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https://mathoverflow.net/questions/239025 | 3 | I was reading some stuff on Hecke characters and came across an issue I have not been able to resolve. I posted it [here on math stack exchange first.](https://math.stackexchange.com/questions/1783909/character-group-of-the-multiplicative-rationals)
Let $\mathbb{Q}^{\times}$ be the multiplicative group of the rationa... | https://mathoverflow.net/users/50426 | Character group of the multiplicative rationals | In Section 8, the notes say that the set of Hecke characters is $\widehat{J/\mathbb{Q}^\times}$ or $\widehat{J^0/\mathbb{Q}^\times}$, which can be identified with $(\mathbb{Q}^\times)^\perp$, but I do not see any reason to think that the Hecke characters should be $\widehat J/(\mathbb{Q}^\times)^\perp$. Indeed, $\wideh... | 2 | https://mathoverflow.net/users/68305 | 239044 | 110,373 |
https://mathoverflow.net/questions/239048 | 4 | Let $O$ be an orthogonal matrix, $O^T O = I$, thus its eigenvalues lie on the unit circle, $\lambda(O)=e^{i\theta}$. Furthermore, assume the form
$O = X Y$, where both matrices satisfy $X^2 = I$ and $X=X^T$, and also $Y^2=I$ and $Y=Y^T$, thus $\lambda(X),\lambda(Y) \in \{-1,1\}$.
(These matrices do not commute $[X,Y] ... | https://mathoverflow.net/users/91760 | About the Eigenvalues of Orthogonal Matrix plus Perturbation | The reason is very simple, actually. Consider the following tautological identity, for some $0\leq z\in \mathbb{R}$,
$$
P^2-2\frac{P+z P^{-1}}{2}P+z I=0.
$$
Denote $\frac{P+z P^{-1}}{2}=B$, then find
$$
P^2-2BP+z I=0.
$$
Note that $[B,P]=0$ and thus if we can diagonalize $P$ and $B$, we can do so simulatneosly, and the... | 5 | https://mathoverflow.net/users/32985 | 239052 | 110,376 |
https://mathoverflow.net/questions/55300 | 14 | As a natural (and expectable) extension of [my earlier question](https://mathoverflow.net/questions/54619/when-does-pa-b0-for-a-ne-b-ensure-p00):
>
> How large must be a set $A\subset F\_2^n$ to ensure that if $P$ is a *cubic* polynomial in $n$ variables over the field $F\_2$, vanishing at every non-zero point of t... | https://mathoverflow.net/users/9924 | When does $P(a−b)=0$ for $a≠b$ ensure $P(0)=0$? (Continued.) | About three years after posting this question, the answer is now known (although there still can be some room for improvements). Namely, Lemma 1 from a [recent joint paper](https://arxiv.org/abs/1605.01506) by Ernie Croot, Peter Pach, and myself reads as follows:
>
> Suppose that $n\ge 1$ and $d\ge 0$ are integers,... | 18 | https://mathoverflow.net/users/9924 | 239053 | 110,377 |
https://mathoverflow.net/questions/237745 | 6 | A famous open problem in Geometric Control Theory and in the study of sub-Riemannian manifolds is whether constant-speed length minimizers in a sub-Riemannian manifold are always smooth (see also [this question](https://mathoverflow.net/questions/156678/why-is-proving-c-infty-regularity-of-sub-riemannian-geodesics-so-h... | https://mathoverflow.net/users/36952 | Are rays in Carnot groups straight? | No, infinite rays are not necessarily one-parameter subgroups.
An example can be found in the paper ["Cut time in sub-riemannian problem on engel group"](https://arxiv.org/abs/1408.6651) by Ardentov and Sachkov.
In the paper the geodesics of the Engel group (a rank 2, step 3 Carnot group) are separated into 7 diffe... | 1 | https://mathoverflow.net/users/91774 | 239070 | 110,384 |
https://mathoverflow.net/questions/239067 | 5 | Let $f: X \rightarrow Y$ be continuous, X,Y pathwise connected and aspherical (i.e. trivial higher homotopy groups). Then $\pi\_1(X)$ acts on the universal cover of $X$ via deck transformations, and on the universal cover of $Y$ via
$$([\sigma],y)\mapsto [\sigma \circ f]y$$
If g is another continuous map whose lif... | https://mathoverflow.net/users/91775 | Are two equivariant maps between aspherical topological spaces homotopic? | In the language of equivariant homotopy theory, your question is as follows. You have a group homomorphism $\phi\colon G\to H$, which you use to make $EH$ into a $G$-space, and then you ask whether $[EG,EH]^G$ is a singleton. In fact, it is true more generally that $[EG,Y]^G$ is a singleton whenever $Y$ is a $G$-space ... | 10 | https://mathoverflow.net/users/10366 | 239072 | 110,385 |
https://mathoverflow.net/questions/239065 | 4 | I was wondering which curves on the $n-1$ sphere arise as the Gauss maps of *closed* paths in $\Bbb R^n$. Necessary conditions are obviously that the path on the sphere is the image of some smooth $S^1\to S^{n-1}$ and that there is no half sphere which contains the complete path (otherwise, if the image of the Gauss ma... | https://mathoverflow.net/users/37059 | Sufficient conditions for a curve on the sphere to be the Gauß map of a closed path | $\def\conv{\mathop{\rm conv}}$Your condition is almost sufficient.
Let $g\colon S^1\to S^{n-1}$ be the given path on the sphere. The `no half-sphere' condition tells that $0$ lies in the convex hull of $g(S^1)$ (if a half-shpere is open) of that $0$ lies in the interior of $g(S^1)$ (if a half-sphere is closed). The ... | 3 | https://mathoverflow.net/users/17581 | 239075 | 110,386 |
https://mathoverflow.net/questions/239064 | 5 | *Context: this question is a translation of a common informal phrasing of the friendship paradox ("Most people have fewer friends than most of their friends"). Note that the question is similar to, but distinct from, [Average degree of neighbors in a simple graph (-> Friendship paradox)](https://mathoverflow.net/questi... | https://mathoverflow.net/users/91772 | Degree of neighbors in a simple graph (friendship paradox variant) | Well, a modification of the previous example works. Take some number $n$ such that there are proper divisors $a\mid n$ and $b\mid n+1$ with $a>b$. Let $V=V\_1\sqcup V\_2$, with $|V\_1|=n$, $|V\_2|=n+1$. Take $n/a$ disjoint copies of $K\_a$ on $V\_1$, and $(n+1)/b$ disjoint copies of $K\_b$ on $V\_2$. Finally, take the ... | 1 | https://mathoverflow.net/users/17581 | 239077 | 110,387 |
https://mathoverflow.net/questions/239073 | 12 | While trying to compute the Matrix Exponential of an $n \times n$ array I decided to take advantage of a Python function called `scipy.linalg.expm()`.
According to the [documentation](http://docs.scipy.org/doc/scipy-0.17.1/reference/generated/scipy.linalg.expm.html), this function adopts the Padé's Approximant to per... | https://mathoverflow.net/users/90217 | What is the time complexity of the matrix exponential? | Moler's paper ["Nineteen dubious ways to compute the exponential of a matrix, twenty-five years later"](http://epubs.siam.org/doi/abs/10.1137/S00361445024180) contains the following extracts:
>
> In estimating the time required by matrix computations it is traditional to estimate the number of multiplications and t... | 12 | https://mathoverflow.net/users/1847 | 239083 | 110,389 |
https://mathoverflow.net/questions/239081 | 7 | I would like to learn this topic of algebraic topology but I cannot find a relevant reference to answer my basic questions on the subject (for example, is there a Hurewicz theorem for regular homotopy groups? under which conditions can one homotop a non-zero class in a non-trivial higher homotopy group to an immersion?... | https://mathoverflow.net/users/56191 | Thorough reference on regular homotopy | The Smale-Hirsch theorem says that the existence of an immersion in a given homotopy class is equivalent to the existence of certain bundle map, in turn equivalent to the existence of a section of a certain bundle. See Hirsch, Morris W. Immersions of manifolds; the Mathscinet review (MR0119214)by Kervaire is a good sum... | 14 | https://mathoverflow.net/users/3460 | 239089 | 110,392 |
https://mathoverflow.net/questions/239087 | 3 | How much is known about the fundamental units in totally real cubic fields? For example, Daniel Shanks has a family of totally real cubic fields for which the fundamental units are known; those with defining polynomial $x^3-ax^2-(a+3)x -1$. Does anyone know of other families of totally real cubic fields for which the f... | https://mathoverflow.net/users/56362 | Fundamental Units in Totally Real Cubic Fields | Here are two references that may be useful.
1. Algebraic number fields with 2 independent units, 1931, by W. E. H. Berwick.
2. The determination of units in totally real cubic fields, 1959, by H. J. Godwin.
| 1 | https://mathoverflow.net/users/40694 | 239117 | 110,402 |
https://mathoverflow.net/questions/239116 | 6 | [This question was asked on [MSE](https://math.stackexchange.com/questions/1788107/on-a-parallelizable-manifold-is-there-always-a-frame-satisfying-x-i-x-j-0), but got no answers, I thought it could be more appropriate here]
Let $M$ be a parallelizable manifold.
1. Is there always a global frame $(X\_i)$ such that $... | https://mathoverflow.net/users/89425 | On a parallelizable manifold, is there always a frame satisfying $[X_i,X_j]=0$? | Suppose that $M$ is compact, such a frame exists implies that the commutative group $R^n$ acts transitively on it, this implies that $M$ is a torus. But a compact Lie group is parallelizable, so that is not always possible.
| 11 | https://mathoverflow.net/users/80891 | 239118 | 110,403 |
https://mathoverflow.net/questions/239079 | 4 | Let $R$ be a quantum R-matrix. Is there a procedure to dequantize $R$ and obtain a classical r-matrix? Thank you very much.
| https://mathoverflow.net/users/11877 | How to obtain a classical r-matrix from a quantum R-matrix? | In the semiclassical (or quasiclassical) limit, by definition, the quantum $R$-matrix contains the classical $r$-matrix in the linear term in an expansion in a small parameter $\hbar$, so that $R \propto 1 + \hbar \, r + O(\hbar^2)$. Expanding the quantum Yang-Baxter equation and collecting terms linear in $\hbar$ yiel... | 7 | https://mathoverflow.net/users/45956 | 239126 | 110,407 |
https://mathoverflow.net/questions/239022 | 4 | Let $G$ be a semi-simple algebraic group over $\mathbb{Q}$, I would like to find an integer $d>0$ only depending on $G$ with the following property. For any two semi-simple $\mathbb{Q}$-subgroups $H\_1,H\_2\subset G$ that are conjugated by an element $g\in G(\mathbb{R})$, there exists a number field $K$ of degree at mo... | https://mathoverflow.net/users/90978 | About the conjugation of semi-simple subgroups | The answer is YES. It suffices to assume that $H\_1$ and $H\_2$ are conjugate over $\mathbb{C}$ or, what is the same, that they are conjugate over $\overline{\mathbb{Q}}$.
>
> **Theorem 1.** Let $G$ be a connected semisimple linear algebraic group defined over a number field $k$.
> There exists a natural number $... | 9 | https://mathoverflow.net/users/4149 | 239129 | 110,408 |
https://mathoverflow.net/questions/239107 | 2 | Study article R. C. Reilly is entitled Applications of Hessian operator in the Riemann manifold had a doubt in the remark, shortly after the theorem 2 of that Article.
The theorem is stated as: *Suppose that $M^{n}$ is compact and $N^{n-1}=\partial M$ is empty. If $f:M \to \mathbb{R}$ is a smooth function, then $S\_{... | https://mathoverflow.net/users/80358 | Applications of Hessian operator in the Riemann manifold. Simple samples $S_{2}(f)$ | On the torus $T^2$ with the coordinates $x,y$ and the flat metric $g= dx^2 + dy^2$ take any function $f(x)$. Its hessian is given, after raising the index, by the (1,1)-tensor $f''(x) dx\otimes \frac{\partial } {\partial x}$ so the operator $S\_2(f)$ (which in dimension two is simply the determinant of this tensor) is ... | 4 | https://mathoverflow.net/users/14515 | 239130 | 110,409 |
https://mathoverflow.net/questions/239035 | 5 | Let $X$ be a compact complex manifold equipped with a holomorphic symplectic form $\omega$. Let $D$ be a smooth divisor on $X$. At each point of
$D$, the restriction of $\omega$ to $D$ has one-dimensional kernel. This gives a non-singular foliation $F$ on $D$. Is it possible that some leaves of $F$ are algebraic while... | https://mathoverflow.net/users/82725 | Algebraicity and non-algebraicity of leaves of the characteristic foliation | Suppose $X$ is projective manifold endowed with holomorphic symplectic form. Let $D$ be smooth divisor on $X$.
1. **Characteristic foliation with algebraic and non-algebraic leaves.**
Suppose that $X$ is a product of two abelian varieties of dimension $n$ ($n\ge 2$) $A\_1 \times A\_2$ in such a way that the fibers o... | 6 | https://mathoverflow.net/users/605 | 239146 | 110,416 |
https://mathoverflow.net/questions/238890 | 3 | It is well known [it's on Wolfram Mathworld, for example] that the probability of no runs of $k$ consecutive $1$'s will occur in a $\{0,1\}$-valued sequence of length $n$ is exactly equal to $$\frac{F^{(k)}\_{n+2}}{2^n}\quad\quad(1)$$ where $F^{(k)}\_l$ is the $l^{th}$ $k-$step Fibonacci number. For example if $k=6,$ t... | https://mathoverflow.net/users/17773 | Probability of no $k$ 1's in arithmetic progression in binary sequence of length $n$ | **Answer added for completeness' sake:**
In the paper "Maximal Arithmetic Progressions in Random Subsets" by Itai Benjamini, Ariel Yadin, and Ofer Zeitouni, arXiv:0707.3888, referred to in the first answer by user91686, the authors show that $$(U^{(N)}- 2 \ln(N))/\ln(2)$$ converges in law to an extreme type (asymmetr... | 2 | https://mathoverflow.net/users/17773 | 239153 | 110,417 |
https://mathoverflow.net/questions/239155 | 3 | Does anyone know definition of Levi-Civita connection map that defined as $K: TTM\to TM$. and how to prove the following theorem:
>
> **Theorem:** If $X\in\mathfrak{X}(M)$ be a vector field over $M$ and $K:TTM\to TM$ Levi-Civita connection map then
> $$K\circ dX=\nabla X$$
> where $dX$ is derivation of vector fie... | https://mathoverflow.net/users/90655 | Definition of Levi-Civita connection map and a theorem about it? | After the comment of Jez, here is the corrected answer:
There is the well-defined vertical bundle $VTM\subset TTM\to TM$ given as the kernel of the (differential of the) projection.
Moreover, for any $v\in TM,$ there is a natural identification $\Phi$ of $V\_vTM$ with $T\_\pi(v)M.$
Using the Levi-Civita connection... | 4 | https://mathoverflow.net/users/4572 | 239156 | 110,418 |
https://mathoverflow.net/questions/239112 | 0 | Is Rademacher complexity defined for any space of functions?
Or are there restrictions on the function space over which this can be defined?
For example is the Rademacher complexity defined or has it been computed over say the space of all polynomials in $n$ variables? I would be most happy to get some references t... | https://mathoverflow.net/users/38852 | Request for references about computing or estimating Rademacher complexity | In order for the Rademacher complexity to be finite, you need to restrict both the domain (say, to a cube or a ball) and the range (say, by restricting the magnitude of the coefficients): [Rademacher complexity of a Lipschitz class: Are the boundedness constraints necessary?](https://mathoverflow.net/questions/186693/r... | 1 | https://mathoverflow.net/users/12518 | 239158 | 110,419 |
https://mathoverflow.net/questions/239148 | 14 | It is known that $\mathbb{R}^4$ has exotic smooth structures, and there are many such examples in higher dimensions, such as the famous 7-sphere. My (probably very naive) question is, for every $n\geq4$, does there exist an $n$-manifold with exotic smooth structures?
In other words, for every $n\geq4$, does there exi... | https://mathoverflow.net/users/824 | Does every dimension $n\geq4$ admit a manifold with an exotic smooth structure? | Yes. For every $n\ge 5$ there are exotic tori.
In fact, the PL-structures on $T^n$ are in one-to-one correspondence with $H^3(T^n;\mathbb{Z}/2)$, and every one of these is smoothable (Reference: "Surgery on Compact Manifolds" by C. T. C. Wall, Chapter 15A). Since any smooth manifold admits a unique PL-structure up t... | 21 | https://mathoverflow.net/users/8103 | 239162 | 110,420 |
https://mathoverflow.net/questions/239168 | 2 | Let $C$ be a hyperelliptic curve of genus $g$ and let $D$ be a divisor on $C$ of degree $g+1$. Assume that the linear system $|D|$ is base-point-free. Now add a $2$-torsion point $[E]$ to $D$. I would like to know if the linear system $|D+E|$ is again base-point-free.
I think of the hyperelliptic curve as the smooth ... | https://mathoverflow.net/users/21778 | Linear systems and 2-torsion shifts on hyperelliptic curves | This will never happen. The linear systems of degree $g+1$ on $C$ with
a base point are of the form $g^1\_2+F$, with $F$ effective of degree $g-1$; in $J^{g+1}C$, they form a divisor $\Delta $ which is a copy of the theta divisor. For any $E\neq 0$ in $JC$ (2-torsion or not), the translate $\Delta -E$ is different fro... | 1 | https://mathoverflow.net/users/40297 | 239176 | 110,423 |
https://mathoverflow.net/questions/233422 | 2 | Let $\Gamma$ be a discrete group.
Q: If $l^\infty(\Gamma)\rtimes \Gamma=l^\infty(\Gamma)\rtimes\_r \Gamma$ canonically, can we conclude that $\Gamma$ is an exact group?
The converse implication is well-known and if we replace $l^\infty(\Gamma)$ by $l^\infty(\Gamma)/C\_0(\Gamma)$ we have indeed an affirmative answer... | https://mathoverflow.net/users/9401 | Uniform Roe algebras and exact groups | See final remark of "SOME REMARKS ABOUT THE WEAK CONTAINMENT PROPERTY FOR
GROUPOIDS AND SEMIGROUPS" by Claire Anantharaman-Delaroche for an affirmative answer of the question.
<https://arxiv.org/pdf/1604.01724.pdf>
edit (by AlexE): the cited argument is flawed, see my comment below.
| 0 | https://mathoverflow.net/users/9401 | 239181 | 110,424 |
https://mathoverflow.net/questions/105864 | 2 | I am trying to implement a finite element scheme using the method of lines (finite difference in time and finite element in space) and enforcing boundary conditions using Lagrange Multipliers. This might be necessary for essential boundary conditions on non-Lagrangrian finite elements, such as Argyris finite elements.
... | https://mathoverflow.net/users/26060 | How to apply Lagrange Multipliers to BCs of Time Dependent problems using finite elements? | In this answer I assume that by finite difference you mean only Forward Euler. Backward Euler and the RK are generalizations of this scheme that I will address at the end of the answer.
Like you said in your question, our goal is to come up with something like
$$
\frac{\mathrm{d}}{\mathrm{d}t}u = F(t; u),
$$
To ac... | 0 | https://mathoverflow.net/users/76602 | 239196 | 110,425 |
https://mathoverflow.net/questions/239208 | 3 | Perhaps this question is too easy for mathoverflow, at least this is how it seems, but I got no answer on stackexchange.
Suppose $T$ is a bounded linear operator on $l\_2$ and $x\in l\_2$ is a vector such that the orbit $(T^{n}x)$ is linearly independent.
Can one find an $\epsilon>0$ such that for all $||y-x||<\e... | https://mathoverflow.net/users/69275 | Behavior of orbits under small perturbations | The first version does not work, as you suspected. For example, if $Te\_n= (1/n)e\_{n+1}$ for $n\ge 0$ on $H=\ell^2(\mathbb Z)$, then the $T^n e\_0$, $n\ge 0$ are linearly independent. However, we can easily arrange matters in such a way that $T^N(e\_0+\delta e\_{-1})=0$ for a small $\delta>0$: $T$ will map $e\_{-1}$ t... | 3 | https://mathoverflow.net/users/48839 | 239213 | 110,432 |
https://mathoverflow.net/questions/239209 | 4 | This question has a subjective component but I would like answers that try to stick to concrete observable facts, such as which papers use which terminology. However, the informed *impressions* of those working with operator spaces or operator algebras is welcome.
The projective and Haagerup tensor products of operat... | https://mathoverflow.net/users/763 | Terminology: jointly completely bounded? | I would say that there is now a consensus to use the terminology of jointly completely bounded for the cb maps $E \to F^\*$ (and to use "completely bounded" for the one that correspond to the Haagerup tensor product, but I know less recent work on this notion).
At least all the papers from the current millenium that ... | 3 | https://mathoverflow.net/users/10265 | 239217 | 110,433 |
https://mathoverflow.net/questions/239194 | 2 | If I'm not wrong, it is easy to prove the following statement:
>
> If $n \leq 4$ is a natural number, if $\mathcal{F}$ is a union-closed family of non-empty sets, if the universe of $\mathcal{F}$ (i.e. the union of all members of $\mathcal{F}$) has exactly $n$ elements, if $\mathcal{F}$ is separating (i.e. for any ... | https://mathoverflow.net/users/82840 | Number of members of a separating union-closed family whose universe has given cardinality | The same statement in fact holds for all $n$.
**Theorem.** Let $\mathcal{F}$ be a separating, union-closed family of non-empty sets on the ground set $[n]$. Then $|\mathcal{F}| \geq n$, and if $|\mathcal{F}|=n$, then there exists some $i \in [n]$ such that $i \in F$ for all $F \in \mathcal{F}$.
*Proof.* Rename el... | 7 | https://mathoverflow.net/users/2233 | 239218 | 110,434 |
https://mathoverflow.net/questions/239222 | 6 | Given finite field $\mathbb{F}\_q$, positive integers $n$ and $k<n$. Given $k$-dimensional subspace $X$ of $\mathbb{F}\_q^n$, for which $m=m(q,k,n)$ may we find for sure a vector in $X$ with at least $m$ non-zero coordinates?
| https://mathoverflow.net/users/4312 | Vector with many non-zero coordinates | By Gauss' method, you may find a basis $v\_1,\dots,v\_k$ of $X$ such that $s(v\_1)<s(v\_2)<\dots<s(v\_k)$, where $s(v)$ is the number of the first nonzero entry in $v$. Using these vectors, you may inductively construct a vector with nonzero entries at positions $s(v\_1)$, \dots, $s(v\_k)$, so at least $k$ nonzero coor... | 6 | https://mathoverflow.net/users/17581 | 239223 | 110,435 |
https://mathoverflow.net/questions/239221 | 2 | Let $G$ be a directed graph on $n$ vertices. Let $H\_1$, ..., $H\_k$ be marked subgraphs of $G$. (Specifically, each $H\_i$ consists of a subset of the vertices of $G$ and a subset of the edges of the induced subgraph.) I am interested in an algorithm for finding the maximum number of edge-disjoint paths from $s \in V$... | https://mathoverflow.net/users/17883 | Edge-disjoint paths avoiding some subgraphs | This seems to be an NP-complete problem. It can be shown by reduction from
3-SAT, as follows.
For a given 3-SAT formula with $p$ variables and $m$ clauses, build a graph consisting of $2p$ internally disjoint paths between vertices $s$ and $t$, each path with $m+1$ internal vertices. For each variable $x\_i$ we have ... | 2 | https://mathoverflow.net/users/24076 | 239226 | 110,437 |
https://mathoverflow.net/questions/239195 | 8 | The question should be an elementary result in the theory of etale cohomology, but I failed to understand it because I am a complete beginner of the theory. So, I should apologise in advance for this rudimenatry question.
Let $C$ be a connected hyperbolic curve defined over an algebraically closed field of characteri... | https://mathoverflow.net/users/44005 | Group cohomology of fundamental group of a curve | My apologies for being terse, I don't have time at the moment for a more in-depth answer. Let me know if anything is unclear.
@1: Let $X$ be a noetherian scheme. Consider the `finite etale' site $FEt(X)$ of $X$: its objects are finite etale maps to $X$, with the etale topology. The forgetful functor to the etale site... | 7 | https://mathoverflow.net/users/3847 | 239229 | 110,438 |
https://mathoverflow.net/questions/235290 | 4 | Fix integers $V$ and $E\_{\text{max}}$, and consider graphs $G$ with $V$ vertices and at most $E\_{\text{max}}$ edges. What is the best lower bound that one can give on the number of distinct independent sets that exist in $G$?
(You can also phrase this in terms of cliques; we're asking if a graph with a large enough... | https://mathoverflow.net/users/3513 | What is the minimum number of independent sets for a graph with fixed numbers of vertices and edges? | I'll phrase the answer in terms of maximizing the number of cliques when $|E| \ge \gamma|V|^2$ with some constant $\gamma > 1/4$. Note that if $|E| \le |V|^2/4$, then by taking a bipartite graph with two (nearly) equal parts, we can make sure that there are no triangles, so that the only cliques are vertex singletons a... | 2 | https://mathoverflow.net/users/8297 | 239234 | 110,439 |
https://mathoverflow.net/questions/238425 | 4 | There is a theorem by Langlands and Shalika ([link](http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=PPN356556735_0038%7CLOG_0006&physid=PHYS_0007)) that the L-function of a cuspidal automorphic representation does not vanish on the line $\mathrm{Re}( s)=1$ (in their normalization which might be uncommon).
Is there... | https://mathoverflow.net/users/39304 | Non-vanishing of L-function of modular form | Yes, for primitive modular forms (both holomorphic and non-holomorphic) you can in fact something much stronger than the non-vanishing result with much (much!) less machinery than Langlands-Shalika.
It follows from the classic analytic properties of the Rankin-Selberg convolution of L-functions (proved in full in Iwa... | 3 | https://mathoverflow.net/users/43108 | 239244 | 110,442 |
https://mathoverflow.net/questions/239219 | 6 | In Huang & Lepowsky's series of papers *A theory of tensor products for module categories for a vertex operator algebra*, they defined for a rational vertex algebra $V$ the $P(z)$ tensor product of two modules $W\_1$ and $W\_2$ to be
$$
W\_1\boxtimes\_{P(z)} W\_2=\coprod\_k{(\mathcal{M}[P(z)]^{M\_k}\_{W\_1~W\_2})}^\*\... | https://mathoverflow.net/users/86652 | Do we have a braided tensor category for vertex algebra modules by using conformal blocks on an arbitary compact Riemann Surface? | In general, you won't get a vertex tensor category, because you don't get well-defined unit behavior when you use conformal blocks on higher genus surfaces.
Huang-Lepowsky assume the vertex operator algebra is rational and $C\_2$-cofinite, and this is conjecturally strong enough to obtain strong factorization propert... | 7 | https://mathoverflow.net/users/121 | 239250 | 110,445 |
https://mathoverflow.net/questions/238933 | 8 | In this [Paper](http://www.ams.org/journals/proc/1976-057-01/S0002-9939-1976-0402611-2/S0002-9939-1976-0402611-2.pdf) there is a proof that a closed plane curve of length
$L$ and curvature bounded by $K$ can be contained inside a circle of radius
$L/4 - (\pi - 2)/2K$. Are there similar results for smooth surfaces in $\... | https://mathoverflow.net/users/80084 | Surfaces contained in a ball | A possible generalization is an upper bound for the diameter of a surface $S \subset \mathbb{R}^3$ in terms of its area $A(S)$ and its gaussian and mean curvatures $K$, $H$. Let me give an answer in the case of positively curved surfaces.
We define an *ovaloid* as a compact connected surface $S \subset \mathbb{R}^3$ ... | 7 | https://mathoverflow.net/users/7460 | 239265 | 110,450 |
https://mathoverflow.net/questions/239275 | 2 | I have already asked this question on math.stackexchange.com
<https://math.stackexchange.com/questions/1789476/is-there-a-matrix-that-converts-the-gradient-of-any-function-to-gradient-of-othe>
Now I realize that Mathoverflow is probably better suited for this question.
The study of hamiltonian mechanics brought me to... | https://mathoverflow.net/users/91886 | Is there a matrix that converts the gradient of every possible function to gradient of other function? | The following considerations hold at least locally (in any sufficiently small open coordinate chart). A vector of functions $g\_i$ is of the form $g\_i = \partial\_i G$ for some function $G$ iff $\partial\_{[j} g\_{i]} = 0$ (Poincaré lemma). A direct calculation gives
$$
\partial\_{[k} g\_{j]}
= \partial\_{[k} (a\_{j... | 5 | https://mathoverflow.net/users/2622 | 239293 | 110,459 |
https://mathoverflow.net/questions/239262 | 9 | Let $G$ be a semisimple algebraic group over $\mathbb{Q}\_p$. Then by definition $G$ admits no non-trivial *algebraic* characters, i.e. homomorphisms $G \to \mathbb{G}\_m$.
However, it is quite possible that $G(\mathbb{Q}\_p)$ admits *topological* characters. E.g. take $G=\mathrm{PGL}\_n$ and consider the composition... | https://mathoverflow.net/users/5101 | Characters of simply connected semsimple algebraic groups over local fields | As I have written in a comment, the answer is YES (any abstract homomorphism into an abelian group is trivial) when $G$ is an *isotropic*, simply connected, simple algebraic group over a nonarchmedean local field $k$. For a proof see the book by Platonov and Rapinchuk, Section 7.2, Theorems 7.1 and 7.6.
Note that any s... | 6 | https://mathoverflow.net/users/4149 | 239307 | 110,464 |
https://mathoverflow.net/questions/239309 | 16 | Let $F\_n$ be the free group on $n$ letters.
The question is as in the title: letting $i:\text{Aut}(F\_n) \hookrightarrow \text{Aut}(F\_{n+1})$ be the natural injection, does there exist a homomorphism $\phi: \text{Aut}(F\_{n+1}) \rightarrow \text{Aut}(F\_n)$ such that $\phi \circ i = \text{id}$? My guess is "no", bu... | https://mathoverflow.net/users/91901 | Does the injection $\text{Aut}(F_n) \hookrightarrow \text{Aut}(F_{n+1})$ split? | [Bridson and Vogtmann](http://arxiv.org/abs/math/0209191) proved a much stronger result. From the abstract: 'If $m$ is less than $n$ then [the image of] a homomorphism $\mathrm{Aut}(F\_n)\to\mathrm{Aut}(F\_m)$ can have cardinality at most 2.'
| 21 | https://mathoverflow.net/users/1463 | 239313 | 110,466 |
https://mathoverflow.net/questions/239149 | 3 | A conjecture by Chen and Chvátal asks for the minimum number of induced "lines" in a metric space, in the same spirit as the [De Bruijn–Erdős theorem](https://en.wikipedia.org/wiki/De_Bruijn%E2%80%93Erd%C5%91s_theorem_(incidence_geometry)).
Though the statement of this problem on Douglas West's page on [the conjectur... | https://mathoverflow.net/users/30994 | Does the Chen-Chvátal Conjecture on metric spaces hold for maximal lines? | The following is the distance function for a metric space with five points.
$$
\begin{array}{r|cccc}
d & 0& 1& 2& 3& 4\\
\hline
0& 0& 3& 2& 3& 2\\
1& 3& 0& 3& 2& 3\\
2& 2& 3& 0& 5& 4\\
3& 3& 2& 5& 0& 3\\
4& 2& 3& 4& 3& 0\\
\end{array}
$$
It has the following three maximal lines:
$$\{\{0, 2, 3, 4\}, \{1, 4\}, \{0... | 2 | https://mathoverflow.net/users/30994 | 239314 | 110,467 |
https://mathoverflow.net/questions/239326 | 20 | I know that (co)ends (i.e. universal wedges) follow Fubini-like relation, i.e.
$$ \int\_{\langle c,d\rangle} F(c,d,c,d) \cong \int\_c\int\_d F(c,c,d,d) \cong \int\_d\int\_c F(c,c,d,d) $$
where we regard $F$ as both a functor $(C\times D)^{op}\times (C\times D)\to X$ and a functor $C^{op}\times C \times D^{op} \times D ... | https://mathoverflow.net/users/91913 | Why do we denote (co)ends with integral notation (beyond Fubini's Theorem)? | It's perhaps not *great* notation, but some of the most common types of coends arising in practice (namely, weighted colimits) can be thought of roughly as "categorified weighted sums".
In enriched category theory (over a complete, cocomplete symmetric monoidal closed category $V$), a *weight* consists of a small $V... | 17 | https://mathoverflow.net/users/2926 | 239331 | 110,473 |
https://mathoverflow.net/questions/239253 | 3 | In the usual setup, consider the category of Harish-Chandra $(\mathfrak{g},K)$-modules with given central character (if the central character is regular, this is equivalent to $K$-equivariant $D$-modules on the flag variety). Assume that everything is split.
I was said that the principal series modules in this catego... | https://mathoverflow.net/users/2095 | Reference request: Principal series are equal in the Grothendieck group | I think the fact that you are referring to is the following, which holds in some generality. Suppose $M$ is a Levi factor in $G$ and $\pi$ is an irreducible representation of $M$. Suppose $P=MN$ is a parabolic subgroup containing $M$. Then the image of $Ind\_{MN}^G(\pi\otimes 1)$ in the Grothendieck group is independen... | 5 | https://mathoverflow.net/users/6030 | 239332 | 110,474 |
https://mathoverflow.net/questions/239312 | 16 | A bit of context for this question: as a project for my master's degree my supervisor asked me to understand the construction of Milnor's exotic spheres. After learning the heavy material (I knew very little algebraic topology so learning about characteristic classes counted as "heavy" for me) the proof of existence of... | https://mathoverflow.net/users/86065 | Nice things that can be proved easily with characteristic classes | In [this blog post](https://qchu.wordpress.com/2014/06/16/hypersurfaces-4-manifolds-and-characteristic-classes/) you'll find a computation of the cohomology ring of a hypersurface of degree $d$ in $\mathbb{CP}^3$ using characteristic classes. This turns out to be a weirdly good exercise in using characteristic classes:... | 16 | https://mathoverflow.net/users/290 | 239335 | 110,475 |
https://mathoverflow.net/questions/239341 | 3 | Let's consider a smooth sphere bundle over a smooth manifold with structure group is equal to the diffeomorphism group of sphere. Then, can we say that this is a double of some disk bundle? Thank you for your helping.
| https://mathoverflow.net/users/85988 | Is it true that all sphere bundles are some double of disk bundle? | No. If a sphere bundle is the double of a disk bundle, then it has a section. You get counterexamples by considering unit sphere bundles of vector bundles with nonvanishing Euler class.
| 14 | https://mathoverflow.net/users/70808 | 239342 | 110,478 |
https://mathoverflow.net/questions/239346 | 8 | Say we have a number field $K$. Let $G\_K = \text{Gal}(\overline{K}/K)$. Let $M$ be a discrete $G\_K$-module. We know that $H^1(K, M) := H^1(G\_K, M)$, i.e. profinite group cohomology. For each place $v$ of $K$, let $K\_v$ be the completion of $K$ at $v$; restriction to a decomposition group $G\_v$ at $v$ defines a hom... | https://mathoverflow.net/users/91923 | Group cohomology question, trivial Galois action on discrete Galois module means we can say what about kernel of map | If the $G\_K$-action on $M$ is trivial, then
$$H^1(K,M)=\mathrm{Hom}(G\_K,M),$$
and by Chebotarev's density theorem
$$ F^1(K,M)=0.$$
For details see Lemma 1.1(i) of Sansuc, J.-J. Groupe de Brauer et arithmétique des groupes algébriques linéaires sur un corps de nombres. J. Reine Angew. Math. 327 (1981), 12–80.
| 8 | https://mathoverflow.net/users/4149 | 239348 | 110,481 |
https://mathoverflow.net/questions/239363 | 4 | It is well known that every local field (i.e. nondiscrete topological field locally compact with respect to the topology) is the completion of some global field. I know the argument, a nice exposition is
<http://math.stanford.edu/~conrad/248APage/handouts/localglobal.pdf>
My question is, does anyone know a *publis... | https://mathoverflow.net/users/50886 | Reference for: Every local field can be realized as the completion of a global field | F. Lorenz: *[Algebra. Volume II: Fields with Structure, Algebras and Advanced Topics](http://www.springer.com/jp/book/9780387724874)*, Theorem 2 p. 78.
| 4 | https://mathoverflow.net/users/7460 | 239366 | 110,486 |
https://mathoverflow.net/questions/239368 | 4 | I am looking for criteria for the irreducibility of monic polynomials with constant term $\pm1$ over $\mathbb Q$. Eisenstein's criterion clearly doesn't apply here.
For instance, for the family of polynomials
$$
p\_n(x)=x^n-x^{n-1}-1
$$
Wolfram Alpha suggests that $p\_n$ is probably irreducible for $n\ge6$. How does... | https://mathoverflow.net/users/8131 | Irreducible monic polynomials | The book by Prasolov "Polynomials" has lot of information that you seek.
To give one specific instance take the infinite series of $e^x$ truncate it. Even though it is not in the form you desire, it is an irreducible polynomial. This result is due to Schur and is very old. This has been generalized, and it may be in ... | 4 | https://mathoverflow.net/users/22878 | 239370 | 110,488 |
https://mathoverflow.net/questions/239381 | 4 | I learned in J. Castillo's Hitchhiker guide to categorical Banach space theory that, by a theorem of Semadeni and Zidenberg, limits and colimits exist in the category $\text{Ban}\_1$ of Banach spaces and contractive maps. I am hoping now that the Banach space dual of a projective limit is the inductive limit of the cor... | https://mathoverflow.net/users/14756 | Dual of colimit in $\text{Ban}_1$ | If I recall correctly, the duality functor $D: {\sf Ban}\_1 \to {\sf Ban}\_1^{\rm op}$ is a left adjoint -- the analogous statement for ${\sf Vect}$ and ${\sf Vect}^{\rm op}$ is in Mac Lane's CFTWM somewhere -- and so $D$ should preserve colimits. That is, if we think of it as a contravariant functor on ${\sf Ban\_1}$,... | 4 | https://mathoverflow.net/users/763 | 239388 | 110,493 |
https://mathoverflow.net/questions/239383 | 29 | I understand that the homotopy category of (pointed) topological spaces and continuous maps is not complete. Nor is it cocomplete. In particular it neither has all pullbacks nor all pushouts.
What are some simple examples of spans $Z\leftarrow X\rightarrow Y$ and cospans $Z\rightarrow X\leftarrow Y$ that cannot be co... | https://mathoverflow.net/users/54788 | The homotopy category is not complete nor cocomplete | I'll work in the based category, and consider $S^1$ as $\{z\in\mathbb{C}:|z|=1\}$. Consider the maps
$$\text{point}\xleftarrow{}S^1\xrightarrow{f}S^1, $$
where $f(z)=z^2$. Suppose that there is a pushout $P$. We would then have a natural isomorphism $[P,X]=\text{Hom}(\mathbb{Z}/2,\pi\_1(X))$. On the other hand, the f... | 38 | https://mathoverflow.net/users/10366 | 239391 | 110,495 |
https://mathoverflow.net/questions/239316 | 3 | A *$\kappa$-tower* in $\mathbb{N}^\mathbb{N}$ is a sequence
$\langle a\_\alpha : \alpha<\kappa\rangle$ in $\mathbb{N}^\mathbb{N}$
that is $\le^\*$-increasing with $\alpha$
and has no $\le^\*$-upper bound.
Piotr Szewczak and I need a reference for the consistency of the existence of a $\kappa$-tower in $\mathbb{N}^\... | https://mathoverflow.net/users/2415 | Reference request: The consistency of a tall tower in $\mathbb{N}^\mathbb{N}$ | A theorem of Hechler says that, given any poset in which every countable subset has a strict upper bound, you can arrange for that poset to be cofinal in the $\leq^\*$ ordering of $\mathbb N^{\mathbb N}$. Apply that to the poset $\omega\_1\times\omega\_2$ (ordered componentwise). Then $\mathfrak b=\aleph\_1$, but $\mat... | 6 | https://mathoverflow.net/users/6794 | 239394 | 110,496 |
https://mathoverflow.net/questions/239374 | 4 | I learnt that $\mathbb{P}^1 \times \mathbb{P}^1$ is rigid, but can be deformed to a non-rigid Hirzebruch surface $S$. Suppose $\pi: M \to B$ is such deformation such that $\mathbb{P}^1 \times \mathbb{P}^1 \cong M\_{t\_0}$ and $S \cong M\_{t\_1}$.
I want to understand the meaning of "rigidity". Does this mean that:
(... | https://mathoverflow.net/users/29730 | Deformation of $\mathbb{P}^1 \times \mathbb{P}^1$ | Rigidity means, if you have a family $F$ of surfaces such that one $f\in F$ of them is $\mathbb P^1 \times \mathbb P^1$, then there is an open set $U\subseteq F, U \ni f$ each of whom is $\mathbb P^1 \times \mathbb P^1$.
The source of confusion, I think, is in mixing up "family of varieties" with "moduli space of com... | 6 | https://mathoverflow.net/users/391 | 239398 | 110,498 |
https://mathoverflow.net/questions/239396 | 8 | Let $k$ be a perfect field, and let $\bar k$ be a fixed algebraic closure of $k$.
Let $\overline{X}$ be a nonempty smooth algebraic variety over $\bar k$.
Does there exist a natural number $d=d(\overline{X})$ with the following property:
>
> For any $k$-form $X$ of $\overline{X}$, the variety $X$ has a $K$-point ov... | https://mathoverflow.net/users/4149 | Algebraic points of uniformly bounded degree on an algebraic variety | No: this fails already when $X=E$ is an elliptic curve and $k=\mathbb{Q}$. This would imply that every element of $H^1(\mathbb{Q},E)$ has order at most $d$, and I'm pretty sure that this cohomology group has elements of arbitrarily large order. Otherwise the Tate-Shafarevich conjecture would be rather trivial, as it is... | 13 | https://mathoverflow.net/users/5101 | 239404 | 110,501 |
https://mathoverflow.net/questions/239399 | 4 | We know from the classy work of Joyce that "any compact Lie group becomes hypercomplex after it is multiplied by a sufficiently big torus". The quote comes from the Wikipedia [page](https://en.wikipedia.org/wiki/Hypercomplex_manifold).
I am asking if it is known that these hypercomplex manifolds are hyper-Kaehler (or... | https://mathoverflow.net/users/42100 | Hyper-Kaehler Strucutre for Compact Lie Groups? | The answer is already 'no' for the simplest case:
$$
S^1\times \mathrm{SU}(2) = (\mathbb{H}{\setminus}\{0\})/\mathbb{Z},
$$
which is clearly hypercomplex, but cannot even be Kähler, much less hyperKähler.
| 8 | https://mathoverflow.net/users/13972 | 239409 | 110,502 |
https://mathoverflow.net/questions/239407 | 10 | I'm a little embarrassed to be asking this, but surely there is a simple argument that I didn't see?
Let $(f\_\lambda)$ be a net in $l^\infty$ which converges weak\* to $f \in l^\infty$. We do not assume the net is bounded. Does the net $(f\_\lambda^+)$ converge weak\* to $f^+$, where $f^+ = \max(f,0)$ is the positiv... | https://mathoverflow.net/users/23141 | Weak* continuity of positive parts | Given a finite set $\cal F$ of functions in $\ell\_1$, choose a function $z\_{\cal F}$ in $\ell\_\infty$ s.t. $\langle z\_{\cal F}, x \rangle =0$ for all $x$ in $\cal F$ s.t. $z\_{\cal F}$ has at least one positive coordinate, and normalized s.t. $\langle z^+\_{\cal F}, u \rangle = 1$, where $u := \sum\_{n=1}^\infty 2^... | 9 | https://mathoverflow.net/users/2554 | 239415 | 110,505 |
https://mathoverflow.net/questions/239406 | 9 | A motivation: The **classical** Stone-Weierstrass theorem says that polynomials are dense among continuous functions (say, on the unit interval), while the **abstract** Stone-Weierstrass theorem (and also the related Kakutani-Krein theorem) give sufficient conditions for a set of continuous functions to be dense.
The... | https://mathoverflow.net/users/1516 | Abstract result on partitions of unity? | I will leave to Yemon Choi discussing the answer from Gelfand-Raikov-Shilov's book (*Commutative Normed Rings*, I suppose?), and restrict myself to more recent discussions on the matter...
There is an extensive discussion on abstract partitions of unity in Chapter III of the book of Andreas Kriegl and Peter Michor, [... | 14 | https://mathoverflow.net/users/11211 | 239418 | 110,506 |
https://mathoverflow.net/questions/239437 | 6 | Is the property of being polynomially convex a topological invariant?
In other words, let $M$ and $N$ be two homeomorphic, compact subsets of $n$-dimensional complex Euclidean space, and assume in addition that $M$ is polynomially convex.
>
> Is $N$ necessarily polynomial convex?
>
>
>
| https://mathoverflow.net/users/36688 | Is polynomial convexity a topological invariant? | The answer is *no*.
In fact, Kallin has shown in **[Kal64]** that the union of three disjoint closed balls is polynomially convex, but the union of three disjoint closed polydisks needs not to be polynomially convex.
Actually, it turns out that polynomial convexity is not even preserved by (real) linear transforma... | 10 | https://mathoverflow.net/users/7460 | 239439 | 110,511 |
https://mathoverflow.net/questions/239345 | 18 | I've heard of this result in a paper on which Yves André proves the p-adic analogue (that is, $\mathrm{Gal}(\overline{\mathbb{Q}}\_p/\mathbb{Q}\_p)=\mathrm{Aut}\underline\pi^{temp}\_{{\mathbb{C}}\_p}$), conditional on Pop's theorem. In there, it is referenced as unpublished at the time (2002):
* F. Pop, A combinatori... | https://mathoverflow.net/users/43108 | Pop's proof that $\mathrm{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})=\mathrm{Aut}\underline\pi^{alg}_{\overline{\mathbb{Q}}}$ | Florian Pop has a manuscript on his web page, [On I/OM](https://www.math.upenn.edu/~pop/Research/files-Res/IOM_29May14.pdf), dated 29 May 2014, where he proves a pro-$\ell$-abelian by central version of the Ihara/Oda-Matsumoto conjecture over any field, and explains how it implies the initial conjecture over the ration... | 7 | https://mathoverflow.net/users/10696 | 239442 | 110,512 |
https://mathoverflow.net/questions/236111 | 4 | Let $G(V,E)$ be a graph. A path whose length is equal to the diameter of a graph is called a **diametral** path. In a cycle graph every vertex has $2$ diametral paths. Now I need to prove that this:
>
> If each $v \in V(G) $ has the same number of diametral paths initiated from it, then $G$ is a regular graph.
>
>... | https://mathoverflow.net/users/78180 | Relation between diametral path and regularity of a graph | The claim is actually **false**, for both versions of the problem. That is, we may define a *diametral path* of $G$ as a path with length equal to the diameter of $G$ (I think this is the OP's intent), or as a path with length equal to the diameter with the additional property that it is a shortest path between its end... | 4 | https://mathoverflow.net/users/2233 | 239446 | 110,513 |
https://mathoverflow.net/questions/239188 | 3 | Let $G$ be a semi-simple algebraic group over $\mathbb{Q}$, I would like to find an integer $d>0$ only depending on $G$ with the following property. For any two semi-simple $\mathbb{Q}$-subgroups $H\_1,H\_2\subset G$ that are conjugated by an element $g\in G(\mathbb{R})$, there exists a number field $K\subset \mathbb{R... | https://mathoverflow.net/users/90978 | A more precise description of conjugation of semi-simple subgroups | The answer is YES.
>
> **Theorem 1.** Let $G$ be a connected semisimple linear algebraic group defined over a number field $k\subset{\mathbb{R}}$.
> There exists a natural number $d=d(G\_{\bar{k}})$ with the following property:
> let $H\_1$ and $H\_2$ be two connected semisimple $k$-subgroups of $G$ that are conjug... | 4 | https://mathoverflow.net/users/4149 | 239450 | 110,515 |
https://mathoverflow.net/questions/239445 | 4 | Let $X$ be a compact complex $n$-manifold and $D$ be a smooth comdimension $1$ submanifold. Also let $U:= X\setminus D$ and $j$ be the inclusion of $U$ in $X$.
Then it is well known that the following (long exact) sequence respects mixed Hodge structures:
$$ \cdots\to H^{k-2}(D)(-1)\stackrel{\gamma\_k}{\to}H^k(X)\st... | https://mathoverflow.net/users/7494 | The compatibility of the Gysin sequence with mixed Hodge structures | Yes, it's fine also if $X$ is not compact. More generally you have for any variety $X$ and subvariety $Z$ a long exact sequence
$$ \ldots \to H^k\_c(X \setminus Z) \to H^k\_c(X) \to H^k\_c(Z) \to H^{k+1}\_c(X \setminus Z) \to \ldots $$
of mixed Hodge structures, which gives your long exact sequence by Poincaré duality ... | 3 | https://mathoverflow.net/users/1310 | 239453 | 110,517 |
https://mathoverflow.net/questions/239202 | 12 | Let $\Gamma$ be a $C^2$ compact submanifold of $\mathbb{R}^n$. Consider the distance function $\delta$ from $\Gamma$. It is well known that, for sufficiently small $\varepsilon>0$, $\delta$ is $C^2$ on $\{ 0<\delta < \varepsilon\}$, and that it satisfies the eikonal equation
$$ \| \nabla \delta \| = 1, \qquad \text{w... | https://mathoverflow.net/users/13915 | Unexpected regularity of the distance from a $C^2$ submanifold | Since this question seems to have attracted some interest, I will post my own answer (which is a proof of the statement in the comments by Anton).
**If $u : M \to \mathbb{R}$ is a $C^2$ solution of the Eikonal equation
$$ \| \nabla u\| = 1, $$
then $\mathrm{Hess}(u)$, which a priori is only continuous, is smooth alon... | 3 | https://mathoverflow.net/users/13915 | 239455 | 110,519 |
https://mathoverflow.net/questions/239459 | 15 | Let $E$ be an elliptic curve over $k=\mathbb{Q}$. Consider $H^1(k,E)$.
In [this answer](https://mathoverflow.net/a/239404/4149) Daniel Loughran writes: "I'm pretty sure that this cohomology group has elements of arbitrarily large order". I would be happy to have an explanation of this fact and/or a reference.
(I usu... | https://mathoverflow.net/users/4149 | Elements of arbitrary large order in the first Galois cohomology of an elliptic curve | Here is the kind of method I had in mind.
We have the elliptic curve Kummer sequence
$$0 \to E[n] \to E \to E \to 0,$$
Here I denote by $E[n]$ the $n$-torsion group scheme of $E$. Applying Galois cohomology we obtain
$$0 \to E(\mathbb{Q})/nE(\mathbb{Q}) \to H^1(\mathbb{Q}, E[n]) \to H^1(\mathbb{Q}, E)[n] \to 0.$$
B... | 9 | https://mathoverflow.net/users/5101 | 239461 | 110,521 |
https://mathoverflow.net/questions/239448 | 4 | Typically the Euler-Lagrange equations are defined for the functional
$$ J[u] = \int\_a^b L(x,u,u') dx. $$
However, I was wondering if anyone knows if they can be solved when the expression involves the inverse of $u$? The way my problem is formulated, it is simplest to write as
$$ J[u] = \int\_{u^{-1}(a)}^{u^{-1... | https://mathoverflow.net/users/91965 | Calculus of variations when functional involves inverse of the function | Probably, the best thing to do would be to write $x = f(u)$ and then use
$$
\int\_{u^{-1}(a)}^{u^{-1}(b)} L(x,u,u') dx = \int\_a^b L\left(f(u),u,\frac{1}{f'(u)}\right)f'(u)\ du
= \int\_a^b M\left(u,f(u),f'(u)\right)\ du
$$
and now compute the Euler-Lagrange equation for $M$, which will give you the differential equatio... | 4 | https://mathoverflow.net/users/13972 | 239476 | 110,523 |
https://mathoverflow.net/questions/239471 | 5 | If $X$ is a separable Banach space and $(x\_n)$ is a basic sequence, then we can define biorthogonal functionals $(x^{\*}\_n)$ in $X^{\*}$ such that $x^{\*}\_n(x\_k)=\delta\_{nk}$.
What about conversely? If $(x^{\*}\_n)$ is a basic sequence in $X^{\*}$, can we always find vectors $(x\_n)$ in $X$ such that $x^{\*}\_n... | https://mathoverflow.net/users/69275 | Biorthogonal functionals | No, that's not true. Let $\mathbb{N}^\* = \mathbb{N} \cup \{\infty\}$ and set $X = C(\mathbb{N}^\*) \cong c$ and $X^\* = l^1(\mathbb{N}^\*) \cong l^1$. Take as the basic sequence of $X^\*$ the vectors $e\_n$ for $n \in \mathbb{N}^\*$. There is no vector $x$ in $c$ with $e\_n(x) = 0$ for all $n \in \mathbb{N}$ but $e\_\... | 4 | https://mathoverflow.net/users/23141 | 239479 | 110,524 |
https://mathoverflow.net/questions/239485 | 4 | This is a follow up to the question: [Biorthogonal functionals](https://mathoverflow.net/questions/239471/biorthogonal-functionals). A positive answer to that question implies a negative answer to this one.
If $X$ is a separable Banach space, can we find a basic sequence $(x^{\*}\_n)$ in $X^{\*}$ with the property t... | https://mathoverflow.net/users/69275 | Trivial intersection of kernels | As I understand you can answer this question in the negative using the notion of $w^\*$-basic sequence introduced by Johnson and Rosenthal (1972), see Lindenstrauss-Tzafriri, Classical Banach spaces, vol. I, p. 11. In the case where the sequence $\{x\_n^\*\}$ is weak$^\*$-convergent to $0$, we can find a $w^\*$-basic s... | 3 | https://mathoverflow.net/users/37822 | 239486 | 110,526 |
https://mathoverflow.net/questions/239482 | 14 | I am interested in seeing examples of a space $X$ (preferably a closed smooth manifold, but any finite-dimensional CW-complex would also be of interest) with a vector bundle $\xi\colon E \to X$ on it, so that there is exactly one index $i$ with $w\_i(\xi) \neq 0$, and $i$ is bigger than $8$. Here are some remarks:
(1... | https://mathoverflow.net/users/14233 | Vector bundles with exactly one nonzero SW-class | For $i = 2^k$ you can get an example with $X = \Bbb{RP}^{m}$ for any $m \geq 2^k$. If $L$ is the canonical line bundle on $\Bbb{RP}^{m}$ then let $E = \bigoplus\_{i=1}^{2^k} L$ be the a sum of several copies of it. Then it has total Stiefel-Whitney class
$$
w(E) = \prod\_{i=1}^{2^k} w(L) = (1+x)^{2^k} = 1 + x^{2^k},
$$... | 12 | https://mathoverflow.net/users/360 | 239491 | 110,527 |
https://mathoverflow.net/questions/239380 | 2 | I understand that the key to retrieve eigenvectors in the non-shifted QR algorithm is to accumulate the transformations at each steps in the following way:
$Q = \Pi\_i Q\_i$
Can we accumulate the transformations in the same way when we are using the shifted QR algorithm?
$(A\_i-\mu I) = Q\_i R\_i$
$R\_i Q\_i + ... | https://mathoverflow.net/users/91940 | How to retrieve eigenvectors from shifted QR algorithm? | At each step, the shifted and unshifted matrices, $A-\lambda I$ and $A$, have the same eigenvectors. So the diagonalizing transformation $A = Q D Q^T$, with $D$ diagonal, is accumulated in the same way as in the unshifted algorithm, $Q = Q\_1 Q\_2 \cdots Q\_k$, where $k$ is the last step of the iteration.
| 2 | https://mathoverflow.net/users/2622 | 239506 | 110,529 |
https://mathoverflow.net/questions/239513 | 4 | Let $E$ be an elliptic curve defined over $\mathbf{Q}$, fix an odd prime $p>3$, let $T\_p$ denote the $p$-adic Tate module of $E$, and let $V\_p = T\_p \otimes \mathbf{Q}\_p$.
If the action of $G\_\mathbf{Q}$ is unramified at $\ell$, it is known that the characteristic polynomial of Frobenius, $\mathrm{Frob}\_\ell$, ... | https://mathoverflow.net/users/10547 | Frobenius at ramified primes | Are you looking for the following sort of answer? $a\_\ell(E)=1$ if $E$ has split multiplicative reduction, $a\_\ell(E)=-1$ if $E$ has non-split multiplicative reduction, and $a\_\ell(E)=0$ if $E$ has additive reduction. This gives the "right" local factors for the $L$-series, i.e., $L\_p=1\pm p^{-s}$ for multiplicativ... | 8 | https://mathoverflow.net/users/11926 | 239521 | 110,533 |
https://mathoverflow.net/questions/239518 | 3 | The definition of a reductive group scheme is as in SGA III. Frankly, I only know that they exist for the adjoint group (the adjoint representation). In SGA III, I could only find a result for general groups over a regular ring of dimension $\leq 2.$ But since reductive groups are especially nice, maybe they do have su... | https://mathoverflow.net/users/91996 | Do all reductive group schemes over semilocal rings admit finite-dimensional free faithful representations? | In Corollary 3.2 of the paper
R. W. Thomason, Equivariant resolution, linearization, and Hilbert’s fourteenth problem over arbitrary base schemes, Adv. Math. 65 (1987), 16–34,
this was proved for semisimple group schemes or, more generally, for reductive group schemes which are either split reductive, or semisimple... | 6 | https://mathoverflow.net/users/89948 | 239522 | 110,534 |
https://mathoverflow.net/questions/239498 | 1 | Tony Huynh gave a nice answer to a question I asked here :
[Number of members of a separating union-closed family whose universe has given cardinality](https://mathoverflow.net/questions/239194/number-of-members-of-a-separating-union-closed-family-whose-universe-has-given-c)
The answer shows in fact that if $\mathc... | https://mathoverflow.net/users/82840 | Intersection of members in a separating union-closed family of sets | Statement (3) is easier to prove directly by induction on $n=|U(\mathcal{F})|$.
The base case $n=1$ is trivial.
To make the induction step for $n>1$, let $x\in U(\mathcal{F})$ be an element that belongs to at least $n$ sets from $\mathcal{F}$, such element exists by (1). Consider two cases:
* if $r=n$, take any $... | 1 | https://mathoverflow.net/users/7076 | 239549 | 110,539 |
https://mathoverflow.net/questions/239547 | 7 | Let $X\subset\mathbb{P}^n$ be a hypersurface singular at finitely many points $p\_i\in X$. We may assume that $X$ has ordinary singularities at the $p\_i$'s.
Does there exists a formula, perhaps in terms of $deg(X)$ of the multiplicities of $X$ at the $p\_i$'s, for the dimension of $H^1(X,T\_X)$, where $T\_X = \mathc... | https://mathoverflow.net/users/nan | Cohomology of tangent sheaf of a singular hypersurface | Put $d:=\deg(X)$. From the exact sequence
$$0\rightarrow \mathcal{O}\_X(-d)\rightarrow \Omega ^1\_{\mathbb{P}^n|X}\rightarrow \Omega ^1\_X\rightarrow 0$$you get an exact sequence $\ 0\rightarrow T\_X\rightarrow T\_{\mathbb{P}^n|X}\rightarrow \mathcal{O}\_X(d)\rightarrow \mathcal{E}xt^1(\Omega ^1\_X,\mathcal{O}\_X)\righ... | 10 | https://mathoverflow.net/users/40297 | 239554 | 110,540 |
https://mathoverflow.net/questions/239567 | 2 | Let $\mathbb{R}P^m$ be the $m$-dimensional real projective space and let $\mathbb{R}P^m\setminus\{\*\}$ be the punctured space. I observe:
1. $\mathbb{R}P^2\setminus\{\*\}$ is homeomorphic to a (open) Mobius strip, that is, the total space of a non-trivial line bundle over $\mathbb{R}P^1$.
2. $\mathbb{R}P^1\setminus\... | https://mathoverflow.net/users/41075 | homeomorphism type of punctured real projective spaces | (i) is true and (ii) is false. In fact, (i) is true for all $m\ge1$. More precisely, dropping a coordinate yields a map from $\mathbb RP^{m+1}\setminus\{\*\}\to\mathbb RP^m$ which is a locally trivial bundle with fiber $\mathbb R$. This bundle is not trivial for $m\ge1$ because, e.g., $\mathbb R P^m$ is orientable if a... | 3 | https://mathoverflow.net/users/89948 | 239568 | 110,542 |
https://mathoverflow.net/questions/219796 | 17 | Suppose that $F(x,y)$ is a binary form of degree $d \geq 3$ with integral coefficients, and non-zero discriminant. It is known (from a paper due to Erdős and Mahler from 1938) that the density of integers in $[1,B]$ which can be represented by $F$ is $\Theta(B^{2/d})$ as $B \rightarrow \infty$, provided that $F(1,0) > ... | https://mathoverflow.net/users/10898 | The density of integers represented by a binary form | This question was open in general for degree $d \geq 5$ at the time this question was posted and in addition to the irreducible cubic case Hooley also dealt with the special case of bi-quadratic quartic forms in a paper in 1986 (bi-quadratic as in forms of the shape $Ax^4 + Bx^2y^2 + Cy^4$); see <http://www.degruyter.c... | 4 | https://mathoverflow.net/users/10898 | 239576 | 110,547 |
https://mathoverflow.net/questions/239161 | 14 | Let $G$ be a compact connected simple Lie group, $T$ a maximal torus, $N(T)$ the normalizer of $T$, and $W=N(T)/T$ the Weyl group. It is well-known that $H^\*(G/T,\mathbb{Q})$ is the regular representation of $W$ induced by the $W$-action on $G/T$, and hence $H^\*(G/N(T), \mathbb{Q})\cong H^\*(G/T, \mathbb{Q})^W\cong\m... | https://mathoverflow.net/users/2306 | Integral cohomology of $G/N(T)$ | There are people on MO more knowledgeable about this topic than I, but since it has been a few days, I will say a few words.
In this case there is a "third isomorphism" of spaces:
**Proposition** $G/N(T)\cong \mathcal{F}/W$
**Proof** The flag variety $\mathcal{F}\cong G/T\cong G\_{\mathbb{C}}/B$ and $W=N(T)/T$ is... | 10 | https://mathoverflow.net/users/12218 | 239588 | 110,550 |
https://mathoverflow.net/questions/239582 | 9 |
>
> **Main Question:** What Is the correpondence between flows and vector
> fields in algebraic geometry?
>
>
>
Here is a more precise statement could be an answer If it was true (I have no idea it is):
>
> **"Proposition": Let $X$ be a *nice enough* (intentionally ambiguous) scheme over a field $k$ with ta... | https://mathoverflow.net/users/22810 | Algebro-geometric version of {vector fields} $\longleftrightarrow$ {flows} correspondence? | You need a characteristic zero assumption, and you need some additional axioms on the homorphism to make it a bijection.
The map from derivations $D$ to homomorphisms sends a function $y \in \mathcal O\_{X,x}$ to $$e^{t D} y = y + (Dy) t + (D^2 y) t^2/2 + (D^3 y) t^3/6 + \dots$$
The inverse map is just going to sen... | 10 | https://mathoverflow.net/users/18060 | 239589 | 110,551 |
https://mathoverflow.net/questions/239578 | 3 | How to compute the value of $$[\gcd(1,x)+\gcd(2,x)+\gcd(3,x)+....+\gcd(x,x)]$$ efficiently?
When x can be as large as million.
| https://mathoverflow.net/users/92042 | Calculating greatest common divisor series: $\gcd(1,x)+\gcd(2,x)+\gcd(3,x)+....+\gcd(x,x)$ | It seems the most efficient way to compute it is though the prime number decomposition of $x$. if $x$ is only "as large as million" it is still relatively effective.
Assume first that $x = a b $ with $a$ and $b$ prime between them. Then:
$$ \sum\_{i=1} ^ x \gcd(x,i) = \sum\_{i \in (\mathbb{Z}/x \mathbb{Z})} \gcd(i... | 2 | https://mathoverflow.net/users/22131 | 239595 | 110,555 |
https://mathoverflow.net/questions/239597 | 8 | Is it true that every holomorphic vector bundle over $\mathbb{C}^{n}\setminus 0$ is trivial? If not true, how can one construct a counterexample?
And just a small note here (**wrong**):
* *For $n\leq 2$, we can push the bundle $\mathcal{F}$ over $\mathbb{C}^{n}\setminus 0$ to be defined on $\mathbb{C}^{n}$, denote... | https://mathoverflow.net/users/nan | Holomorphic vector bundles over $\mathbb{C}^{n}\setminus 0$ | For $n\geq 3$, there exists such non-trivial bundles. For example, take the bundle on $\mathbb{C}^n-\{0\}$ given by the quotient of $\mathcal{O}^n$ by the subbundle generated by the vector $(x\_1,\ldots,x\_n)$, where the $x\_i$s are the coordinate functions. Then, for $n\geq 3$, this bundle is not trivial. The non-triv... | 11 | https://mathoverflow.net/users/9502 | 239601 | 110,557 |
https://mathoverflow.net/questions/238375 | 8 | Let $(M,g)$ be a simply-connected compact surface with boundary $\partial M$ and metric $g$. Let $N$ denote the outward unit normal on $\partial M$, $\nabla$ the Levi-Civita connection and $\Delta\_g$ the corresponding connection Laplacian (i.e. the trace of the Hessian).
I am interested in the properties of the vect... | https://mathoverflow.net/users/91411 | Properties of connection Laplacian on vector fields | I don't think such a result is possible without any assumptions on the geometry of the boundary.
Consider the scalar case. If $M$ is a closed manifold with $\operatorname{Ric} \geq (n-1)k g$ for some positive constant $k$, then $\lambda\_1(M) \geq nk$. The proof (which is due to Lichnerowicz and is summarized in Obat... | 2 | https://mathoverflow.net/users/91324 | 239604 | 110,559 |
https://mathoverflow.net/questions/239607 | 4 | Working over a field $k$, Nagata's compactification theorem implies that any separated scheme $X$ of finite type over $k$ admits a compactification (a dense open immersion $i \colon X\hookrightarrow\bar{X}$ into a proper scheme $\bar{X}$ over $k$).
Let $\mathbf{V}/k$ be the category of separated schemes of finite typ... | https://mathoverflow.net/users/91373 | Minimal Nagata-like compactification | The following argument shows that the very best thing you could hope for does not work. I'm not sure whether I need all of my assumptions.
>
>
> >
> > **Claim:** There does not exist a compactification functor $(C,\eta)$ such that $\eta\_{\mathbb A^2}$ is a [dense] open immersion into a smooth proper [hence proje... | 6 | https://mathoverflow.net/users/82179 | 239622 | 110,565 |
https://mathoverflow.net/questions/239565 | 4 | I have a random walk on $\mathbb{Z}$ with starting point $0$ and with length $n$ and possible steps to right, left or stay where you are, all with the same probabilities. I am interested in exact probability that the walk will visit $k$ distinct values.
I think this should be known but I cannot find such result.
| https://mathoverflow.net/users/nan | Range of random walk | Just an addition to Pablo Lessa's comment. If probability to stay at a point is zero, then you have a simple random walk and the Reflection Principle is valid. So, your question will follow if you find the joint distribution of the maximum and minimum, as the simple random walk visits all points between them. The exact... | 2 | https://mathoverflow.net/users/85303 | 239625 | 110,566 |
https://mathoverflow.net/questions/239623 | 7 | Let $X$ be a **positive** real-valued random variable. Let $Y$ be an independent copy of $X$ and assume that the equality $X+Y=2X$ holds in distribution. Does this imply that $X$ is constant?
| https://mathoverflow.net/users/56188 | Distributional equation X+Y=2X | Yes, it does. As the random variable is positive, consider the Laplace transform $f(\lambda)=E \exp(-\lambda X)$ (instead of the Fourier one).
Then, one has $f^2(\lambda)= f(2\lambda)$.
Taking the logarithm, one gets $2\log f(\lambda)=\log f(2\lambda)$, and for the function $g(\lambda)=\frac{\log f(\lambda)}{\lambd... | 13 | https://mathoverflow.net/users/31371 | 239635 | 110,568 |
https://mathoverflow.net/questions/239493 | 13 | Let $k$ be a number field and $M$ be a nonzero finite discrete $\mathrm{Gal}(\bar k/k)$-module. Is it true that $H^1(k,M)$ is infinite?
This would complete the [answer of Daniel Loughran](https://mathoverflow.net/a/239461/4149). There is a comment of nfdc to that answer, but I think that this question deserves an ans... | https://mathoverflow.net/users/4149 | Infiniteness of the Galois cohomology over a number field with coefficients in a finite Galois module | In fact, $H^1(K,M)$ is infinite not only to the family of number fields, but also to the more general family of Hilbertian fields (those fields which satisfies the Hilbert irreducibility theorem). Below is a proof based on embedding problems:
Let $K$ be Hilbertian with absolute Galois group $G\_K$ and $M$ a finite no... | 9 | https://mathoverflow.net/users/2042 | 239636 | 110,569 |
https://mathoverflow.net/questions/239637 | 5 | I have heard that Progress towards the Inverse Galois problem over $\mathbb{Q}$
is very well documented for sporadic groups ($M\_{23}$ is the only case open) and for $PSL\_n(q)$.
From where I can read more about these results in details. Like papers, articles, books, etc.
| https://mathoverflow.net/users/92070 | On progress towards inverse Galois problem over rationals | All the sporadic groups except for $M\_{23}$ and $M\_{24}$ are realized over $\mathbb{Q}$ using the rigidity criterion. This technique is explained in all three main textbooks on inverse Galois theory:
* Jean-Pierre Serre - Topics in Galois theory
* Helmut Volklein - Groups as Galois Groups
* Gunter Malle & B. H. Mat... | 11 | https://mathoverflow.net/users/43108 | 239642 | 110,570 |
https://mathoverflow.net/questions/239641 | 2 | In their paper 'Sign changes of Hecke eigenvalues', Matomaki and Radziwill established in Lemma 6.2 the following result: There exists absolute positive constants $c$ and $\eta$ such that uniformly in $h \leq X^{\eta},$ we have
$$\int\_{X}^{2X} \left|\sum\_{x\leq n\leq x+hk(X)} sign(\lambda\_f(n)) w\_n \right|^2 dx \le... | https://mathoverflow.net/users/76102 | Clarification request on sign changes of Hecke eigenvalues | The constant $c>0$ in Lemma 6.2 is absolute, while $K>0$ is arbitrary in Proposition 3.4. There is no relation between these two quantities.
| 2 | https://mathoverflow.net/users/11919 | 239647 | 110,573 |
https://mathoverflow.net/questions/239653 | 12 | It is usually told that Birch and Swinnerton-Dyer developped their famous conjecture after studying the growth of the function
$$
f\_E(x) = \prod\_{p \le x}\frac{|E(\mathbb{F}\_p)|}{p}
$$
as $x$ tends to $+\infty$ for elliptic curves $E$ defined over $\mathbb{Q}$, where the product is defined for the primes $p$ where $... | https://mathoverflow.net/users/nan | Clarification on the weak BSD conjecture | In regards to question 2, in 1982 Goldfeld proved that if $f\_{E}(x) \sim C (\log x)^{r}$, then (i) $L(E,s)$ has no zeroes with ${\rm Re}(s) > 1$, and (ii) the order of vanishing at $L(E,s)$ is equal to $r$. I do not know if the converse is true (even assuming GRH for $L(E,s)$), as I don't have a copy of Goldfeld's pap... | 7 | https://mathoverflow.net/users/48142 | 239656 | 110,577 |
https://mathoverflow.net/questions/239633 | 3 | Let $X$ be a noetherian,integral,separated scheme which is nonsingular in codimension $1$. Let $Z\_1, \ldots, Z\_k$ be a fixed set of prime divisors. Now given a prime divisor $Y$, does there exist prime divisors $W\_1, \ldots, W\_m$, none of them equaling any of the $Z\_j$s, such that $Y +\sum n\_i W\_i = 0$ in $\text... | https://mathoverflow.net/users/23927 | Extending a prime divisor to a principal divisor | It is easier to write this as an answer than as a comment. A good resource for some questions of this type is "Ample subvarieties of algebraic varieties" by Robin Hartshorne.
Let $A$ be an ample divisor class. Then there exists an integer $n\_0$ such that for every integer $n\geq n\_0$ and for every $j=1,\dots,k$,
bo... | 5 | https://mathoverflow.net/users/13265 | 239666 | 110,581 |
https://mathoverflow.net/questions/239670 | 8 | There are suggestions that says that Grothendieck developed (in some sense) a theory of Motivic homotopy types or at least named it.
I would like to know the reference in which Grothendieck did it, and some references on subsequently development.
| https://mathoverflow.net/users/83957 | What are Motivic homotopy types? | Not sure about later developments, but the idea is mentioned in a famous passage of Grothendieck's *Récoltes et Semailles*. I quote from Roy Lisker's translation:
>
> Thus, the motive presents itself as the deepest "form invariant" which
> one has been able to associate up to the present moment with an
> algebrai... | 13 | https://mathoverflow.net/users/43108 | 239672 | 110,582 |
https://mathoverflow.net/questions/239650 | 8 | I have been considering a (definability-free) weak form of the constructibility axiom, which is intended to capture the coarse structure of the constructible hierarchy. This means that this weak form is intended to capture the ordinal pattern abstracted from the constructible hierarchy if the fine behavior of the ordin... | https://mathoverflow.net/users/9825 | Is this weak form of $V=L$ (in)consistent with large cardinals? | This is consistent with very large cardinals. It follows from the work of Friedman and Holy, who showed that an abstract version of local club condensation and acceptability are simultaneously consistent with the existence of very large cardinals. See their papers:
(1) Condensation and Large Cardinals.
Fundamenta Mat... | 6 | https://mathoverflow.net/users/11115 | 239690 | 110,586 |
https://mathoverflow.net/questions/239696 | 0 | Let $A\subseteq B$ be commutative Noetherian rings such that $A$ is a regular ring, i.e., $A\_{\mathfrak{m}}$ is a regular local ring for all maximal ideals $\mathfrak{m}$ of $A$ and $B$ is a finite $A$ module. If $\mathfrak{p}$ varies over the set of prime ideals of $A$, does the vector space-dimension of $B\otimes\_A... | https://mathoverflow.net/users/66365 | Invariance of the fiber-dimension of a finite map | That's false. In general the dimension is only upper semicontinuous. The simplest counterexample which comes to my mind is $A=\mathbb C[x^2,x^3]\subseteq B=\mathbb C[x]$. Then the dimension over the maximal ideal $(x^2,x^3)$ is $2$. All other dimensions are $1$ since $A$ and $B$ become equal after inverting $x^2$. For ... | 4 | https://mathoverflow.net/users/89948 | 239701 | 110,589 |
https://mathoverflow.net/questions/239686 | 2 | I am wondering whether anyone knows the following integration has a named special function or a reference
$$
F\_{a,b}(z) :=\frac{2}{\sqrt{\pi}} \int\_0^z \text{erf}(a+b y)\: e^{-y^2} \text{d}y
$$
for $a,b, z\in \mathbb{R}$. We are in particular interested in the case when $a\ne 0$.
This function is well defined.... | https://mathoverflow.net/users/36814 | Ask for a special function related to the error function | For $a=0$ it reduces to [Owen's T-function:](https://en.wikipedia.org/wiki/Owen%27s_T_function)
$$\frac{2}{\sqrt{\pi}} \int\_0^z \text{erf}(b y)\: e^{-y^2} \text{d}y=4T\left(\sqrt{2} \,b z,1/b\right)+\text{erf}\,(z) \,\text{erf}\,(b z)-\frac{2}{\pi}\, \text{arccot}\, b$$
(here's an amusing [commentary](http://blog.... | 3 | https://mathoverflow.net/users/11260 | 239712 | 110,593 |
https://mathoverflow.net/questions/239462 | 8 | **(A)** In this really stylish [answer](https://mathoverflow.net/questions/220214/is-there-an-integrable-complex-structure-on-su3/220230#220230) it is shown that one can define a family of complex structures $J\_{\lambda}$ on the Lie group SU(3), dependent on the parameter $\lambda \in {\mathbb C}\backslash {\mathbb R}... | https://mathoverflow.net/users/42100 | The Hypercomplex Structure of $SU(3)$ | I'm assuming that you have a copy of Dominic Joyce's 1992 JDG article, "Compact hypercomplex and quaternionic manifolds" handy. Write $\frak{g} = \frak{su}(3)$ as a direct sum
$$
\frak{su}(3) = \frak{b}\ \oplus\ \frak{d}\ \oplus\ \frak{f}\ ,
$$
where
$$
{\frak{b}} = \left\{\begin{pmatrix}ia&0&0\\0&ia&0\\0&0&-2ia\end{pm... | 11 | https://mathoverflow.net/users/13972 | 239717 | 110,596 |
https://mathoverflow.net/questions/156656 | 21 | Does it exist online and where can one find it? (For example, these [two](http://arminstraub.com/math/what-is-column) [sources](http://www.math.cornell.edu/~whieldon/What_is....html) are not official; is the longer one complete?)
| https://mathoverflow.net/users/4721 | Complete (possibly official) list of "What is..." articles from the Notices of the AMS | It seems they made a list in the meantime:
<http://www.ams.org/publications/notices/whatis/noticesarchive>
| 15 | https://mathoverflow.net/users/92120 | 239740 | 110,602 |
https://mathoverflow.net/questions/239725 | 1 | Let $L$ be a complete lattice with an involution operation $\*$ (a unary operation such that for any $x, y \in L$, $x \leq y$ implies $x^{\*} \geq y^{\*}$). Now, suppose that there is an element of $L$ that satisfies $x = x^{\*}$. How much does that tell us about $L$? Clearly, it means that excluded middle and non-cont... | https://mathoverflow.net/users/45570 | What does the existence of self complemented elements tell us about a complete lattice? | It's hard to imagine what the existence of $x=x^\*$ could possibly tell us. To wit, there is a simple way to take any bounded lattice $L$ with involution $\*$ and adjoin a new element $x$ with $x^\*=x$. In detail, put $L':= L\cup\{x\}$ with ordering such that $0\leq x\leq 1$, but $x$ is not ordered relative to any othe... | 3 | https://mathoverflow.net/users/27013 | 239742 | 110,604 |
https://mathoverflow.net/questions/239709 | 4 | Let $\chi$ and $\psi$ be two quadratic Dirichlet characters and let $L(s,\chi)$ and $L(s,\psi)$ their associated Dirichlet $L$-functions.
Is there a realtion between these two Dirichlet $L$-functions: $L(s,\chi)\times L(s,\psi)$ and $L(s,\chi\times \psi)$?
I mean can we derive an expression which relates these two ... | https://mathoverflow.net/users/76102 | Relation of these two Dirichlet $L$-functions | Using the perspective of "pretentious" multiplicative number theory, one can see that these $L$ functions are correlated with each other through the Deuring-Heilbronn repulsion phenomenon, but there is not a simple identity connecting the two (unless one is willing to introduce zeta functions of biquadratic fields, as ... | 9 | https://mathoverflow.net/users/766 | 239745 | 110,606 |
https://mathoverflow.net/questions/239677 | 19 | Every smooth manifold admits a complete Riemannian metric. In fact, every Riemannian metric is conformal to a complete Riemannian metric, see [this note](http://www.ams.org/journals/proc/1961-012-06/S0002-9939-1961-0133785-8/S0002-9939-1961-0133785-8.pdf). What about in the Kähler case?
>
> Does a Kähler manifold a... | https://mathoverflow.net/users/21564 | Does a Kähler manifold always admit a complete Kähler metric? | Grauert proved that a relatively compact domain with real analytic boundary in C^n has a complete Kahler metric iff the boundary is pseudoconvex .For any
relatively compact domain in C^n which is the interior of its closure ,with a complete Kahler metric, Diederich and Pflug showed that it is locally Stein.
See the pa... | 21 | https://mathoverflow.net/users/4696 | 239759 | 110,612 |
https://mathoverflow.net/questions/239753 | 1 | In the definition of operads, if we restrict our attention to $\mathbb{S}$-modules where the action by the symmetric groups is free, then the *free operads*
have still an underlying free $\mathbb{S}$-module? Even the colimits over this kind of operads have still an underling free $\mathbb{S}$-module?
And finally, in... | https://mathoverflow.net/users/86066 | Free Symmetric Operads and $\mathbb{S}$-modules | 1. What you say about free operads on free $\mathbb{S}$-modules being themselves free $\mathbb{S}$-modules and being describable in terms of the free non-symmetric operad is correct.
In section 5.9.11 of Loday–Vallette, they construct adjoint functors between ns operads and (symmetric) operads. The free ns operad and... | 4 | https://mathoverflow.net/users/3075 | 239811 | 110,630 |
https://mathoverflow.net/questions/239803 | 8 | I want to know whether a continuous nowhere differentiable function $f: \mathbf{R} \to \mathbf{R}$ can map Lebesgue measurable sets to Lebesgue measurable sets. More generally I'm interested to know if there are any necessary conditions for a continuous functions to preserve measurability.
| https://mathoverflow.net/users/nan | Can a nowhere differentiable function preserve measurability? | The answer is "no".
As it was noted by Martin Sleziak in order to preserve measurability, your function has to satify Luzin N property.
Let me show that this is not the case.
That is, any continuous nowhere differentiable function $f$ maps a set of zero measure maps to a set of positive measure.
Note that for fi... | 4 | https://mathoverflow.net/users/1441 | 239813 | 110,631 |
https://mathoverflow.net/questions/239808 | 12 | I'm looking for a closed form for the expression
$$
\sum\_{k=1}^{\infty}\frac{1}{(2k)^5-(2k)^3}
$$
I know that Ramanujan gave the following closed form for a similar expression
$$
\sum\_{k=1}^{\infty}\frac{1}{(2k)^3-2k}= \ln(2)-\frac{1}{2}
$$
I wonder if it is possible to find such a similarly simple and nice closed f... | https://mathoverflow.net/users/6842 | Finding a closed form for $\sum_{k=1}^{\infty}\frac{1}{(2k)^5-(2k)^3}$ | $$ \frac{1}{(2k)^5 - (2k)^3} + \frac{1}{(2k)^3} = \frac{1 + (2k)^2 - 1}{(2k)^5 - (2k)^3} = \frac{1}{(2k)^3 -2k}$$
So by Ramanujan's result:
$$\sum\_{k=1}^{\infty} \frac{1}{(2k)^5 - (2k)^3} = \ln(2) - \frac{1}{2} - \frac{1}{8}\zeta(3)$$
| 28 | https://mathoverflow.net/users/22131 | 239816 | 110,632 |
https://mathoverflow.net/questions/239812 | 1 | Let $E$ be an elliptic curve over $\mathbb{Q}$.
Then how to compute the $p$-torsion elements of $E$ over the $p$-adic field $\mathbb{Q}\_p$ using SAGE or any other means ?
At least can we say whether $E(\mathbb{Q}\_p)[p]=0$ or not ?
| https://mathoverflow.net/users/30999 | Finding out $p$-torsion elements of an elliptic curve $E$ over $\mathbb{Q}_p$ | Suppose $p>2$ and that $E$ has good reduction. If the reduction $\tilde E(\mathbb{F}\_p)$ has no $p$-torsion then there is no $p$-torsion in $E(\mathbb{Q}\_p)$. Otherwise look at the exact sequence
$$ 0\to E(\mathbb{Q}\_p)[p]\to \tilde E(\mathbb{F}\_p)[p]\to \hat E(p\mathbb{Z}\_p)/p\hat E(p\mathbb{Z}\_p).$$
Here $\hat ... | 9 | https://mathoverflow.net/users/5015 | 239818 | 110,633 |
https://mathoverflow.net/questions/239806 | 0 | Let $G=(V,E)$ be a finite, simple, unconnected graph. We define the *total graph* $T(G)$ of $G$ as follows:
* $V(T(G)) = (V\times\{0\}) \cup (E\times\{1\})$,
* $E(T(G)) = E\_v \cup E\_e \cup E\_{v+e}$, where
+ $E\_v = \big\{\{(v,0), (w,0)\}: \{v,w\}\in E\big\}$,
+ $E\_e = \big\{\{(e,1), (f,1)\}: (e,f\in E) \land (e... | https://mathoverflow.net/users/8628 | Total chromatic number and total clique number | Counterexamples are, for example, complete graphs with even number $2k$ of vertices. If we manage to color such a graph with $2k$ colors, then all vertices have different colors, and from each vertex of color, say, $s$, there is exactly one edge of each color except $s$. Therefore total number of edges of each color is... | 3 | https://mathoverflow.net/users/4312 | 239821 | 110,635 |
https://mathoverflow.net/questions/239824 | -2 | Consider a continously differentiable non-constant function $f:\mathbb{R}^2\to\mathbb{R}$. Define
$$
K=\{x\in\mathbb{R}^2|f(x)=0\}.
$$
I wish to know whether there is a continuously differentiable function $g:\mathbb{R}\to\mathbb{R}^2$ such that
$$
\forall k\in K,\quad\exists u\in\mathbb{R}, \quad g(u)=k.
$$
So it ... | https://mathoverflow.net/users/45305 | Is every implicit function reparametrized? | Under your hypotheses, the answer is : no.
Basically because the existence of such function $g$ would mean that the zeroes of your function $f$ is a one-dimensional manifold, which is not true in general.
Take for instance $f(x)=\text{dist}^2(x;C)$, where $C$ is the unit disc: $f$ is smooth, and here $K=C$. But yo... | 2 | https://mathoverflow.net/users/73608 | 239826 | 110,636 |
https://mathoverflow.net/questions/239825 | 6 | For a monic polynomial with integer coefficients $f$ where $\partial f = 2$, we have
$$
\textrm{inf}(f(x)) > 0 \implies
\textrm{inf}(f(x)) \geq \frac{3}{4} .
$$
Could we generalize this (for common $\partial f$)?
| https://mathoverflow.net/users/22954 | On property of monic polynomial with integer coefficients | No. If $p,q$ satisfy Pell's equation $p^2-2q^2=1$, then minimum of $f(x)=(x^2-2)^2+(qx-p)^2$ is at most $f(p/q)=1/q^4$.
| 9 | https://mathoverflow.net/users/4312 | 239827 | 110,637 |
https://mathoverflow.net/questions/239822 | 2 | I am considering an ODE $\dot{x}=f(x)$, with $x\in\mathbb{R}^d$ and $d<\infty$. $f$ is a $C^k$ function and I denote by $\Phi\_t x$ be the value of the solution at time $t$.
I assume that my ODE has a unique attractor that is exponentially stable: $\|\Phi\_t x - x^\*\| \le \beta e^{-\alpha t}$, with $\alpha>0$ and I... | https://mathoverflow.net/users/90045 | Smooth dependence on the initial condition of the integral of an ODE | I don't know a reference, but here's the rough proof that $G \in C^1$.
First, from the integral equality
$$
\Phi\_t(x+h) - \Phi\_t(x) = \int\_0^1 D\Phi\_t(x+\eta h) \cdot h \,\mathrm{d}\eta
$$
follows the mean-value estimate
$$
\| \Phi\_t(x+h) - \Phi\_t(x) - D\Phi\_t(x) \cdot h \|
\le \sup\_{\eta \in [0,1]} \| D\Phi\... | 4 | https://mathoverflow.net/users/3928 | 239842 | 110,641 |
https://mathoverflow.net/questions/239823 | 7 | In Representation Theory, the theme of the existence of a canonical basis has been explored quite a lot. I will limit myself in this question to the kind of canonical bases that arise from the Geometric Satake Correspondence (due to Mirkovic-Vilonen, followed up in works by Anderson, Kamnitzer and many others). Here, o... | https://mathoverflow.net/users/26208 | Motivating the existence of Canonical Bases for Representations | I don't really see how to get there from just compact groups, so in that sense this is not an answer. My take on the question is something like: how might one have guessed the existence of canonical basis theories\*, knowing only the Lie theory of, say, 1975? (The type A case was well under way by then, motivating [sta... | 6 | https://mathoverflow.net/users/391 | 239847 | 110,644 |
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