q$ (so identify the zero of the monoid with the zero of the algebra). I claim that $\mathbb QP(Q)$ is redu... | 1 | https://mathoverflow.net/users/15934 | 180079 | 90,446 |
https://mathoverflow.net/questions/180027 | 1 | I want to try to implement a functional encryption scheme proposed in <http://eprint.iacr.org/2011/410>. The first problem I faced with is a TrapGen algorithm. In the paper theorem 3.1 states that:
>
> Let $q, n, m \in \mathbb Z$ with $q \geq 2$ and $m \geq 6n\log q$. There is a p.p.t. algorithm $\mathrm{TrapGen}(q... | https://mathoverflow.net/users/57518 | Functional Encryption for Inner Product Predicates | Your algorithm will not work, because simply choosing uniformly random $\mathbf{A}$ over $\mathbb{Z}\_q$ makes it computationally hard to find *any* short nonzero vector in $\Lambda^\perp(\mathbf{A})$ -- much less a full basis of short vectors.
The most efficient and implementation-friendly trapdoor generator is the ... | 4 | https://mathoverflow.net/users/57858 | 180099 | 90,455 |
https://mathoverflow.net/questions/180092 | 7 | Let $G$ be a finite perfect group. Are there any results on the number of elements of $G$ which can be written as a commutator? When $G$ is finite non-abelian simple group, then every element can be written as a commutator.
| https://mathoverflow.net/users/57278 | Percentage of elements that can be written as commutators | The proportion of elements of a finite group $G$ which are commutators is at least $\frac{1}{k(G)}$ (and the inequality is strict for non-Abelian groups) where $k(G)$ is the number of conjugacy classes of $G,$ because there are $|G|^{2}$ expressions of the form $[a,b]$ and every element of $G$ has at most $k(G)|G|$ exp... | 8 | https://mathoverflow.net/users/14450 | 180101 | 90,457 |
https://mathoverflow.net/questions/180093 | 3 | Let $M$ be a smooth compact closed manifold.
Let $u \in H^1(0,T;H^{-1}(M)) \cap L^2(0,T;H^1(M))$ be a solution of
$$u\_t - \Delta u - u = 0$$
$$u(0)=u(T)$$
satisfying $\int\_M u(t) = 0$ for all $t$. Is there any way to show that $u$ must be zero (i.e. solutions are unique)?
The problem is the $-u$ term. We have Poin... | https://mathoverflow.net/users/57856 | Uniqueness of weak solutions of a heat equation | Your claim holds true if and only if $-1$ is not an eigenvalue of $\Delta$.
If there is a nontrivial function $v:M\to\mathbb R$ with $\Delta v=-v$ (then necessarily $\int\_Mv=0$), then $u(x,t)=v(x)$ solves your PDE with all conditions but does not vanish identically.
Suppose then that $-1$ is not an eigenvalue.
The... | 1 | https://mathoverflow.net/users/55893 | 180104 | 90,459 |
https://mathoverflow.net/questions/180110 | 18 | Let $O\_n$ be the (real) orthogonal group of $n$ by $n$ matrices.
I am interested in the following sequence which showed up in a calculation I was doing
$$a\_k = \int\_{O\_n} (\text{Tr } X)^k dX$$
where the integral is taken with respect to the normalized Haar measure on $O\_n.$ I noticed some work of Dianconis et al. ... | https://mathoverflow.net/users/32470 | Moments of the trace of orthogonal matrices |
>
> [Pastur and Vasilchuk](https://link.springer.com/article/10.1007/s00220-004-1231-3) have extended the result of [Diaconis and Evans](https://statistics.stanford.edu/sites/default/files/2000-24.pdf) for $a\_{2k}$ from $2k\leq n/2$ to $2k\leq n-1$:
>
>
> $$a\_{2k}=\pi^{-1/2}2^{k}\Gamma(k+1/2)\;\;\text{for}\;\;2k\... | 15 | https://mathoverflow.net/users/11260 | 180112 | 90,461 |
https://mathoverflow.net/questions/180109 | 4 | Something came up yesterday in a referee request and I was surprised to find that I did not know the facts in full generality. This is about positive quadratic forms in three variables with integer coefficients, primitive, meaning the GCD of coefficients is 1.
In the case of the sum of three squares, if $R(n)$ is th... | https://mathoverflow.net/users/3324 | Simple comparison of positive ternary quadratic form representation counts | No, this isn't true in general. You could, for example, take $f(x,y,z) = x^{2} + y^{2} + 13z^{2}$. Then $R(1) = 4$ and $R(25) = 12$. The form is isotropic at $5$ (since $5$ doesn't divide the discriminant). Why might you get the idea that it would be true based on the sums of three squares form? The reason is that the ... | 7 | https://mathoverflow.net/users/48142 | 180113 | 90,462 |
https://mathoverflow.net/questions/180121 | 1 | Let $f:X\rightarrow Y$ be a morphism of schemes and let $\mathcal{F}$ be a sheaf on $Y$. Then there is a natural map $$\Psi:\mathcal{F} \rightarrow f\_{\ast}f^{\ast}(\mathcal{F})$$ and localizing $$\Psi\_p:\mathcal{F}\_p \rightarrow f\_{\ast}f^{\ast}(\mathcal{F})\_p=f\_{p,\ast}f\_p^{\ast}(\mathcal{F}\_p)$$
I'm intere... | https://mathoverflow.net/users/54631 | Sheaf isomorphism | If $f\_\*O\_X=O\_Y$ then this is true for any locally free (or flat) sheaf (and if $f$ is flat then for any quasicoherent sheaf) by projection formula.
| 4 | https://mathoverflow.net/users/4428 | 180128 | 90,466 |
https://mathoverflow.net/questions/180129 | 2 | Let $K$ be a global field and set $O := \prod\_{v\nmid \infty} O\_v$ where $v$ runs over the finite places of $K$. Equip $\mathrm{GL}\_n(O) = \prod\_v \mathrm{GL}\_n(O\_v)$ with the product of the $v$-adic topologies and suppose that $H \le \mathrm{GL}\_n(O)$ is a subgroup of finite index.
1. Is $H$ necessarily open... | https://mathoverflow.net/users/53197 | Openness of finite index subgroups of $\mathrm{GL}_n(\prod O_v)$ | The answer to 1 is no. For example each $GL\_n(O\_v)$ has a continuous map to $\{ \pm 1\}$ so your product has a quotient isomorphic to $F\_2^{\mathbf{N}}$. Zorn's lemma will give you a non continuous linear form on that, and the pullback of the quotient is a non closed subgroup of index $2$.
Edit: the pullback of t... | 8 | https://mathoverflow.net/users/5743 | 180134 | 90,468 |
https://mathoverflow.net/questions/180102 | 2 | I was reading a paper and they mentioned the Rosati form. Particularly, what they said was:
Let $A$ be an abelian surface defined over $k$ such that $ST\_A^0$ (the connected component of the Sato-Tate group) $ = U(1) \times U(1)$. Then the matrices in $M\_4(\mathbb{C})$ commuting with $U(1) \times U(1)$ are
$$\Bigg... | https://mathoverflow.net/users/57859 | What is Rosati Form | the Rosati form is defined in definition 2.18 of [*Sato-Tate distributions and Galois endomorphism modules in genus 2*](http://arxiv.org/abs/1110.6638)
| 7 | https://mathoverflow.net/users/11260 | 180135 | 90,469 |
https://mathoverflow.net/questions/180046 | 11 | The sum of the degrees of the irreducible complex characters (not the square sum which is the group order) is relevant to determine the dimension of a maximal torus of the Lie algebra associated to the group algebra of a finite group over the complex field.
I have investigated certain group classes as extra-special g... | https://mathoverflow.net/users/57804 | Sums of degrees of irreducible complex characters | I just realised that a result of mine ("On the minimal norm of a non-regular generalized character of a finite group" Bull LMS 2010) can be used to give a slightly better upper bound for the sum of the irreducible character degrees of finite non-Abelian group $G$.
For such a group $G,$ with irreducible characters $\ch... | 6 | https://mathoverflow.net/users/14450 | 180146 | 90,474 |
https://mathoverflow.net/questions/180164 | 0 | Suppose $f(t,x)\in \mathcal{C}^0([0,1]\times \mathbb{R}^n)$. Further suppose that for each $t$
$$ C(t):= \sup\_{x\in\mathbb{R}^n} |f(t,x)|<\infty \, .$$
Does it follow that $f$ is bounded?
Note that $C(t)$ need not be continuous as is evidenced by $f(t,x)= \sin(t x)$.
| https://mathoverflow.net/users/57697 | Extending point-wise bound to uniform bound | It does not. Start from any non-zero continuous function $\phi$ with compact support in $\mathbb{R}\setminus\{0\}$ and define
$f(t,x)=\phi(tx)/t$ for $t>0$ and $f(0,x)=0$.
| 3 | https://mathoverflow.net/users/6101 | 180166 | 90,476 |
https://mathoverflow.net/questions/180149 | 7 | There is a theorem :
1) 2-dim (pseudo-)Riemannian manifold must be local conformal flat;
2) 3-dim (pseudo-)Riemannian manifold is local conformal flat iff the Cotton tensor vanishes.
3) n-dim (n>3) (pseudo-)Riemannian manifold is local conformal flat iff the Weyl tensor vanished.
Then I'm curious about the nece... | https://mathoverflow.net/users/43941 | The necessary and sufficient condition for $\textbf{global}$ conformal flatness of a n-dim (pseudo-)Riemannian manifold | The questions you are asking have been carefully studied in the literature, but, usually, with carefully chosen global hypotheses so that reasonable results can be achieved. You should look at the works of William Goldman, Mischa Gromov, and Scott Adams, just to name a few, as well as more recent authors studying parab... | 7 | https://mathoverflow.net/users/13972 | 180174 | 90,479 |
https://mathoverflow.net/questions/179444 | 2 | On $X=CP^2\#k{(-CP^2)}$ in $k$ generic points, let $h^i=\dim H^i(T^{1,0}X)$, for $i\ge 0$. First, we know $h^i=0$ for $i\ge 2$. By Riemann–Roch formula, I obtain that $h^0-h^1 = 8-2k$. Would someone be kind to show me how to compute the general formula for $h^0$? Thank you in advance.
| https://mathoverflow.net/users/40220 | Holomorphic vector fields on blow-ups of CP^2 | On CP^2, assume the cover is given by $U\_i=\{[z\_0, z\_1, z\_2]| z\_i\ne 0\}$ where $i=0,1,2$, then one can compute that on $U\_0$, $H^0(T^{1,0}CP^2)$ is spanned by $\frac{\partial}{\partial z\_1}$, $\frac{\partial}{\partial z\_2}$, $z\_1\frac{\partial}{\partial z\_1}$, $z\_1\frac{\partial}{\partial z\_2}$, $z\_2\frac... | 0 | https://mathoverflow.net/users/40220 | 180175 | 90,480 |
https://mathoverflow.net/questions/180156 | 8 | This question is about $\ell$-adic monodromy theorems for families over a number field. ($\ell$-adic analogues of Corollaries 6.2.8 and 6.2.9 in [BBD].)
Notation
========
$H$ denotes étale cohomology.
Let $f \colon X \to Y$ be a proper morphism of finite type schemes over $\mathbb{C}$.
Let $\mathcal{F}$ be in $\m... | https://mathoverflow.net/users/21815 | $\ell$-adic monodromy theorems (over $\mathbb{C}$) | The sages (for whom I am but an unworthy mouthpiece) say:
Tell the OP to look at 6.2.9 in Deligne’s Weil II paper, it’s quite close to answering
what he is asking in his Q2.
| 8 | https://mathoverflow.net/users/11142 | 180184 | 90,483 |
https://mathoverflow.net/questions/180185 | 5 | Let $K$ be a field of characteristic $0$, $X\_K$ a smooth projective curve over $K$. Denote by $\bar{K}$ the algebraic closure of $K$. The base change morphism $X\_{\bar{K}} \to X\_K$, induces via the pull-back map, a morphism of the Picard groups, $\mbox{Pic}(X\_K) \to \mbox{Pic}(X\_{\bar{K}})$. Is this map injective?... | https://mathoverflow.net/users/54369 | Injectivity under flat base change of the Picard group on smooth projective curves | This map is injective. There is a Hochschild-Serre spectral sequence with $E^{pq}\_2=H^p(\mathrm{Gal}(\bar{K}/K), H^q(X\_{\bar{K}},\mathbb{G}\_m))$ converging towards $H^{p+q}(X\_{K},\mathbb{G}\_m)$. This gives a first terms exact sequence
$$0\rightarrow H^1(\mathrm{Gal}(\bar{K}/K), \bar{K}^\*)\rightarrow \mathrm{Pic}(... | 7 | https://mathoverflow.net/users/40297 | 180189 | 90,486 |
https://mathoverflow.net/questions/180120 | 2 | This is a follow up of [this question](https://mathoverflow.net/questions/178276/r-is-a-right-multiplier-and-rab-a-overset-implies-a-is-unital). Let $I$ be closed left ideal of $C^\*$-algebra $A$.
Assume we are given a sequence of left $A$-module morphisms $R\_n:I\to A$ with $\sum\_n \Vert R\_n\Vert<\infty$ and a seq... | https://mathoverflow.net/users/19593 | Is this left ideal of C*-algebra principal? | Yes, $I=Ap$ for some projection $p$. More generally, suppose that a closed left ideal
$I$ contains a sequence $c\_1,c\_2,\dots$ such that
$
\sum\_{i=1}^\infty \|c\_i\|<\infty
$
and for any $x\in I$ there exist $x\_1,x\_2,\dots$ such that $\|x\_i\|\leq \|x\|$ and
$$
\sum\_{i=1}^\infty x\_ic\_i=x.
$$
Then $I$ is gener... | 3 | https://mathoverflow.net/users/13381 | 180193 | 90,488 |
https://mathoverflow.net/questions/180191 | 8 | A colleague in my department posed the following question...
Let $A=(0,0)$, $B=(1,0)$, and $C=(1/2,\sqrt{3}/2)$. Then $\Delta ABC$ is an equilateral triangle with sides of length 1. Let $B\_{\epsilon}({\bf x}) = \{ {\bf y} \in \mathbb{R}^2 \;|\; \mbox{distance}({\bf y},{\bf x})<\epsilon \}$ be an epsilon ball centere... | https://mathoverflow.net/users/17263 | Rational distance from vertices of an equilateral triangle | Yes. In fact, more is true. T. G. Berry (["Points at rational distance from the vertices of a triangle"](http://matwbn.icm.edu.pl/ksiazki/aa/aa62/aa6246.pdf), *Acta Arith.* **62** (1992), no. 4, 391–398) proved:
>
> **Theorem.** Let
> $ABC$
> be a triangle such that the length of at least one side
> is rational ... | 15 | https://mathoverflow.net/users/3106 | 180195 | 90,489 |
https://mathoverflow.net/questions/180202 | 1 | Let $M$ be a hypersurface in $\mathbb{R}^{n+1}$ with bounded second fundamental form
$|A|\leq C$. Does intrinsic distance satisfy $d\_g(p,q)\leq C'|p-q|$, where $C'$ only depends on $C$. Here $d\_g(p,q)$ is the intrinsic metric on $M$, and $|p-q|$ is the extrinsic metric in $\mathbb{R}^{n+1}$.
| https://mathoverflow.net/users/48506 | Is the extrinsic metric and intrinsic metric equivalent if second fundamental form is bounded | No, this is not true.
For $L>0$, consider two lines in $\mathbb{R}^2$:
$\{(x,0) : 0 \leq x \leq L\}$
and
$\{(x,2) : 0 \leq x \leq L\}$.
These can be "capped" at either end with half-circles of radius $1$. This produces a closed curve in $\mathbb{R}^2$. If you want, you can smooth things to make the curve $C^... | 5 | https://mathoverflow.net/users/1540 | 180205 | 90,494 |
https://mathoverflow.net/questions/180199 | 1 | Let $K$ be a field of characteristic zero and $X\_K$ a smooth projective curve on $K$. Denote by $\bar{K}$ the algebraic closure of $K$ and $X\_{\bar{K}}$ the base change of $X\_K$ to $\bar{K}$. Under the natural morphism from $X\_{\bar{K}} \to X\_K$, we have an induced morphism $\mbox{Pic}^0(X\_K) \to \mbox{Pic}^0(X\_... | https://mathoverflow.net/users/54369 | Jacobian of a curve and field extension | First of all, the image of your homomorphism is invariant under the Galois group $G:=\mathrm{Gal}(\bar{K}/K)$. So the right question is to ask whether the induced homomorphism $\mathrm{Pic}(X\_{K})\rightarrow \mathrm{Pic}(X\_{\bar{K}})^G$ is surjective.
In our answer to [this question](https://mathoverflow.net/questi... | 5 | https://mathoverflow.net/users/40297 | 180212 | 90,496 |
https://mathoverflow.net/questions/180228 | 1 | Let $X\subset\mathbb{P}^n$ be an hypersurface given by the vanishing of a polynomial $F\in k[x\_0,...,x\_n]\_d$. Let $Y\subset X$ be a subvariety. Then $X$ has multiplicity $m$ along $Y$ if all the partial derivatives of order $m-1$ of $F$ vanish on $Y$ but there is at least one partial derivative of $F$ of order $m$ n... | https://mathoverflow.net/users/nan | Multiplicity of a variety along a subvariety | The comment of Joe Silverman is correct. You have to consider the order of vanishing of $r\times r$ minors of $Jac(F\_1,...,F\_r)$. For instance, consider $F\_1 = xw-yz$ and $F\_2 = w$ the intersection of a quadric with its tangent plane in $[1:0:0:0]\in\mathbb{P}^3$. Note that $\frac{\partial F\_2}{\partial w} \neq 0$... | 1 | https://mathoverflow.net/users/14514 | 180234 | 90,499 |
https://mathoverflow.net/questions/180236 | 15 | The only really "economic" cell structures for $K(\pi,n)$'s that I know is the one with a single cell in each dimension for $K(\mathbb Z/n\mathbb Z,1)$ and the one with a single cell in each even dimension for $K(\mathbb Z,2)$.
Vaguely I remember attending a talk where some lower bounds on numbers of cells in each di... | https://mathoverflow.net/users/41291 | "Economic" CW-structure for Eilenberg-MacLane spaces? | You can build them using a homology decomposition and read off the number of cells in each dimension from the homology groups.
For any simply-connected space, this will give you a construction by iterated cofiber sequences $M\_n \to X(n-1) \to X(n)$ where $M\_n$ is a Moore space and $X = \mathrm{colim} X(n)$.
If you... | 10 | https://mathoverflow.net/users/3634 | 180242 | 90,502 |
https://mathoverflow.net/questions/180237 | 4 | Empirically, the Legendre functions of second kind, $Q\_n(x)$, appear to be of form
$$
Q\_n(x)=\frac{P\_n(x)}{2} \cdot\ln(\frac{1+x}{1-x})+p\_n(x),
$$
with $P\_n(x)$ the Legendre polynomials of first kind and $p\_n(x)$ some rational polynomial of degree $n-1$.
This observation came up with my current reimplementatio... | https://mathoverflow.net/users/41285 | Legendre Q(n,x) function coefficients in terms of P(n,x) coefficients | The formular you search for is really known. It is on p. 360 of the NIST Handbook of Mathematical Functions, formulas (14.7.2)--(14.7.7). $p\_n(x)$ is in fact an explicit polynomial (not rational) with coefficients depending on $\psi$--function.
| 3 | https://mathoverflow.net/users/49208 | 180257 | 90,509 |
https://mathoverflow.net/questions/180251 | 3 | As far as I understand, *embedded resolution of singularities* means the following: given a variety $X$ over an algebraically closed field, and a closed subvariety $Y$, there exists a birational map $f:X'\to X$ such that $X'$ and the proper transform $Y'$ of $Y$ are smooth. Sometimes a bit more is required (e.g. $Y'$ t... | https://mathoverflow.net/users/3847 | Embedded resolution of curves on smooth varieties | This should be fine at least for the weaker notion you mention. One can always resolve the singularities of a curve by blowing up a sequence of closed points (on the curve). Hence if you embed the curve in a regular ambient space $X$, you can always do the following. Choose the point on $Y \subseteq X$ you were going t... | 4 | https://mathoverflow.net/users/3521 | 180259 | 90,510 |
https://mathoverflow.net/questions/180169 | 0 | Can anyone give me suggestions for new books about Besicovitch's almost periodic functions? Thanks a lot.
| https://mathoverflow.net/users/57892 | A book about almost periodic functions | What do you mean "a new", of what years?
Classics are always new-the books of Bohr, Levitan and Zhikov, Besikovich himself.
There is also a book of Corduneanu with standard name "almost periodic function" of 1989. Look also for Diagana 2013, Albrecht Böttcher et al 2002, Gaston M. N’Guerekata 2001, Wolfgang Schwarz 199... | 2 | https://mathoverflow.net/users/49208 | 180261 | 90,511 |
https://mathoverflow.net/questions/180231 | 12 | Given integers $a, b, c$ all at least 2, I would like to identify the smallest group with an element $x$ of order $a$, element $y$ of order $b$ such that the product $yx$ has order $c$. For example, if $(a,b,c)=(2,3,7)$ the smallest group with elements $x, y, yx$ of orders $2,3,7$, respectively, is the simple group of ... | https://mathoverflow.net/users/11124 | What are smallest finite images of triangle groups? | Here's an elaboration of my comment above in the pairwise coprime case, in an even more special case (the fact that in a finite solvable group we can't have three elements of pairwise coprime orders whose product is the identity is, I believe, due to P. Hall). If one insists, as in the question, on the minimality of th... | 9 | https://mathoverflow.net/users/14450 | 180268 | 90,514 |
https://mathoverflow.net/questions/119082 | 2 |
>
> BIG EDIT of the previous question "Coverings of the free Burnside groups", never answered.
>
>
>
In the paper <http://link.springer.com/article/10.1007/BF00046586> (last section) there is an interesting statement concerning the finiteness of the 2-generator Burnside groups $B(2,n)$ (let's assume $n\geq 4$). ... | https://mathoverflow.net/users/22606 | Can the finiteness of a Burnside group with two generators be checked algorithmically by using Fuchsian von Dyck groups? | The answer to your Question 1 is certainly 'yes', for the reason you already explained: $\mathbb{H}^2/K\_4$ is a Riemann surface, and its quotient by the finite group $B(2,4)$ is compact.
I don't know about Question 2, and I haven't looked at the paper that you link to, but it seems worth pointing out that the algori... | 1 | https://mathoverflow.net/users/1463 | 180284 | 90,520 |
https://mathoverflow.net/questions/180241 | 6 | Let $q$ be a power of $p$, let $l$ be a prime different from $p$, and let $H\_1$ and $H\_2$ be two subgroups of $GL\_n(\mathbb F\_q)$ that are $l$-groups.
If for all characteristic $0$ representations $V$ of $GL\_n(\mathbb F\_q)$, we have
$$\dim V^{H\_1} \leq \dim V^{H\_2}$$
does it follow that:
$$\dim \left(\m... | https://mathoverflow.net/users/18060 | An inequality on representations and subgroups of general linear groups over finite field | The following is a series of counterexamples: Let $q$ be a prime power with $q\equiv1\pmod{8}$, and pick $\omega\in\mathbb F\_q^\star$ with order $8$. Set $G=\text{GL}\_2(\mathbb F\_q)$, and let $H\_1$ and $H\_2$ be the cyclic subgroups generated by $\begin{pmatrix}\omega & 0\\0 & 1\end{pmatrix}$ and $\begin{pmatrix}\o... | 6 | https://mathoverflow.net/users/18739 | 180287 | 90,521 |
https://mathoverflow.net/questions/180275 | 3 | Is the property of left invariant orderability for finitely generated groups preserved by quasi-isometrics? More precisely, if $G$ is a left orderable (finitely generated) group and $H$ is a torsion-free group quasi-isometric (in the sense of Gromov) to $G$, can we conclude $H$ is left orderable?
If the answer is no i... | https://mathoverflow.net/users/38190 | Quasi-isometry and left invariant orderability for groups | This answer adds some details, and some references, following Ian's comment.
First the references:
<https://arxiv.org/abs/math/0211110>,
<https://arxiv.org/abs/math/0203192v2>
Now, for the answer: In the Boyer, Rolfsen, Wiest paper (first reference) we find:
>
> Theorem 1.8 For each of the eight 3-dimensio... | 6 | https://mathoverflow.net/users/1650 | 180288 | 90,522 |
https://mathoverflow.net/questions/179618 | 15 | I am interested in realizing linear orders as orderings of abelian groups. In particular, can Suslin lines (and other classes of line) be realised in this way?
Let $\mathcal{C}$ be a class of (torsion-free) abelian groups, and $L$ be the first-order language of linear orderings. Let $LO(\mathcal{C})$ be the theory of... | https://mathoverflow.net/users/57583 | Can Suslin (or Aronszajn) lines ever be orderings of abelian groups? | Let me give a simple direct proof that there is no ordered abelian group $\langle S,+,<\rangle$ whose order type is a Suslin line (Dedekind-complete, densely ordered, and without a countable dense subset; the argument doesn’t need ccc).
Since $S$ is nontrivial and densely ordered, we can find a strictly decreasing se... | 9 | https://mathoverflow.net/users/12705 | 180294 | 90,523 |
https://mathoverflow.net/questions/180300 | 12 | The [heuristic](http://www.jstor.org/stable/2152950) from circle method for integral points on diagonal cubic surfaces $x^3+y^3+z^3=a$ ($a$ is a cubic-free integer) seems to fit well with numerical computations by [ANDREAS-STEPHAN ELSENHANS AND JORG JAHNEL](http://www.ams.org/journals/mcom/2009-78-266/S0025-5718-08-021... | https://mathoverflow.net/users/18286 | Failing of heuristics from circle method | The heuristic fails for precisely the reason you state; there is a parametric family of solutions which makes it fail.
People often use heuristic arguments to predict the number of integral/rational solutions of bounded height to diophantine equations. These heuristics are often true, provided one allows a small marg... | 18 | https://mathoverflow.net/users/5101 | 180302 | 90,524 |
https://mathoverflow.net/questions/168482 | 4 | I'm very sorry if this is the wrong place to ask this question, but I've asked it on StackExchange and received no answers. ( <https://math.stackexchange.com/questions/813063/convergence-to-a-schwartz-distribution> )
Let $M$ be a smooth manifold with countable atlas, and define the distributions $\mathscr{D}'(M)$ as ... | https://mathoverflow.net/users/51382 | Extension of pseudodifferential operators | Your operator $A$ is continuous from $C\_0^\infty(M)$ into $C^\infty(M)$, so the adjoint $A^\*$
is continuous from $\mathcal E'(M)$ into $\mathscr D'(M)$. Now, the operator $A^\*$ is also a pseudodifferential operator of the same order than $A$, whose principal symbol is the complex conjugate of the symbol of $A$: this... | 3 | https://mathoverflow.net/users/21907 | 180307 | 90,526 |
https://mathoverflow.net/questions/180289 | 9 | I would like to know what a Penrose transform is, or more precisely, what is it intended to be - I'm interested in ideas, intuition and some examples of application.
My knowledge of differential geometry and homological algebra are basic (at the moment, I am working on a graduation paper on derived categories), so plea... | https://mathoverflow.net/users/45668 | Idea and intuition behind Penrose transform | well, if you are interested in developing some basic intuition on the Penrose transform, you could go back to Harry Bateman's 1904 paper *The solution of partial differential equations by means of definite integrals*, where he derived what is essentially the same representation of a harmonic function $\phi(w,x,y,z)$ of... | 5 | https://mathoverflow.net/users/11260 | 180311 | 90,527 |
https://mathoverflow.net/questions/175872 | 3 | According to several articles I could find, a canonical algebra of type $(2,2,r)$ is derived equivalent to a path algebra of type $\tilde{D}\_{r+2}$, where $r \geq 2$.
I don't know how to obtain this statement. The Happel-Rickard-Schofield theorem entails that the previous statement is equivalent to saying that $C(2... | https://mathoverflow.net/users/34106 | A canonical algebra of type $(2,2,r)$ is derived equivalent to a path algebra of type $\tilde{D}_{r+2}$ (references) | This result is attributed to Thomas Hübner in Thm. 3.5 of Lenzing's ICRA 2010 proceedings "Weighted projective lines and applications", available [here](http://www.ems-ph.org/books/show_abstract.php?proj_nr=135&vol=1&rank=5 "here") with subscription and [here](http://books.google.co.jp/books?id=WrgIgVpJnrcC&lpg=PA153&d... | 3 | https://mathoverflow.net/users/10724 | 180338 | 90,536 |
https://mathoverflow.net/questions/180339 | 1 | Numerous papers are referring to the following one
*R. M. Blumenthal and R. K. Getoor, Sample functions of stochastic processes with stationary independent increments, J. Math. Mech. 10 (1961), 493–516*
in the literature, where they authors define what is now referred as the Blumenthal-Getoor index. However, I coul... | https://mathoverflow.net/users/39261 | Reference request: seminal paper on the Blumenthal-Getoor index | The journal changed name and Mathscinet doesn't yet link to its archive -- where your paper is [freely available](http://www.iumj.indiana.edu/IUMJ/fulltext.php?year=1961&artid=10031&volume=10).
| 2 | https://mathoverflow.net/users/19276 | 180340 | 90,537 |
https://mathoverflow.net/questions/173589 | 6 | Higman's Criterion is a fundamental result in the representation theory of a finite group $G$ over a field $k$ of characteristic $p>0$. One version is the following (I hope the notation is standard enough):
Let $M$ be a $kG$-module and let $H$ be a subgroup of $G$. Then $M$ is a direct summand of ${\rm Ind}\_H^G\,{\r... | https://mathoverflow.net/users/20764 | Higman's Criterion | This has the rather easy answer: No!
Let $G$ be a finite group of order divisible by $p$ that has a $p$-block $B$ of defect $0$ and let $M$ be the unique simple $kG$-module in $B$. Take $H$ to be any nontrivial $p$-subgroup of $G$. Then $k\_H$ is a submodule of Res$\_H^G(M)$, as $k\_H$ is the unique simple $kH$-modul... | 2 | https://mathoverflow.net/users/20764 | 180348 | 90,538 |
https://mathoverflow.net/questions/180353 | 1 | I am reading the book Index Theorem and the Heat Equation written by Peter.B.Gilkey. Here is my question:
Let E be a hermitian vector bundle on a compact smooth manifold M. Let $D : \mathcal{C}^{\infty}(E) \rightarrow \mathcal{C}^{\infty}(E)$ be an elliptic operator of positive order. Then it extends to the completion ... | https://mathoverflow.net/users/57965 | Elliptic operator are unbounded | The fact that (non-trivial) elliptic operators are not bounded on $L^2$ is a special case of the fact that differential operators generally are not continuous (i.e., not bounded) on $L^2$. This much has little to do with the compactness of the manifold.
Note that such operators do not extend "to the completion of $C^... | 3 | https://mathoverflow.net/users/15629 | 180354 | 90,542 |
https://mathoverflow.net/questions/180319 | 2 | *Rounge's Theorem* states that any meromorphic function on a domain inside $\mathbb{C}$ can be approximated (over compact subsets) by a sequence of rational functions (meromorphic functions on $\mathbb{C}P^1$).
Donaldson wrote a small paper *"Approximation of Instantons"* based off of Taubes' gluing construction for... | https://mathoverflow.net/users/12310 | Approximation theorem for Anti-Self-Dual Metrics | I think that your point 2) is more or less correct. In his proof Taubes first constructs approximately ASD metrics $g\_N$ on $X \#\_N CP^2$, starting with a metric $g$ on $X$, and this step would not touch the metric on $U$ if $g$ is already ASD there. These will be metrics with
$$ \lim\_{N\to \infty} \Vert W^+(g\_N)\... | 3 | https://mathoverflow.net/users/57966 | 180356 | 90,543 |
https://mathoverflow.net/questions/180359 | 6 | Let $G$ be a discrete, finitely generated, and amenable group. Let $H$ be a group which is quasi- isometric to $G$. Is $H$ amenable?
| https://mathoverflow.net/users/21985 | Amenability as a geometric property | Yes. This is essentially an immediate consequence of the [Folner sets](http://en.wikipedia.org/wiki/F%C3%B8lner_sequence) definition of amenability. You can find references at Theorem 10.23 of the article by Ghys and de la Harpe, "[Infinite groups as geometric objects](http://perso.ens-lyon.fr/ghys/articles/infinitegro... | 11 | https://mathoverflow.net/users/20787 | 180363 | 90,544 |
https://mathoverflow.net/questions/180203 | 5 | At
[Simple comparison of positive ternary quadratic form representation counts](https://mathoverflow.net/questions/180109/simple-comparison-of-positive-ternary-quadratic-form-representation-counts)
Jeremy answered:
"The reason is that the theta series for the sums of three squares form is an eigenfunction for all the... | https://mathoverflow.net/users/3324 | Ternary quadratic form theta series as Hecke eigenforms and class number one | There are many specific questions raised in this post. I will address the main one and show that if a $Q$ is a ternary quadratic form and the theta series $\theta\_{Q}$ is a Hecke eigenform, then $Q$ has class number one.
The form $\theta\_{Q}$ is a weight $3/2$ modular form and any such form has a decomposition as $... | 6 | https://mathoverflow.net/users/48142 | 180366 | 90,546 |
https://mathoverflow.net/questions/180334 | 1 | I first asked the question below at math.stackexchange.com ( <https://math.stackexchange.com/questions/920442/number-of-points-in-an-intersecting-linear-hypergraph> ) but somebody suggested I ask it in mathoverflow.net instead. Here's the question:
An *intersecting linear hypergraph* is a pair $H=(P,\mathcal{L})$ whe... | https://mathoverflow.net/users/8628 | Number of points in an intersecting linear hypergraph | The usual proof of Fisher's inequality is very finite. I will call things in $\mathcal{P}$ points and those in $\mathcal{L}$ lines although the reverse might be more natural. Let $L=|\mathcal{L}|,P=|\mathcal{P}|$ and let line $\ell\_i$ have $c\_i+1$ points. Consider $A$, the $L \times P$ $0,1$ matrix giving incidence b... | 1 | https://mathoverflow.net/users/8008 | 180368 | 90,547 |
https://mathoverflow.net/questions/180372 | 2 | Is there a way to compute/estimate the following integral?
$\int\_0^\infty e^{-(x/c)^2}\left|\cos{x}\right|dx$
where $c$ is a real constant. I would like to know if it is of order $e^{-c^2/4}$ like the integral without absolute value. Or is there a sharp upperbound?
Thank you very much in advance.
| https://mathoverflow.net/users/57976 | Integration of gaussian times absolute value of cosine | I presume you're interested in $c \to \infty$. Without absolute value you get
$\dfrac{\sqrt{\pi}}{2} c \exp(-c^2/4)$, which is not **quite** of order $\exp(-c^2/4)$. With absolute value, a lower bound is
$$ \int\_0^\infty \exp(-(x/c)^2) \cos^2 x\; dx = \sqrt{\pi} c (1 + \exp(-c^2))/4$$
so it is not even $O(1)$. An upp... | 6 | https://mathoverflow.net/users/13650 | 180377 | 90,552 |
https://mathoverflow.net/questions/180378 | 0 | This question originates an engineering application.
There is a certain process that is presumed to be a sequence of diffusions and is usually modelled as a sum of Gaussians:
$$\Sigma\_n w\_ne^{-\frac{x^2}{\sigma\_n^2}}$$
It is known that the function $f(x)$ that is being approximated is monotone exponentially d... | https://mathoverflow.net/users/38448 | Can a monotone exponentially decreasing function be uniformely approximated bt Gaussians? | You want the $w\_n$ to be nonnegative, right?
With that interpretation, the answer is no. Just substitute $x^2=y$, and the problem becomes one of approximating by a sum of decaying exponentials. Such functions are well studied; they are called completely monotone.
| 5 | https://mathoverflow.net/users/12120 | 180379 | 90,553 |
https://mathoverflow.net/questions/176411 | 3 | I am trying to formulate the measure of event
$E=\{B[0\infty)\cap A,B \neq \varnothing$ and $B[0\infty)\cap C= \varnothing\}$,
where $B[0\infty)$ is a Brownian path and $A,B,C$ are pairwise disjoint compact non-empty sets.
Q1:
I am looking for $\mu\_{W}(E)$. Any solutions?
One guess is:
$\mu\_{W}(E)=\int\_... | https://mathoverflow.net/users/40793 | Wiener measure of hitting sets A,B but not C (or easier hitting A but not C) | This is a variation on [Brownian motion and hitting a Quadrilateral](https://mathoverflow.net/questions/180360) and can be reduced in a similar way to the solution of a problem in electrostatics. Let me assume that the Brownian motion starts at the origin and ask for the probability $P\_{A|C}$ that the particle eventua... | 1 | https://mathoverflow.net/users/11260 | 180388 | 90,556 |
https://mathoverflow.net/questions/171499 | 7 | Let $(X,\mu)$ be a standard probability space, and $(Y,\tau)$ an uncountable Polish space. Then the set $L^0(X,\mu,Y)$ of measurable maps from $X$ to $Y$ identified up to measure 0 is Polish w.r.t. the topology of convergence in measure.
It is then not hard to see that the subset of maps $X\to Y$ with countable rang... | https://mathoverflow.net/users/52833 | Is the set of measurable maps with countable range Borel? | The set of maps in $L^0(X,\mu,Y)$ with countable range is always Borel, and the way to see it is to correctly reformulate the question: we are looking at the set of elements $f\in L^0(X,\mu,Y)$ such that the pushforward measure $f\_\*\mu$ is completely atomic.
Now, if we let $(A\_s)\_{s\in \mathbb N^{<\mathbb N}}$ b... | 1 | https://mathoverflow.net/users/52833 | 180389 | 90,557 |
https://mathoverflow.net/questions/180387 | 6 | How many binary cyclic sequences of length $n$ exist, where ones only appear in blocks of length at least $k$? We do **not** consider sequences that result from each other by a cyclic shift equivalent.
**Example:** Let $n=6$ and $k=2$, i.e. we have no isolated one. Then
[0,0,0,0,0,0]
[1,1,0,0,0,0] and 5 cyclic ... | https://mathoverflow.net/users/57980 | Counting cyclic binary sequences of length $n$ where ones appear in blocks of length at least $k$ | Consider the monoid $M\_k$ in the generators $A\_k=\lbrace 0, 01^k,
01^{k+1}, 01^{k+2},...\rbrace$, where $1^r$ denotes a string of $r$
ones. This contains all binary sequences with no strings of 1's of
length less than $k$, except for the words $1^r$ for $r\geq k$. The
monoid $M\_k$ is freely generated by $A\_k$. Henc... | 6 | https://mathoverflow.net/users/2807 | 180404 | 90,563 |
https://mathoverflow.net/questions/180400 | 17 | The (general) analytic class number formula gives a value for the residue of the Dedekind zeta function of a number field at the point $s=1$ (or, as I prefer, the leading Taylor coefficient at $s=0$). To whom should this formula be attributed?
My usual go-to place for such history questions is Narkiewicz's book *Elem... | https://mathoverflow.net/users/1021 | History of the analytic class number formula | Do you insist that the formula be interpreted as the value of a residue, hence requiring that it be known that the zeta-function of every number field is meromorphic around $s = 1$? It goes back to Dedekind that for every $K$ the limit $\lim\_{s \rightarrow 1^+} (s-1)\zeta\_K(s)$ exists and is given by the standard for... | 24 | https://mathoverflow.net/users/3272 | 180408 | 90,565 |
https://mathoverflow.net/questions/180391 | 4 | Does any one know how to write the Maxwell equations as an equation on a principal $U(1)$-bundle?
In Freed & Uhlenbeck's *Instantons and Four manifolds*, the authors claim that the Maxwell equations can be written in a gauge theoretic way: the electro-magnetic field can be viewed as the curvature $F$ of a $U(1)$-bun... | https://mathoverflow.net/users/37354 | Gauge-theoretic formulation of Maxwell equations | I don't have my copy handy, but I think this is all worked out in chapter 2 of Naber's "Topology, Geometry, and Gauge Fields: Interactions". The book is essentially a textbook on differential geometry with a view toward physics, so the exposition is very detailed; I just can't remember how much time is spent on Maxwell... | 4 | https://mathoverflow.net/users/4362 | 180416 | 90,569 |
https://mathoverflow.net/questions/180405 | 1 | Let $\Omega \subset \mathbb{R}^d$ be compact and convex, and let $f \in C^{k+1}(\Omega)$. Let $P\_k$ be the set of multivariate polynomials up to degree $k$ on $\Omega$. I am looking for any results in the literature that allow us to estimate
$$ \inf\_{p \in P\_k} \|f - p\|\_{L^2(\Omega)}$$
in terms of norms on the $(k... | https://mathoverflow.net/users/7378 | Error estimates for $L^2$ approximation of multivariate $C^k$ functions in terms of norms on derivatives | Check out Bramble-Hilbert Lemma: <http://en.wikipedia.org/wiki/Bramble%E2%80%93Hilbert_lemma>
| 1 | https://mathoverflow.net/users/35593 | 180428 | 90,571 |
https://mathoverflow.net/questions/180426 | 4 | Is it true that a smooth quadric hypersurface has a rational point if and only if it has an odd degree $0$-cycle?
I think this is true. If so, can someone give a (geometric) proof?
| https://mathoverflow.net/users/57994 | odd degree $0$-cycles and rational points on a quadric hypersurface | Yes. It was a conjecture of Witt, proved by Springer, that a quadric hypersurface, over a field of characteristic not equal to $2$, has a rational point if and only if it has a point over a field extension of odd degree. This was proved in:
Springer, Tonny Albert - Sur les formes quadratiques d'indice zéro. (French) ... | 5 | https://mathoverflow.net/users/5101 | 180430 | 90,573 |
https://mathoverflow.net/questions/180207 | 2 | Let $X$ be a Banach space and $(x\_{n})\_{n=1}^{\infty}$ be a $p-$summable sequence in $X$. My basic questions are the following:
1. For any $\epsilon>0$, is there a sequence $(\xi\_{n})\_{n=1}^{\infty}$ such that $1\leq \xi\_{n}\rightarrow \infty(n\rightarrow \infty)$ and $\|(\xi\_{n}x\_{n})\_{n=1}^{\infty}\|\_{p}\l... | https://mathoverflow.net/users/41619 | Two basic questions on $p-$summable sequences | I assume "$p$-summable" means "absolutely $p$-summable", i.e., $\sum \|x\_n\|^p < \infty$.
Since all that matters about $x\_n$ is its norm, the question reduces to the scalar case.
1. Yes. Find a strictly increasing sequence of indices $(n\_k)$, $k \geq 1$, such that $\sum\_{i \geq n\_k} \|x\_i\|^p < \frac{\epsilon}{... | 2 | https://mathoverflow.net/users/23141 | 180442 | 90,574 |
https://mathoverflow.net/questions/180361 | 8 | Let $X,Y$ be CW complexes. By Kunneth formula, we have a group isomorphim
$$ H^n(X\times Y;G) \cong \oplus\_{p+q=n} H^p(X;H^q(Y;G))$$
Is there a natural map realizing this isomorphism?
| https://mathoverflow.net/users/37354 | Naturality of a Kunneth formula for cohomology | I came across this question in my 1961 DPhil Thesis; this was written up in two papers which are available from my [Publication List](http://pages.bangor.ac.uk/~mas010/publicfull.htm),
[3]. ``Cohomology with chains as coefficients'', *Proc. London Math. Soc*. (3) 14 (1964), 545-565.
[4]. ``On K\"{u}nneth suspensi... | 8 | https://mathoverflow.net/users/19949 | 180449 | 90,576 |
https://mathoverflow.net/questions/175980 | 2 | Suppose we have coprime integers $a$ and $b$ with $p \mid a$ but $p^2 \nmid a$ for some prime $p\geq 5$. We can write $a=px$ and $b=pr+\hat{b}$. Suppose also that $a$ and $\hat{b}$ are coprime; that is to say, $\hat{b}$ and $x$ are coprime. Consider the set:
$$S=\left\{k<a\mid\{\tfrac{kr}{x}\}+\{\tfrac{k\hat{b}}{a}\}... | https://mathoverflow.net/users/45594 | Fractional Part Problem | It seems that the counting of elements in $S\_i$ can be made in the same way as the counting of those in $S$.
For simplicity, I rename $y=r$, $z=\hat b$.
Let $T=\{k<a\}\setminus S$. Then it is easy to see that
$$
S=\left\{k<a\colon \left\{k\cdot \frac{py}{px}\right\}+\left\{k\cdot \frac{z}{pr}\right\}
=1+\left\{k... | 1 | https://mathoverflow.net/users/17581 | 180462 | 90,579 |
https://mathoverflow.net/questions/180452 | 1 | I cannot seem to find stated the following fact, which is surely well known to experts.
Let (S,L) be a polarized K3 surface. Then $M = L^{\otimes 3}$ is very ample and we can consider the embedding in the corresponding projective space $S \to P^N$.
Question: is the image of $S$ in $P^N$ a complete intersection? I s... | https://mathoverflow.net/users/46690 | are K3 surfaces complete intersections in their polarization? | Suppose $X$ is a smooth surface complete intersection in $\mathbb{P}^n$, given by equations of degrees $d\_1,\ldots ,d\_{n-2}\ $ . I assume $d\_i\geq 2$, because the image of a variety by a complete linear system cannot be contained in a hyperplane. By the adjunction formula the canonical divisor of $X$ is $(\sum d\_i-... | 5 | https://mathoverflow.net/users/40297 | 180463 | 90,580 |
https://mathoverflow.net/questions/180470 | 3 | I am reading (or trying to read :)) *"One parameter semigroups for Linear Evolution equations"* by Klaus-Jochen Engel and Rainer Nagel. I was wondering if someone was aware of a good set of lecture notes for a course given at a university using this particular book or some such similar textbook. I feel that the informa... | https://mathoverflow.net/users/58011 | Lecture notes on semigroup theory for linear evolution equations | The lecture notes of the various Internet Seminars on Evolution Equations (<http://www.math.kit.edu/iana3/page/isem/en>) sound like something you'll enjoy. The topics vary with each seminar, but all are based on semigroup theory.
I participated in the 16th internet seminar, and I found the lecture notes from this sem... | 7 | https://mathoverflow.net/users/nan | 180474 | 90,585 |
https://mathoverflow.net/questions/180455 | 21 | Say that a polynomial in an indeterminate $x$ with real coefficients of degree $d$ *has positive coefficients* if each of the coefficients of $x^d,\ldots,x^1,x^0$ is (strictly) positive.
For $f$ a **monic** univariate polynomial with real coefficients, if there exists a positive integer $m\_0$ such that its $m\_0$th... | https://mathoverflow.net/users/nan | Stability of real polynomials with positive coefficients | Yes. For nonnegative (as opposed to strictly positive) coefficients, a more general result is available (in several variables). See the article by me on powers of polynomials in Symbolic Dynamics and Its Applications (Editors, Adler and Walters, AMS 1992 (unfortunately, I can't find a link to it) [I have three articles... | 18 | https://mathoverflow.net/users/42278 | 180475 | 90,586 |
https://mathoverflow.net/questions/180496 | 2 | Inspired by [this question](https://mathoverflow.net/questions/180490/curve-through-a-point-avoiding-an-hypersurface):
>
> Suppose given an algebraic curve $C \subset \mathbb{A}^2$, and a point $x \in C$. Can you find another (closed) curve $D \subset \mathbb{A}^2$ such that $C \cap D = x$?
>
>
>
| https://mathoverflow.net/users/4707 | curve through a point avoiding an hypersurface, II | Let $C$ be a smooth curve which is the zero set of a function $y^2-x^3-ax-b$ in $\mathbb{A}^2$. I.e. $C$ is a Weierstrass elliptic restricted to the x-y plane. Let $x \in C$ be a chosen point. Suppose $D$ is another curve in $\mathbb{A}^2$ such that $D \cap C = {x}$ and the defining equation for $D$ has degree $d$.
T... | 5 | https://mathoverflow.net/users/17630 | 180499 | 90,598 |
https://mathoverflow.net/questions/180490 | 0 | Let $H$ be a closed hypersurface in $\mathbb{A}^{n}$, $n$ big enough over $\mathbb{C}$. Let $U$ be the complementary open subset.
Let $x\in H$, Is it possible to find an curve $C\subset\mathbb{A}^{n}$ such that and $C\cap H=\{x\}$.
| https://mathoverflow.net/users/27398 | curve through a point avoiding an hypersurface | In general no, thanks to [this example](https://mathoverflow.net/questions/180496/curve-through-a-point-avoiding-an-hypersurface-ii). Take the curve $E\subset \mathbb{A}^2$ of the example (given by $y^2=x^3+ax+b$), and take $H=E\times \mathbb{A}^{n-2}\subset \mathbb{A}^{2}\times \mathbb{A}^{n-2}=\mathbb{A}^n$. Let $p:\... | 1 | https://mathoverflow.net/users/40297 | 180508 | 90,603 |
https://mathoverflow.net/questions/180506 | 4 | For each $i$ (up to infinity), let $u\_i \in C^1(0,T)$ satisfy
$$\frac{d}{dt}u\_i(t) + \sum\_{j=1}^\infty b(t;w\_j,w\_i)u\_j(t) = 0$$
$$u\_i(0) = u\_i(T)$$
where $b(t;\cdot,\cdot)$ is an inner product on some (infinite-dimensional) Hilbert space $H$ (which has an inner product $(\cdot,\cdot)\_H$) for each $t$ and $w\_i... | https://mathoverflow.net/users/58028 | Uniqueness of solutions to an ODE system | I'll presume that $H$ is finite dimensional. Write $b(t;\mathbf{u},\mathbf{v}) = (\mathbf{u},B(t)\mathbf{v})\_H$ for some positive symmetric linear operator $B(t)\colon H\to H$. $B(t)$ is positive symmetric because $b(t;\cdot,\cdot)$ is itself an inner product. Then your system of equations is equivalent to $$\frac{\pa... | 6 | https://mathoverflow.net/users/2622 | 180516 | 90,604 |
https://mathoverflow.net/questions/179123 | 37 | What might it mean for a relation $R\subset X\times Y$ to be continuous, where $X$ and $Y$ are topological spaces? In topology, category theory or in analysis? Is it possible, canonical, useful?
I have a vague idea of the possibility of using continuous relations in science, for example in biochemistry or celestial m... | https://mathoverflow.net/users/57255 | Continuous relations? | Here's a different and quite generic approach: Let $X,Y$ be topological spaces. Then we topologize $\mathcal{P}(Y)$ and say that $R\subseteq X\times Y$ is continuous if and only if the function $f\_R: X \to \mathcal{P}(Y)$ defined by $x\mapsto \{y\in Y: (x,y) \in R\}$ is continuous.
As for topologizing $\mathcal{P}(Y... | 9 | https://mathoverflow.net/users/8628 | 180527 | 90,608 |
https://mathoverflow.net/questions/180520 | 4 | This question was inspired by [Can we build a continuous function from "fibers"/preimages defined over a topological base?](https://mathoverflow.net/questions/180282/can-we-build-a-continuous-function-from-fibers-preimages-defined-over-a-topolo)
Let $X,Y$ be sets and $L\subseteq \mathcal{P}(Y)$. Suppose $L$ has the f... | https://mathoverflow.net/users/8628 | Constructing a function from preimages | This is true. Note that it suffices to show that for any $x\in X$, the set $U\_x:=\bigcap\limits\_{U\in L,x\in F(U)}{U} \cap \bigcap\limits\_{U\in L,x\notin F(U)}{(Y\setminus U)}$ is nonempty, since any map $f:X\rightarrow Y$ sending $x\in X$ to some element from $U\_x$ does the job.
To show the nonemptiness of $U\_x... | 6 | https://mathoverflow.net/users/57975 | 180530 | 90,611 |
https://mathoverflow.net/questions/180536 | 4 | This is in a sense a follow up on the popular question [Induction and Coinduction of Representations](https://mathoverflow.net/questions/1534/induction-and-coinduction-of-representations), where this particular question is one of several points, and it is neglected.
It seems that the restriction functor of representa... | https://mathoverflow.net/users/13767 | When are induction and coinduction of representations of Lie groups isomorphic? When they are compact? Semisimple? | The inclusion $H \to G$ induces a map of classifying stacks $f \colon BH \to BG$ (note that these are *stacks*, not the "topologists $BG$"). As remarked in the question you linked to, induction and coinduction correspond to the functors $f\_\ast$ and $f\_!$ respectively. These functors will be equal if $f$ is proper. T... | 3 | https://mathoverflow.net/users/1310 | 180539 | 90,614 |
https://mathoverflow.net/questions/19079 | 67 | For $r > 0$, let $L(r) = \# \{ (x,y) \in \mathbb{Z}^2 \ | \ x^2 + y^2 \leq r^2\}$ be the number of lattice points lying on or inside the standard circle of radius $r$. It is easy to see that $L(r) \sim \pi r^2$ as $r \rightarrow \infty$. The **Gauss circle problem** is to give the best possible error bounds: put
$E(r... | https://mathoverflow.net/users/1149 | What is the status of the Gauss Circle Problem? | [This](https://www.arxiv.org/pdf/1409.2446.pdf) recent arxiv posting by Shaneson claims that one may take $\epsilon > 1153/9750$, improving on Huxley's bound. It also includes the passage
>
> In 2007 Cappell and the author posted a paper on the arXiv claiming to obtain
> the estimate [in my notation -- PLC] $O(r^{1... | 16 | https://mathoverflow.net/users/1149 | 180549 | 90,618 |
https://mathoverflow.net/questions/180551 | 5 | Let $M$ be a module over an algebra or a group. I am interested the following decreasing filtration:
$F^0M=M$;
$F^iM$ is the smallest sub-module of $F^{i-1}M$ such that the quotient is *semi*-simple. This filtration is unique.
Does it have a specific name?? Harder-Narasimhan?? Any reference??
(Jordan-Holder is... | https://mathoverflow.net/users/48866 | Jordan-Holder vs Harder-Narasimhan | If your algebra is finite-dimensional, then this is the Loewy-Filtration and explicitly $F^{k}M = J(A)^k M$ where $J(A)$ is the Jacobson-radical of $A$.
Also note: Without any finiteness condition your filtration is not well-defined. Consider for example $A=K[X]$ for some field $X$ and choose pairwise distrinct, moni... | 4 | https://mathoverflow.net/users/3041 | 180553 | 90,619 |
https://mathoverflow.net/questions/180542 | 2 | Here's a numerical analysis question which may not be very important, especially in practice, but has been bugging me.
Suppose $f$ is a continuous function on an interval $[a,b]$. Let $T\_n(f)$ be the
approximation to $\int\_a^b f$ using the trapezoid rule with $n$ subintervals of equal length, and $E\_n(f)$ the erro... | https://mathoverflow.net/users/19444 | Monotonicity of Trapezoid Approximations | For convenience let the interval be $[0,1]$, and let's look at
the case $f(t) = t^p$. We have $E\_n(t^p) = 0$ for $p \le 1$, while by Faulhaber's formula we have
$$\eqalign{E\_n(t^p) &= \dfrac{1}{p+1} + \dfrac{1}{2n} - \dfrac{1}{n^{p+1}} \sum\_{i=1}^{n} i^p\cr
&= - \sum\_{j=0}^{p-2} {p \choose j} \dfrac{B\_{p-j}}{j+1}... | 4 | https://mathoverflow.net/users/13650 | 180555 | 90,620 |
https://mathoverflow.net/questions/180511 | 14 | (copied from [math.SE](https://math.stackexchange.com/questions/916884/stromquists-3-knives-procedure))
BACKGROUND: A cake has to be divided among 3 people with possibly different tastes, such that each person receives a single connected piece, and no person prefers another person's piece. In other words, no particip... | https://mathoverflow.net/users/34461 | Stromquist's 3 knives procedure | Edit: The below does rely on the assumption that knives move continuously, see the comments.
I think the procedure is "safe": Each player can guarantee not to envy either of the others by following the suggested protocol. I am assuming that the rules are that players must keep their knives to the right of the sword. ... | 8 | https://mathoverflow.net/users/29697 | 180556 | 90,621 |
https://mathoverflow.net/questions/180540 | 4 | It is known after Stallings that a group can have 0, 1, 2 or infinitely many ends. Are there known results on the space of ends of a Coxeter group?
| https://mathoverflow.net/users/21985 | Ends of Coxeter Groups | The following book has a wealth of material on this topic:
>
> *The geometry and topology of Coxeter groups* by Michael Davis.
>
>
>
By way of example, here is one result from the book:
>
> **Theorem 8.7.1** Let $(W,S)$ be a Coxeter System:
>
>
> 1. $W$ is one-ended if and only $H\_c^1(\Sigma)=0$.
> 2. $... | 2 | https://mathoverflow.net/users/801 | 180557 | 90,622 |
https://mathoverflow.net/questions/180254 | 11 | Let $k$ be a number field and $S$ be a finite set of places of $k$.
Let $G$ be a connected semisimple algebraic group over $k$.
Let $k\_S=\prod\_{v\in S}k\_v$
where $k\_v$ is the completion of $k$ at $v$.
**Question:** Is maximal compact subgroup of $G(k\_S)$ unique up to conjugation?
If it is not unique, are there ... | https://mathoverflow.net/users/11056 | Maximal compact subgroup of p-adic lie groups | Since the maximal compact subgroup question has a complicated history, and is treated at very different levels of generality in the literature (Bruhat-Tits papers in particular), it may be helpful to fill in Paul's answer a bit. There was early work in special cases by Bruhat over half a century ago, in the aftermath o... | 11 | https://mathoverflow.net/users/4231 | 180562 | 90,625 |
https://mathoverflow.net/questions/180514 | 11 | Let $X$ be a (smooth) Fano variety over $\mathbb{C}$. If $\dim(X)=3$, inspection of the Iskovskikh-Mori-Mukai lists seems to indicate that $H^3(X,\mathbb{Z})$ is torsion free. Is there a theoretical reason for that? What happens in higher dimension?
| https://mathoverflow.net/users/40297 | Does there exist a Fano variety with torsion in $H^3$? | There are smooth Fano $5$-folds $\widetilde Y$ with non-zero $2$-torsion in the Brauer group (i.e., $2$-torsion in $H^3(\widetilde Y,\mathbb Z)$); the examples that follow are directly inspired by Beauville's exposition of the Artin--Mumford examples in terms of Reye congruences [Springer LNM $997$, pp. $28-30$].
Sta... | 12 | https://mathoverflow.net/users/58044 | 180568 | 90,627 |
https://mathoverflow.net/questions/180434 | 2 | $Φ\_n(1)$ and $Φ\_n(−1)$ for the cyclotomic polynomials are well-known.
I am now looking for
$$Φ\_n(i)$$
and/or
$$Φ\_n(−i)$$
with i the complex unit.
At this moments my endeavours result in intricate categorizing of values
for $n$ e.g. as $4$, $2^k$, $4p$, $p^k$ with $p$ prime and considerations about
the prime... | https://mathoverflow.net/users/49056 | Looking for ways how to calculate $\Phi_n(i)$ | The question has already been answered by Vladimir Dotsenko, but as Wolfgang Tintemann is still interested in a solution using Möbius inversion, let me expand the comments I made above.
Since $\Phi\_n(-i)=\overline{\Phi\_n(i)}$, it is enough to deal with $i$. We are going to prove
$$\Phi\_n(i)=\begin{cases}
i-1&n=1,\... | 6 | https://mathoverflow.net/users/12705 | 180579 | 90,631 |
https://mathoverflow.net/questions/180575 | 1 | This is a very vague question.
Is there any example of spin structures on a moduli space? References are requested.
I have vaguely heard that Witten discussed when a sigma model is spin. Somehow I cannot locate it in the literature.
Thanks!
| https://mathoverflow.net/users/41734 | When does a moduli space admit a spin structure? | You didn't specify what kind of moduli space you meant, but there certainly are examples coming from gauge theory. The paper [Spin structures on the Seiberg-Witten moduli spaces](http://arxiv.org/abs/math/0404275), by H. Sasahira shows that under some topological hypotheses on a 4-manifold M, the Seiberg-Witten moduli ... | 3 | https://mathoverflow.net/users/3460 | 180582 | 90,632 |
https://mathoverflow.net/questions/180567 | 8 | Let $f$ be a weight 1 modular form (let's say cuspidal, new, normalized, and a Hecke eigenform). Then there's an associated Artin representation $\rho\_f: \operatorname{Gal}(\overline{\mathbf{Q}} / \mathbf{Q}) \to GL\_2(\mathbf{C})$.
The character of this representation is determined by $f$, so it takes values in the... | https://mathoverflow.net/users/2481 | Field of definition of Galois representations of weight 1 modular forms | No. The fact that $\rho(c)$ has eigenvalues +1 and -1 implies that $\rho$ is defined over $L$. Here is a low-tech, longish argument (there are shorter ones using slightly more technology), which is essentially from an old paper of Wiles.
Work in a basis $(e\_1,e\_2)$ where $\rho(c)$ is the diagonal matrix $(1,-1)$. N... | 9 | https://mathoverflow.net/users/9317 | 180591 | 90,635 |
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