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https://mathoverflow.net/questions/72240 | 7 | Let $K$ be a field of characteristic $0$. Let $C/K$ be a be a quasi-projective conic defined over $K$. Let $L/K$ be a finite dimensional field extension of **odd** dimension. Assume that
$C(L)$ is not empty.
**Q**: Then is true that $C(K)$ is not empty? If so, then how does one prove it?
| https://mathoverflow.net/users/11765 | On rational points of conics | You can find an elementary proof in
[Un théorème arithmétique sur les coniques](https://doi.org/10.1007/BF02590881 "zbMATH review at https://zbmath.org/0048.02702") by Trygve Nagell in Ark. f. Mat. 2 (1952). This is
a quadratic analogue of what Birch called Heegner's Lemma, which deals with the solvability of certain q... | 5 | https://mathoverflow.net/users/3503 | 72262 | 44,092 |
https://mathoverflow.net/questions/72259 | 14 | I just ran into this deceptively simple looking question.
>
> Is it always possible to partition $\mathbb{R}$ (or any other standard Borel space) into precisely $\aleph\_1$ Borel sets?
>
>
>
On the one hand, this is trivial if the Continuum Hypothesis holds. Less trivially, this also follows from $\mathrm{cov}... | https://mathoverflow.net/users/2000 | Partitioning $\mathbb{R}$ into $\aleph_1$ Borel sets | It suffices to express $\mathbb R$ as the union of $\aleph\_1$ (not necessarily disjoint) Borel sets such that no countably many of them cover $\mathbb R$, because then you can list them in an $\omega\_1$-sequence and subtract from each one the union of the previous ones. Partition $\mathbb R$ into a non-Borel $\Pi^1\_... | 23 | https://mathoverflow.net/users/6794 | 72263 | 44,093 |
https://mathoverflow.net/questions/72266 | 0 | This is a question that originated with World of Warcraft. I have a solution, but I don't know where to look up other problems of the same kind for a better explanation.
There is a plane with p axis and q axis.
There is a function f(x,y) = 2px+qy^2 such that f(x,y) is only defined on the curve x^3+y^3=1.
Claim: T... | https://mathoverflow.net/users/16996 | Is this a Karusch-Kuhn-Tucker method or something else? | It is using KKT. For details, I'd recommend the Convex Optimization book by Boyd and Vandenberghe.
As a side note, this is not research level mathematics, and I would like to vote to close, but since I am still a new user, I don't know how to do it and I just leave this comment as an answer (can't comment yet either... | 1 | https://mathoverflow.net/users/12425 | 72267 | 44,096 |
https://mathoverflow.net/questions/72268 | 7 | Suppose that I want to know whether a polynomial $P(z)$ has a root with multiplicity at least three. This is obviously an algebraic condition, but is there some reasonably concise set of conditions defining the variety (in the space of coefficients)? This must have been studied by the ancients. It is clearly necessary ... | https://mathoverflow.net/users/11142 | Less discriminating discriminants | Here's one way to think of this. Let $\lambda$ be a new variable and let $Q(z) = \lambda P'(z) + P''(z)$. Let $R(\lambda)$ be the resultant of $P$ and $Q$. Then we want $R$ to identically vanish (i.e. want to see if there's value of $z$ for which $P$ and $Q$ have a common zero, for every value of $\lambda$). So the coe... | 11 | https://mathoverflow.net/users/2698 | 72269 | 44,097 |
https://mathoverflow.net/questions/72251 | 11 | I know that in the smooth category the following is true. There are at most countable many disjoint embedded moebius bands in euclidean 3-space. Is this also true in topological category?
| https://mathoverflow.net/users/3097 | Packing moebius bands | There are at most countably many disjoint embeddings of homeomorphic images of a non-orientable hypersurface in $\mathbb R^k$. This is theorem 2 in ["An uncountable family of disjoint spatial continua in Euclidean space"](https://doi.org/10.1007/BF00971401 "Sib Math J 34, 848–851 (1993). zbMATH review at https://zbmath... | 11 | https://mathoverflow.net/users/2384 | 72271 | 44,099 |
https://mathoverflow.net/questions/72273 | 6 | Let $V$ be a virtually cyclic group.
Then is $Aut(V)$ also a virtually cyclic group?
This is true when $V$ is a finite group (zero-ended) and when $V = C\_\infty, D\_\infty$ (both two-ended).
| https://mathoverflow.net/users/16862 | Automorphism groups of virtually cyclic groups | In [Finitely generated groups with virtually free automorphism groups](http://journals.cambridge.org/action/displayAbstract?fromPage=online&aid=3078536), by M.R. Pettet, it is proved in theorem 3.4 that the automorphism group $Aut(G)$ of a finitely generated group is virtually cyclic if and only if both $Z(G)$ and $G/Z... | 8 | https://mathoverflow.net/users/2384 | 72276 | 44,101 |
https://mathoverflow.net/questions/72281 | 1 | (Note: I have a very little knowledge in the area related to this question. (My research is more related to Combinatorics.) So, I apologize in advance if I say something wrong or trivial.)
One version of the Brouwer fixed point theorem states that any continuous function from the closed unit ball in n-dimensional Eu... | https://mathoverflow.net/users/5016 | Does convex continuous mapping have a unique fixed point? | As to the specific question in the next to last paragraph, the identity function is convex. Except for the zero dimensional case there are many fixed points.
| 5 | https://mathoverflow.net/users/8684 | 72283 | 44,104 |
https://mathoverflow.net/questions/72050 | 5 | I am currently doing research in non-linear dynamical systems, and I require to calculate Lyapunov exponents from time series data frequently. I found a MatLab program lyaprosen.m that does this for me, but I am not very sure of its validity, as I do not get the same results from it, as some results in some papers. Doe... | https://mathoverflow.net/users/16883 | Calculation of Lyapunov Exponent from Time Series | TSTOOL is the state of the art: <http://www.physik3.gwdg.de/tstool/index.html>
| 3 | https://mathoverflow.net/users/11260 | 72285 | 44,106 |
https://mathoverflow.net/questions/4802 | 36 | Let $z^i(X, m)$ be the free abelian group generated by all codimension $i$ subvarieties on $X \times \Delta^m$ which intersect all faces $X \times \Delta^j$ properly for all j < m. Then, for each i, these groups assemble to give, with the restriction maps to these faces, a simplicial group whose homotopy groups are the... | https://mathoverflow.net/users/733 | What do higher Chow groups mean? | I believe Bloch's original insight was something like the following:
First, if $X$ is a regular scheme, you can filter $K\_0$ by ``codimension of support''; that is, view $K\_0(X)$ as the Grothendieck group of the category of all finitely generated modules and let $F^iK\_0(X)$ be the part generated by modules with co... | 23 | https://mathoverflow.net/users/10503 | 72291 | 44,107 |
https://mathoverflow.net/questions/72288 | 12 | I would like to ask if there exist pedagogical expositions of the Mordell-Weil theorem ([wikipedia](http://en.wikipedia.org/wiki/Mordell%E2%80%93Weil_theorem)). What parts of number theory (algebraic geometry) one should better learn first before starting to read a proof of Mordell-Weil?
| https://mathoverflow.net/users/13441 | Proofs of Mordell-Weil theorem | J. Silverman and J. Tate "The rational points on elliptic curves" is a wonderful introduction to elliptic curves over rational numbers. It covers topics such as Mordell-Weil, Nagell-Lutz Theorem, elliptic curves over finite fields, etc.
For more advanced treatment of Mordell-Weil, I suggest the following textbook:
... | 17 | https://mathoverflow.net/users/16995 | 72294 | 44,109 |
https://mathoverflow.net/questions/72242 | 1 | Can you point me in the direction of any research done on the spectral theory (i.e. eigenvalues and eigenvectors) of real symmetric matrices with random (Gaussian or Levy) diagonal elements and fixed off-diagonal ones? Any links to papers, theorems or books will be appreciated.
| https://mathoverflow.net/users/1580 | Spectral theory of real symmetric matrices with random diagonal elements | There are two cases to consider depending on how your off-diagonal looks like:
If only two off-diagonal are non-zero, you are in the realm of random Schr\"odinger operators or random Jacobi operators. Then the special structure of the random variables is preserved up to a renormalization. In particular, the eigenvalu... | 5 | https://mathoverflow.net/users/3983 | 72296 | 44,111 |
https://mathoverflow.net/questions/36767 | 5 | I'm looking for an explicit description of all the finite dimensional irreducible representation of the Lie group $SO(n,1)(\mathbb{R})$. Can you tell me, where I can find this description ? Thank you.
| https://mathoverflow.net/users/8801 | Finite dimensional spherical representation of $SO(n,1)(\mathbb{R})$ | Bonsoir Ludo! I am puzzled by the fact that your title asks something more restrictive than the OP, since the latter does not contain the word "spherical". Let me answer the latter first. Any finite-dimensional representation of $SO(n,1)(\mathbb{R})$ extends to a representation of the complexification, which is $SO\_{n... | 6 | https://mathoverflow.net/users/14497 | 72307 | 44,120 |
https://mathoverflow.net/questions/71584 | 4 | The first version of this question received a helpful answer but was too vague to fully convey what I intended. I hope this version remedies that problem. For any hyperarithmetic set of integers $S$, does there exist a single recursive process that can determine if $s\in S$ for all integers provided it also has access ... | https://mathoverflow.net/users/16554 | Hyperarithmetic statements decidable by induction up to a recursive ordinal | Well this has gone unanswered for a while, so I'll give a pseudo-answer. First I'll answer yes to a slightly different question, then I'll guess that the answer to the question as stated is no, then I'll make some random comments.
---
My proposed alterations are: (1) instead of a three-valued computable function,... | 3 | https://mathoverflow.net/users/6787 | 72322 | 44,126 |
https://mathoverflow.net/questions/72308 | 9 | Consider a $(p,q)$ torus knot $K$ in 3-dimensional Euclidean space $\mathbb R^3$ where $p,q \geq 2$ and $\operatorname{GCD}(p,q)=1$.
Let $\operatorname{Isom}(\mathbb R^3,K)$ be the isometries of $\mathbb R^3$ that preserve $K$.
It's a fairly standard argument using theorems about uniqueness of Seifert fiberings t... | https://mathoverflow.net/users/1465 | Torus knots in Euclidean space -- a symmetry argument | The answer no. Neither quantum invariants nor the Alexander polynomial sees the difference between a knot in the three sphere and a knot in Euclidian three space. In the case of the Alexander polynomial the missing point does not interfere with the first homology of the infinite cyclic cover. In the case of quantum inv... | 4 | https://mathoverflow.net/users/4304 | 72323 | 44,127 |
https://mathoverflow.net/questions/72328 | 2 | Let $D$ be an effective $ \mathbb{Z}$-divisor on $ \mathbb{P}^1$. Is there a form to associate a curve $C$, and morphism $C \to P^1$ to the divisor $D$ ? For example, let $Y$ be a singular plane curve, sometimes, we can built a cover of $ \mathbb{P}^2$ that branches along $Y$. Is there a similar construction for covers... | https://mathoverflow.net/users/16409 | About maps induced for a divisor D in P^1 | Yes, this is the famous Riemann Existence Theorem.
In its general form, it can be stated as follows.
**Theorem.** Let $Y$ be a compact Riemann surface, and $D \subset Y$ be an effective reduced divisor. Then there is a $1$-$1$ correspondence between the following sets:
$\mathbf{1)}$ finite covers $f \colon X \to ... | 3 | https://mathoverflow.net/users/7460 | 72331 | 44,131 |
https://mathoverflow.net/questions/72314 | 1 | (i) Are there limits on how many numbers must be in the set? { 1, 2 } or { 1, 5, 7, 8 , 9}
(ii) Are there limitations on how diverse or similar the numbers in the set can be? Coprime? Pairwise? { 1, 3, 9, 81 } (essentially powers of 3)
(iii) Is there any limitations on the relationship between the numbers of the s... | https://mathoverflow.net/users/17007 | Knapsack Problem Specifics | Let me try to give (very partial) answers. I think that your questions need some clarifications if you want more *useful* answers.
(i) I do not think your question is well-posed. As already mentioned in comments, you cannot fix the instance size and speak about NP-completeness. Nevertheless, you could define a new pr... | 0 | https://mathoverflow.net/users/16178 | 72337 | 44,137 |
https://mathoverflow.net/questions/65963 | 11 | The following theorem, due to Regev, is one of the cornerstones of the theory of PI algebras (i.e., associative algebras satisfying a nontrivial polynomial identity):
Let $A$, $B$ be two PI algebras over a field $K$. Then their tensor product $A \otimes\_K B$ is PI.
Consider the following "proof" of this theorem. ... | https://mathoverflow.net/users/1223 | Deceptively short proof of Regev's $A \otimes B$ theorem | I spoke about this with Louis Rowen (who, among other, wrote a few books on the subject) and here is what I got from this conversation:
1. There is no circular dependency in this argument.
2. The Razmyslov-Kemer-Brown theorem is more difficult and complicated result than Regev's theorem, so there is no much point to ... | 2 | https://mathoverflow.net/users/1223 | 72346 | 44,139 |
https://mathoverflow.net/questions/72312 | 4 | To each closed $3$-manifold $N$, there is a corresponding Witten--Reshetikhin--Turaev invariant $Z\_k(N)$ depending on an integer $k$ (the level) and a Lie group $G$ (and perhaps we'll just concentrate on $G=SU(2)$ for the moment). Based on the perturbation analysis of the path integral "definition" of $Z\_k(N)$, one p... | https://mathoverflow.net/users/35353 | What is known rigoruously about the semiclassical (k to infinity) limit of WRT invariants? | The answer is no. Rigorous results on the asymptotic expansion are known only for limited classes of 3-manifolds and/or leading terms of the expansion. See the papers cited by Charlie Frohman and Stefan Behrens in the comments to your question.
There's also a useful (though perhaps a little dated) summary of the pro... | 2 | https://mathoverflow.net/users/284 | 72355 | 44,145 |
https://mathoverflow.net/questions/72339 | 5 | Let $H$ be the $n$-dimensional hypercube, i.e. $\{0,1\}^n$ with edges between two vertices if and only if they differ in exactly one co-ordinate. We say that an edge is in direction $i$ if its endpoints differ in exactly the $i$'th co-ordinate. Suppose $V$ is a subset of $H$ such that $|V| > 2^{n-1}$. Is it true that a... | https://mathoverflow.net/users/17015 | Connected components of large induced subgraphs of hypercubes | Yes, this is true. Thanks to Sukhada Fadnavis and Seva for pointing out in the comments that the argument I had written here was wrong. Instead I will point you to the paper where this is proved
>
> ["Bulky subgraphs of the hypercube"](http://www.sciencedirect.com/science/article/pii/S0195669899903621), by Andrei K... | 4 | https://mathoverflow.net/users/2384 | 72363 | 44,149 |
https://mathoverflow.net/questions/72364 | 13 | I have a, probably very simple, question: My intuition tells me that the following statement should be true, but I couldn't find it anywhere and I wanted to make sure I am not missing something.
Let $\pi:Y\to X$ be a finite, étale morphism of nonsingular varieties over some algebraically closed field $\Bbbk$. Is it t... | https://mathoverflow.net/users/9947 | Finite, Étale Morphism Of Varieties | No, it's not true. Consider the map $x\mapsto x^2$ as a map from $X=\mathbb A^1-\{0\}$ to itself. The "problem" is that Zariski neighborhoods are too big. *Any* open subset of $X$ has exactly one irreducible component (in general, an open subset cannot have more components than the ambient space), so there is no hope t... | 21 | https://mathoverflow.net/users/1 | 72366 | 44,150 |
https://mathoverflow.net/questions/72368 | 4 | For a boolean algebra B, let d(B) be the least cardinality of a dense subset of B. Let A be a (non-regular) subalgebra of a boolean algebra B. Is it possible that d(A) > d(B)? What if d(B) = $\aleph\_0$?
| https://mathoverflow.net/users/11145 | density of boolean algebras | Yes, it is possible.
Let $\mathbb{B}$ be any complete Boolean algebra with density $\aleph\_0$. For example, we could use the power set Boolean algebra $\mathbb{B}=P(\mathbb{N})$, which has density $\aleph\_0$ in light of the singleton sets, since every nonempty set contains a singleton.
Meanwhile, the [Balcar-Fra... | 6 | https://mathoverflow.net/users/1946 | 72378 | 44,154 |
https://mathoverflow.net/questions/72361 | 9 | Let $M\_n$ be the $n\times n$ matrix
$$
(M\_n)\_{ij}=\begin{cases}1 & i\leq j,\\\\ 0 &i>j.\end{cases}
$$
Is there around an explicit expression or at least an asymptotic for $\left\Vert M\_n \right\Vert$? The norm is the usual Euclidean induced norm $\left\Vert M \right\Vert=\rho(M^TM)^{1/2}$.
I apologize if this a s... | https://mathoverflow.net/users/1898 | 2-norm of the upper triangular "all-ones" matrix | The eigenvalues of $M^{\rm T}M$ are $1 / (4 \phantom. \cos^2\frac{k\pi}{2n+1})$ for $k=1,2,\ldots,n$. The largest of these arises for $k=n$ and equals $1/(4\phantom.\sin^2\frac{\pi}{4n+2})$. Hence $\|M\| = 1 / (2 \phantom.\sin\frac{\pi}{4n+2})$, which is asymptotic to $2n/\pi$. This is easier to see if we work not with... | 20 | https://mathoverflow.net/users/14830 | 72383 | 44,158 |
https://mathoverflow.net/questions/72158 | 2 | Hello everyone,
it seems to be "well-known" that $H^0(X;End(V))$ only contains isomorphisms where $X$ is a Riemann surface and $V$ a stable (algebraic) vector bundle over $X$. The usual proof considers (roughly) the (coherent) image sheaf of a non-zero vector bundle morphism $\varphi:V\to V$ and one obtains a contra... | https://mathoverflow.net/users/16969 | Endomorphisms of stable vector bundles over a Riemann surface | Consider the two following potential definitions of stability of a locally free coherent sheaf $E$.
(A) Every subbundle (i.e. locally free subsheaf) has strictly smaller slope.
(B) Every subsheaf has strictly smaller slope.
I claim that over a curve $X$ these two definitions are equivalent. Indeed, suppose (A) ho... | 2 | https://mathoverflow.net/users/7399 | 72389 | 44,162 |
https://mathoverflow.net/questions/72376 | 13 | In his solution of the equation $x^3 + dy^3 = 1$, Nagell
comes across the equation
$$ u^3 + 6u^2v + 3uv^2 - v^3 = w^3. $$
He then observes that
$$ (u^3 + 6u^2v + 3uv^2 - v^3) U^3 = V^3 + W^3 $$
for
$$ U = u^2+uv+v^2, \quad
V = u^3+3u^2v-v^3, \quad
W = 3u^2v+3uv^2, $$
and then appeals to Fermat's Last Theorem for the... | https://mathoverflow.net/users/3503 | Sums of two cubes | This seems to be two separate questions, one for writing a given cubic form $C(u,v)$ as the sum of two *polynomial* cubes, the other for a sum of two *rational* cubes of given form.
In the polynomial case we want $C=x^3+y^3$ with $x,y$ linear. Assume we're not in characteristic $3$ (maybe I also have to exclude chara... | 15 | https://mathoverflow.net/users/14830 | 72391 | 44,163 |
https://mathoverflow.net/questions/72344 | 4 | Given a proper algebraic surface $S$, the Picard group $Pic(S)$ is endowed with the (symmetric) intersection form. We can therefore talk about reflections in the classes of $-2$-curves. These will preserve the intersection form and therefore the positive cone inside the lattice $Pic(S)$. Many times I have come across t... | https://mathoverflow.net/users/6254 | Fundamental domain for group generated by reflections in -2 curves | This is a general fact about groups generated by reflections acting on lattices. The reflections in $-2$ curves generates an action on $H^2(S;\mathbb{Z})$ by reflections, with quadratic form given by the cup product. Restricting this to $H^{1,1}(S)\cap H^2(S;\mathbb{Z})$, one gets an action on an integral lattice of si... | 6 | https://mathoverflow.net/users/1345 | 72392 | 44,164 |
https://mathoverflow.net/questions/21524 | 4 | There are many representation theorems which state that the dual space of a Banach space $X$ has a particularly concrete form. For example, if $X = C([0,1],\mathbb R)$ is the space of real-valued continuous functions on $[0,1]$, then $X^\*$ is the space of Radon measures on $[0,1]$. When you are confronted with some Ba... | https://mathoverflow.net/users/238 | Examples of Banach spaces and their duals | In wikipedia, there is a [list of banach spaces](http://en.wikipedia.org/wiki/List_of_Banach_spaces) with its dual space and it also tells you if it is reflexive, for example.
| 1 | https://mathoverflow.net/users/17009 | 72403 | 44,169 |
https://mathoverflow.net/questions/72353 | 5 | Let $X$ be a smooth projective algebraic surface (over $\mathbb{C}$ ). For all $L\in \mathrm{Pic}(X)$, we have
$$\chi(L)=\chi(\mathcal{O}\_X)+\frac{1}{2}(L^2-L\cdot \omega\_X).$$
This is the famous Riemann-Roch theorem in the flavour I like the most. It usually comes together with the following two formulas:
$$\chi(\ma... | https://mathoverflow.net/users/1887 | Generalisations of Riemann-Roch for surfaces | If $X$ is proper with rational singularities (and quotient and A-D-E (=Du Val) singularities are rational), then you can do most cohomology computations on a resolution.
Let $\pi:Y\to X$ be a resolution of singularities (not necessarily minimal). Then if $X$ has rational singularities, then $R^i\pi\_\*\mathscr O\_Y=0... | 13 | https://mathoverflow.net/users/10076 | 72404 | 44,170 |
https://mathoverflow.net/questions/72395 | 1 | Let $(X,\tau)$ be a $T\_1$ topological space and $Y\subset X$ a dense subspace which is completely metrizable. Are there any sufficient conditions to ensure that $(X,\tau)$ is Hausdorff using the known facts?
EDIT: Here is an example of such a topological space which isn't Hausdorf, as Valerio asked.
Take $(Y,\|\cd... | https://mathoverflow.net/users/17009 | Sufficient conditions for Hausdorffness | This answer begins with an easily understood fact and example followed by a more complicated example serving to illustrate why convenient answers to dan232's question can be challenging to find.
FACT: If the 0-dimensional space X is T1, then X is T2.
Pf. Fix distinct points x and y. Since X is T1, X\y is open, and... | 2 | https://mathoverflow.net/users/17029 | 72410 | 44,173 |
https://mathoverflow.net/questions/72418 | 8 | For example, if $n = 10$ and $k = 3$, then the legal partitions are
$$10 = 7 + 2 + 1 = 6 + 3 + 1 = 5 + 4 + 1 = 5 + 3 + 2$$
so the answer is $4$. By choosing $k$ random elements of $\{1,\ldots,2n/k\}$, one can easily construct about $(n/k^2)^k$ such partitions. For $k \approx \sqrt{n}$ this is not far from best possible... | https://mathoverflow.net/users/17008 | What are the best known bounds on the number of partitions of $n$ into exactly $k$ distinct parts? | In the 1990 paper by Charles Knessl and Joseph Keller, the authors prove the asymptotic result (for $n>>1, k=O(1)$, your number is *asymptotic* to:
$\dfrac{n^{k-1}}{k[{k-1]!}^2}.$
They show a number of other related asymptotic results.
**EDIT** for $k \ll n,$ they have the asymptotic too painful to typeset, but y... | 6 | https://mathoverflow.net/users/11142 | 72421 | 44,176 |
https://mathoverflow.net/questions/72419 | 54 | I'm a student (I've been studying mathematics 4 years at the university) and I like functional analysis and topology, but I only studied 6 credits of functional analysis and 7 in topology (the basics). What I am looking for is good books that I could understand to go deeper in this areas, what do you recommend? (I can ... | https://mathoverflow.net/users/17009 | A good book of functional analysis | I am an algebraist and not an analyst, however my favourite book on this area is "Walter Rudin: Functional Analysis".
| 50 | https://mathoverflow.net/users/14653 | 72422 | 44,177 |
https://mathoverflow.net/questions/70447 | 6 | If one talks about homogeneous Markov diffusion
$$
\mathrm dX\_t = \mu(X\_t)\mathrm dt+\sigma(X\_t)\mathrm dw\_t
$$
with $\mu,\sigma$ sufficiently differentiable and of appropriate dimensions, there is nice equation for a function $m\_f(x,t) = \mathsf E\_x f(X\_t)$ for $f\in C^2(\mathbb R)$:
$$
\begin{cases}
\frac{\par... | https://mathoverflow.net/users/11768 | Time-dependent Markov process: infinitesimal generator | You can directly deal with inhomogeneous Markov processes through the Kolmogorov backward and forward equations. I suppose you are asking for the forward equation (i.e. derivative with respect to the time in the future). Let me discuss things on a formal level, which means here I ignore regularity conditions to ensure ... | 5 | https://mathoverflow.net/users/12425 | 72426 | 44,181 |
https://mathoverflow.net/questions/72375 | 3 | Let $R=\mathbb{Q}[e^{2\pi i /3}]$. Does $H\_3(GL(R))$ have $\mathbb{Z}$-rank $1$?
If so, what is the index of the map:
$$
\mathbb{Z}\cong K\_3(R)/{\rm Torsion} \to H\_3(GL(R))/{\rm Torsion}\cong \mathbb{Z}?
$$
| https://mathoverflow.net/users/5394 | K-theory of number field | Let $F = \mathbb{Q}(\zeta\_3)$ and $R = \mathbb{Z}[\zeta\_3]$ with a third root of unity $\zeta\_3$. Then the Hurewicz homomorphism
$$h\_3: K\_3(R) \to H\_3(GL(R); \mathbb{Z})$$
induces an isomorphism on the torsion-quotients und the index of the map is $1$. This can be seen as follows:
1) rank $K\_3(R) = r\_2 = 1... | 6 | https://mathoverflow.net/users/10194 | 72432 | 44,186 |
https://mathoverflow.net/questions/72380 | 2 | Hello,
I am considering the following non-linear heat equation
$$
\left(\frac{\partial}{\partial t}-\nu\: \Delta \right) u(t,x) = F(t,x) \sigma(u(t,x)),\qquad (t,x)\in R\_+\times R^d
$$
where $F(t,x)$ is certain external heat source and $\sigma$ is a Lipschitz continuous function. Let's call $\nu$ the diffusion coe... | https://mathoverflow.net/users/36814 | Are there any physical phenomena of the heat transfer critically depending on diffusion coefficient? | What you describe is very much expected from the statistical physics principle "there is no phase transition in one-dimensional systems with short-range interactions at $T>0$." See [Lower Critical Dimension](http://en.wikipedia.org/wiki/Critical_dimension#Lower_critical_dimension) in Wikipedia. Since you have a PDE you... | 4 | https://mathoverflow.net/users/7949 | 72435 | 44,187 |
https://mathoverflow.net/questions/72408 | 10 | I am currently preparing a talk that revolves around the triangle inequality.
Because this inequality is so well-established, I do not want to (in my talk) belabor too much upon the importance it enjoys. For example, I learned some useful views [here](https://mathoverflow.net/questions/65513/why-the-triangle-inequali... | https://mathoverflow.net/users/8430 | History of the triangle inequality | There is a discussion of this issue in Dieudonné's *History of Functional Analysis*, p. 115:
>
> It may seem obvious to us that the results of Hilbert are but one step removed from what we now call the theory of Hilbert space; but if, in fact, the birth of that theory almost immediately followed the publication of ... | 7 | https://mathoverflow.net/users/290 | 72436 | 44,188 |
https://mathoverflow.net/questions/72111 | 5 | Suppose $j\colon V\to M$ is an elementary embedding and $\kappa$ is the critical point of $j$, then $\kappa$ is measurable, and we can define the ultrafilter $U$ over $\kappa$ as: $$A\in U\iff \kappa\in j(A)$$
This is a normal ultrafilter. Despite not seeing an actual example, I am aware that $\operatorname{Ult}(V,U)... | https://mathoverflow.net/users/7206 | Ultrapowers by normalized ultrafilters | I would answer question 1 by saying that
$M=\text{Ult}(V,U)$ if and only if $j$ is isomorphic to an
ultrapower by *some* normal measure. This is another way of
saying that normal measures are minimal with respect to the
Rudin-Kiesler order. The point is that an embedding is the
ultrapower by a normal measure if and onl... | 4 | https://mathoverflow.net/users/1946 | 72461 | 44,201 |
https://mathoverflow.net/questions/72472 | 2 | Let $\textbf{BG}$ be the pseudovariety of block groups, also known as $\textbf{EJ}, \textbf{PG},\ldots,\text{etc.}$(see [1]), and let $\textbf{R}$ be the pseudovariety of R-trivial monoids, by the Green's R relation.
Is the pseudovariety $\textbf{BG}\cap\textbf{R}$ well known in the field of semigroup&monoid theory?... | https://mathoverflow.net/users/16758 | The intersection of Block Groups and R-trivial (finite) monoids | I believe the answer is the pseudo variety of J-trivial monoids. Each regular J-class of an R-trivial monoid is a left zero semigroup. The block group condition allows only the trivial left zero semigroup. So each regular J-class is trivial. Thus all J-classes are trivial. The corresponding languages are the piecewise ... | 2 | https://mathoverflow.net/users/15934 | 72477 | 44,207 |
https://mathoverflow.net/questions/72467 | 5 | Let $k$ be an algebraically closed field and let $G\leq\rm{GL}\_n(k)$ be a linear group. Assume that $M< G$ is a maximal subgroup (in the abstract group sense). Denote by $\bar{G}^Z$ the Zariski closure of $G$ in $\rm{GL}\_n(k)$. Is it true that if $\bar{M}^Z\neq \bar{G}^Z$ then it is a maximal subgroup in the algebrai... | https://mathoverflow.net/users/6227 | Does the Zariski closure of a maximal subgroup remain maximal? | The answer is no. Assume that $G=\text{SO}(2,\mathbf{R})\ltimes\mathbf{R}^2\subset\text{GL}\_3(\mathbf{C})$ and $M=\text{SO}(2,\mathbf{R})$. Then $M$ is maximal in $G$. However the Zariski closures are $\text{SO}(2,\mathbf{C})\subset\text{SO}(2,\mathbf{C})\ltimes\mathbf{C}^2$, so $\text{SO}(2,\mathbf{C})$ is not maxima... | 10 | https://mathoverflow.net/users/14094 | 72479 | 44,209 |
https://mathoverflow.net/questions/72490 | 33 | The question is not about where operads are used, I know that. It is about what makes them useful. For example, van Kampen diagrams are useful in combinatorial group theory because these are planar graphs and so one can use planar geometry (say, the Jordan lemma) to investigate the word problem in complicated groups. S... | https://mathoverflow.net/users/nan | Why are operads useful? | Here are a couple 2-3 line answers to your question:
1) They allow you to treat various algebraic problems uniformly. For example, Commutative, Associative, and Lie algebras all have their own cohomology theories (Harrison, Hochschild, and Chevalley-Eilenberg respectively). These can all be seen as instances of a sin... | 24 | https://mathoverflow.net/users/5323 | 72496 | 44,218 |
https://mathoverflow.net/questions/72495 | 5 | Hi,
I am a physicist and currently doing my bachelor thesis about geometric quantization.
In the book by Bates and Weinstein I encountered the Maslov index, which seems to be very important :-).
But unfortunately my education didn't include anything in the direction of algebra beyond the scope of basic linear algeb... | https://mathoverflow.net/users/17047 | Where to start with research regarding maslov index/class | May I recommend self-study with <http://www.maths.ed.ac.uk/~aar/maslov.htm> ?
| 11 | https://mathoverflow.net/users/732 | 72503 | 44,221 |
https://mathoverflow.net/questions/72492 | 6 | Let $X$ be an algebraic variety over some algebraically closed field $\Bbbk$ and let us assume $\dim(X)=2$, i.e. $X$ is an algebraic surface.
First, I would like to know the definition of an **ordinary $n$-uple singular point** on $X$, because in the Literature I know, it is only defined with respect to curves. Wolfr... | https://mathoverflow.net/users/9947 | Ordinary n-uple Points and Resolution of Singularities on a Surface | A germ $(X, p)$ of isolated surface singularity is called an *ordinary n-tuple point* if
$$\hat{\mathcal{O}}\_p=\mathbb{C}[[ x^n, x^{n-1}y, \ldots, xy^{n-1}, y^n]],$$
see for instance Miyaoka's paper [The maximal number of quotient singularities on surfaces with given numerical invariants](https://doi.org/10.1007/BF014... | 5 | https://mathoverflow.net/users/7460 | 72504 | 44,222 |
https://mathoverflow.net/questions/72469 | -1 | I have developed an algorithm, Sieves like, that only pick every integer once. The Sieve algorithm of course selecting integers in as many iterations as the integer has distinct prime factors.
Is this seen before (I haven't found it) ?
| https://mathoverflow.net/users/17039 | Sieve algorithm that only pick every integer once? | It may interest you that the asymptotically fastest prime sieve (of Atkin-Bernstein) spends less than unit time on each number in the interval, by considering increasingly thin congruence classes as its input increases. (Of course it spends positive time, $O(\log n/\log\log n)$, on each *prime*.)
| 1 | https://mathoverflow.net/users/6043 | 72509 | 44,226 |
https://mathoverflow.net/questions/72508 | 2 | Consider the Torelli morphism $T$: $M\_{g}$-----------> $A\_{g}$. The differential of this morphism is the map
$dT$ : $ T\_{M\_{g}}$---------> $T\_{A\_{g}}$
between the tangent bundles. Now at the point $[C]$ $\in$ $M\_{g}$, this becomes the map
$H^{1}(T\_{C})$ -----> $Sym^2 H^{1}(O\_{C})$ .
On the other hand... | https://mathoverflow.net/users/14181 | Differntial of the Torelli morphism and the multiplication map | $\DeclareMathOperator{\Jac}{\mathrm{Jac}}\DeclareMathOperator{\Sym}{\mathrm{Sym}}$
The answer is yes.
First of all let's see how to identify the tangent space of a point $[\Jac C] \in A\_g$ with $\Sym^2 H^1(C,O\_C)$.
The 1st order deformations of an abelian variety $A$ (not necessarily preserving the polarization)... | 4 | https://mathoverflow.net/users/1310 | 72516 | 44,232 |
https://mathoverflow.net/questions/72525 | 1 | Hello,
When I read Stein's book of [Singular Integrals](http://rads.stackoverflow.com/amzn/click/0691080798), at p. 118, there is an obvious mistake:
$$
\int\_{R^n} |x-y|^{-n+\alpha} |y|^{-n+\beta}=\frac{\gamma(\alpha)\gamma(\beta)}{\gamma(\alpha+\beta)},\quad 0<\alpha, 0<\beta,
$$
with $\alpha+\beta < n$ and
$$
\... | https://mathoverflow.net/users/36814 | High dimensional beta integral (a typo in Stein's book "singular integrals") | How is that integral infinite? If $x\neq 0$, then the function is locally integrable near $y=x$ and locally integrable near $y=0$ and it is integrable at $\infty$ since the integrand behaves like $|y|^{-2+\alpha+\beta}$ which decays like $y^{1+\epsilon}$. So the integral is not infinite unless $x=0$.
There is also th... | 2 | https://mathoverflow.net/users/5751 | 72528 | 44,238 |
https://mathoverflow.net/questions/72498 | 20 | Let us assume someone is interested in the study of Hamiltonian mechanics.
What are good examples to illustrate him of the usefulness of contact geometry in this context?
On one hand the Hamiltonian mechanics was time ago expressed in the language of symplectic geometry, but, on the other hand, the contact geometr... | https://mathoverflow.net/users/12617 | What is the role of contact geometry in the hamiltonian mechanics? | In mechanics you often want to study systems whose Hamiltonian function depends on time (explicitly). For example, you can look at the motion of a charged particle in a *time-dependent* electric field. In such cases you are solving an ODE in the "extended phase space" ($\mathbb{R}^7$ in the above example), and not in $... | 14 | https://mathoverflow.net/users/6278 | 72533 | 44,242 |
https://mathoverflow.net/questions/72536 | 8 | We consider the sequence $R\_n=p\_n+p\_{n+1}+p\_{n+2}$, where $\{p\_i\}$ is the prime number sequence, with $p\_0=2$, $p\_1=3$, $p\_2=5$, etc..
The first few values of $R\_n$ are:
10, 15, 23, 31, 41, 49, 59, 71, 83, 97, 109, 121, 131, 143, 159, 173, 187, 199, 211, 223, 235, 251, 269, 287, 301, 311, 319, 329, 349, 371... | https://mathoverflow.net/users/16855 | Asymptotics for primality of sum of three consecutive primes | I believe proving (or disproving) such a statement is beyond current technology. On the other hand, by a crude heuristics the conjecture must be right: $R\_1,\dots,R\_n$ are $n$ odd numbers up to $\sim 3n\ln(n)$ which are rather evenly distributed in size and in residue classes. In this range the density of primes amon... | 9 | https://mathoverflow.net/users/11919 | 72539 | 44,247 |
https://mathoverflow.net/questions/72519 | 10 | [Mark Sapir's question](https://mathoverflow.net/questions/72490/why-are-operads-useful) inspired me to ask the question in the title. A lot of mathematicians who have done work related to mathematical physics (e.g Kontsevich, Stasheff, Getzler, Manin, etc.) have done work with operads, but I don't really grok why oper... | https://mathoverflow.net/users/9417 | Why are operads so closely connected to mathematical physics? | I'm not a mathematical physicist, so parts of this may be wrong.
In quantum field theory, one encounters operators that are supported at points in spacetime, or at least are very local. For example (if we ignore uncertainty), one may have a "photon creation at $x$ with momentum $p$ and helicity $\xi$" operator, which... | 9 | https://mathoverflow.net/users/121 | 72541 | 44,249 |
https://mathoverflow.net/questions/72434 | 2 | Let's say I have a Brownian particle of some radius $r\_b$ and coefficient of diffusion $D$, freely moving about in a toroidal/doughnut-shaped chamber with inner and outer radius $R\_{inner}$ and $R\_{outer}$, respectively. As such, the circle that can be rotated to generate the torus has radius $R\_{chamber} = \frac{1... | https://mathoverflow.net/users/3248 | Inferring the location of a reflecting boundary in a toroidal cage with a Brownian particle | If a Brownian motion hits a reflecting barrier like that, it is almost surely going to hit multiple times. In fact, it will almost surely hit in a Cantor set of times and locations, just as an unreflected one-dimensional Brownian motion almost surely returns to the value at $t=0$. This type of behavior almost surely do... | 3 | https://mathoverflow.net/users/2954 | 72551 | 44,254 |
https://mathoverflow.net/questions/72451 | 3 | Euclid's proof is typically regarded as very weak, because it produces a density which is extremely thin. My question is how far the most obvious generalizations of Euclid's algorithms (for definiteness, algorithm E2) partially fix this problem. In particular, does E2 produce a smooth monotone $\pi(n)$ which asymptotic... | https://mathoverflow.net/users/14689 | How many primes does Euclid's prime generating algorithm really produce? | You might be interested in the papers Guy, Lacampagne, and Selfridge, Primes at a glance, Math. Comp. 48 (1987) 183-202, and Agoh, Erdos, and Granville, Primes at a (somewhat lengthy) glance, Amer. Math. Monthly 104 (1997) 943-945. In the second paper, it is proved that, given $p\_1=2\lt p\_2=3\lt\cdots\lt p\_k$, the f... | 3 | https://mathoverflow.net/users/3684 | 72552 | 44,255 |
https://mathoverflow.net/questions/72527 | 4 | Let's start with a product manifold $M\times N$, with a product Riemannian metric $g\_M\oplus g\_N$. Consider a generic Morse function $f(x, y)$ on $M\times N$. Then its critical points are discrete and don't depend on the metric. But its gradient vector field does depend on the metric.
Now we would like to change t... | https://mathoverflow.net/users/2555 | Morse theory and adiabatic limits | Don't you want to take $\epsilon\to\infty$ to make the $N$ component of the gradient small?
Anyway, here is roughly what I think happens. For each $y\in N$ there is a function $f(\cdot,y)$ on $M\times\{y\}$ with some critical points. The unions of these critical points form submanifolds (maybe with singularities, but... | 4 | https://mathoverflow.net/users/6670 | 72553 | 44,256 |
https://mathoverflow.net/questions/72341 | 7 | Let $V\_{i,j}=x\_i^j$ where $x\_i\in\mathbb F\_q$ for $1\le i\le n,1\le j\le n$ be a Vandermonde matrix over finite field $\mathbb F\_q$.
I wish to know the currently known fastest algorithms for computation of
1) $Vx$ where $x\in\mathbb F\_q^{n\times1}$;
2) $V^Tx$ where $V^T$ is the transpose of $V$;
3) $V^{-1}x$;
4... | https://mathoverflow.net/users/17016 | Fast Vandermonde matrix multiplication over finite field | All problems can be solved in $O(M(n)\log(n))$ base field operations, where $M(n)$ is the time it takes to multiply polynomials in degree $n$ (so using FFT, this is quasi-linear).
This is in Chapter 3 of Pan's *Structured Matrices and Polynomials*.
| 5 | https://mathoverflow.net/users/17054 | 72556 | 44,258 |
https://mathoverflow.net/questions/72554 | 6 | In Segal's paper on $\Gamma$-spaces, he gives a functor $Spectra \rightarrow \Gamma-Spaces$ defined by taking a functor $E$ and sending it to the $\Gamma$-space $AE$ with $AE(n) = Mor(S \times \cdots \times S, E)$, where $S$ is the sphere spectrum. Now, since this is supposed to define a $\Gamma$-space, in particular t... | https://mathoverflow.net/users/6936 | What is the topology on hom-sets of spectra? | If $X = (X\_n)$ and $Y = (Y\_n)$ are spectra, one can define a morphism just to be a collection of maps $x\_n \to Y\_n$ commuting with the suspensions. Thus the set of morphisms between $X$ and $Y$ is a subset of $\prod\_n Map(X\_n,Y\_n)$ - and we give it just the subspace topology.
An alternative way is the simplic... | 9 | https://mathoverflow.net/users/2039 | 72564 | 44,264 |
https://mathoverflow.net/questions/72568 | 2 | It is well-known that there exist rational curves on Fano manifold. The only existing proof is due to Mori. His proof uses geometry of characteristic p.
My question is: For a hypersurface of degree $\leq n$ in $CP^n$, it is Fano by the adjunction formula. Does there exist an "elementary" proof of existence of rationa... | https://mathoverflow.net/users/15882 | existence of rational curves on hypersurface of degree<=n in CP^n | I don't know if there is an elementary proof out there which works for all hypersurfaces of degree $d\ge n$, but if $X$ is general, you can see quite easily that they contain projective lines - or equivalently, the Fano variety of lines is of positive (expected) dimension. This is proved in Joe Harris' algebraic geomet... | 3 | https://mathoverflow.net/users/3996 | 72572 | 44,267 |
https://mathoverflow.net/questions/72579 | 3 | I think there are references for this question, but I didn't find it. We know that for a simple abelian variety $A/k$, the rign $\mathrm{End}^0 (A)$ is a division algebra. One use the fact that every homomorphism $\phi : A \rightarrow A$ is either an isogeny or $0$. To see this, one consider the identity connected comp... | https://mathoverflow.net/users/5482 | geometrical reducedness of the identity connected component (reference request) | You can find a proof [here](http://www.math.u-bordeaux.fr/~liu/Notes/sub-abelian_varieties.pdf) following hints of Raynaud. The main idea is that the prime-to-$p$ torsion of $G^0$ is dense in $G^0\_\mathrm{red}$ and its Zariski closure (with reduced structure) is geometrically reduced.
| 5 | https://mathoverflow.net/users/3485 | 72586 | 44,273 |
https://mathoverflow.net/questions/72609 | 4 | Let $K$ be a finite type field extension of $\mathbf{Q}$ and let
$Y$ be a smooth quasi-projective variety over $K$. Let $\Omega\_{Y/K}^{\bullet}$ denote the complex
of sheaves of (algebraic) regular differential forms on $Y$. Recall that the $k$-th
De Rham cohomology group of $Y$ is defined as
$$
H^k(Y):=\mathbb{H}^k... | https://mathoverflow.net/users/11765 | On the finitess of algebraic De Rham cohomology of smooth quasi-projective variety | $\def\HH{\mathbb{H}}$The following sketch of an argument is taken from Grothendieck's [On the de Rham cohomology of algebraic varieties](http://www.ams.org/mathscinet-getitem?mr=199194). You can also find a good discussion in Voisin's Hodge Theory book, volume 1, chapter 8. I talked about it a little on the [last day](... | 12 | https://mathoverflow.net/users/297 | 72615 | 44,284 |
https://mathoverflow.net/questions/72616 | 5 | I have a system to solve, set up as :
$$Ax = b$$
with a square rank deficient matrix $A$. The paper suggests to use a Moore Penrose pseudo inverse, which in my case can be computed using the traditional inverse :
$$ A^+ = (A+\frac{ee^T}{n})^{-1} - \frac{ee^T}{n} $$
where $e$ is a vector containing only ones, and $n$ i... | https://mathoverflow.net/users/8646 | Solving for Moore Penrose pseudo inverse | For black-box linear algebra (GMRES and the like) you don't need "sparse", you only need "can compute products quickly". If you check the docs for your sparse solver, I'm sure there'll be a version where you can provide directly the function $v\mapsto Av$ (and sometimes $w\mapsto A^Tw$ is needed, too). In your case, yo... | 6 | https://mathoverflow.net/users/1898 | 72621 | 44,287 |
https://mathoverflow.net/questions/72614 | 4 | Greetings,
I am currently working on a paper that involves an upper bound of the largest root of a polynomial. With the help of the Mean Value Theorem, I believe a colleague and I have proved the following: if f^(n)(p) > 0 (the n-th derivative evaluated at "p", for n=0,1,2..d, where d is the degree of the polynomial)... | https://mathoverflow.net/users/17074 | A known Lemma on the largest root of a polynomial and its derivatives? | For me, it is a folklore result for people interested in Descartes' rule of signs, even though I am not aware of any paper stating it explicitly. It follows from Descartes' rule of signs.
**Theorem** (Descartes' rule of signs). The number of positive roots of a real polynomial is not larger than the number of sign ch... | 5 | https://mathoverflow.net/users/16178 | 72623 | 44,289 |
https://mathoverflow.net/questions/72628 | 6 | What are some of the natural number theory problems that are np-complete? I am looking for examples not in lattices and geometric number theory. Examples in analytic/algebraic number theory are ok.
| https://mathoverflow.net/users/16007 | Number theory and NP-complete | You can take a look at the papers by Adleman and Manders (not always in this order) from the 70s (at least "Computational complexity of decision procedures for polynomials", "NP-complete decision problems for quadratic polynomials", "Diophantine complexity"), and the references therein.
One example of the problems t... | 15 | https://mathoverflow.net/users/12082 | 72633 | 44,296 |
https://mathoverflow.net/questions/72600 | -12 | Product $a\times b$ of filters $a$ and $b$ is defined as the filter (on the set of binary relations) defined by the base $\{ A\times B | A\in a,B\in b \}$.
I will denote the principal filter corresponding to a set $X$ as $\uparrow X$.
Let $a$ and $b$ are filters.
Suppose for some filter $M$ on binary relations we... | https://mathoverflow.net/users/4086 | Direct product of filters | I'm not entirely sure I understand the notation, because it seems $a\times b$ would be a filter on the set of ordered pairs, not on the set of binary relations (though it would be a filter *in* the Boolean algebra of binary relations). But, assuming my best guess for what was intended, here's a counterexample. Let both... | 30 | https://mathoverflow.net/users/6794 | 72638 | 44,299 |
https://mathoverflow.net/questions/72416 | 1 | *EDIT (28.11.2013).*
This math.SE [question about Cesaro operators](https://math.stackexchange.com/questions/119964/cesaro-operator-is-bounded-for-1p-infty) should also be of interest (it discusses upper bounds; Noam has produced lower bounds below)
---
Federico's recent question [Norm of upper triangular matrix ... | https://mathoverflow.net/users/8430 | Norm of Cesaro matrix | Here there is likely no closed form for $\|C\_n\|\_2$, but I can show $\|C\_n\|\_2 > 2 - O(1/\log n)$; given that also $\|C\_n\|\_2 \leq 2$ (is this easy to prove?) it follows that $\|C\_n\|\_2 \rightarrow 2$ as $n \rightarrow \infty$.
In general we can consider, for fixed $p \in [1,\infty]$, the $l^p$-operator norm ... | 4 | https://mathoverflow.net/users/14830 | 72639 | 44,300 |
https://mathoverflow.net/questions/72635 | 10 | Hi,
It's shown by an easy cardinality argument that there are complete second-order theories that are not categorical (have more than one model up to isomorphism). Anyone knows of a concrete example of such a theory?
Thanks in advance
| https://mathoverflow.net/users/7409 | Categoricity in second order logic | I won't actually write down a concrete example (too much work), but here's how to get one. Work with the vocabulary (= language = signature) that has a constant symbol 0, a unary function symbol $S$, a binary predicate symbol $\in$, and two unary predicate symbols $N$ and $P$. The structures I want to consider look lik... | 10 | https://mathoverflow.net/users/6794 | 72647 | 44,306 |
https://mathoverflow.net/questions/72546 | 3 | Let's say I have a motive in $\mathcal{M}\_{num}(K)$ ($K$ a number field). For each prime $l$ there is a realization of this motive in terms of etale cohomology with coefficients in $\mathbb{Q}\_l$. This has more structure than being a mere vector space: it is a representation of $Gal(K)$! This representation has an $L... | https://mathoverflow.net/users/5756 | Is the "L-function of the complex cohomology" of a motive equal to the L-function of its l-adic realization? | As far as I can tell, the Galois action that you assert exists doesn't in fact exist.
First of all, to define the motivic Galois group, you have to choose a realization; it seems
that you are choosing the Betti realization, so the motivic Galois group is a pro-algebraic group $G$ over $\mathbb Q$ which acts on $H^i(... | 10 | https://mathoverflow.net/users/2874 | 72652 | 44,309 |
https://mathoverflow.net/questions/72440 | 5 | **Note.** *I have edited my question to make it more transparent, following some very good comments that I received. I am sorry if it is a bit long.*
A local homomorphism of local rings $(A,\mathfrak{m})\stackrel{\varphi}{\longrightarrow}(B,\mathfrak{n})$ is called a *scalar extension* (terminology due to Hans Schout... | https://mathoverflow.net/users/16046 | On the functoriality of scalar extensions of local rings (edited) | I post this as an answer since it is too long, actually answers Question B and sheds some light on Question A. The example is taken from Eisenbud, *Commutative Algebra*, Exercise 7.17b.
Let $A\_1=\mathbf F\_p(t)$, $A\_2=\mathbf F\_p(u)[[x]]$, $\psi\colon A\_1\to A\_2$, $t\mapsto u^p+x$. On the residue fields, $\psi$ ... | 4 | https://mathoverflow.net/users/2035 | 72657 | 44,314 |
https://mathoverflow.net/questions/70611 | 2 | The following theorem seems to have folk status:
The topological dual of the space $C\_b(X)$ of bounded continuous functions on a topological space $X$ is isomorphic to the space $rba(X)$ of finite, regular, finitely additive Borel set functions.
This fact is often mentioned (for instance in the answer to [Dual of ... | https://mathoverflow.net/users/16530 | Reference for proof that $C_b^* = rba$ | The topological dual of the space of bounded continuous functions on a topological space X is isomorphic to the space of finite, zero set regular, finitely additive *Baire* set functions; see: R. F. Wheeler, A survey of Baire measures and strict topologies, Exposition. Math. 2 (1983), 97–190 (a proof is on pp. 115-117)... | 3 | https://mathoverflow.net/users/17082 | 72662 | 44,317 |
https://mathoverflow.net/questions/72658 | 1 | While working on a project for mathematics I came across the following lemma: [Kock]
If $X$ is a curve defined over an algebraically closed subfield $N$ of $\mathbb{C}$ and let $t:X\rightarrow \mathbb{P}\_{\mathbb{C}}^1$ be a finite morphism defined over $N$, then the critical values of $t$ are $N$-rational.
I hav... | https://mathoverflow.net/users/17081 | Branch points of a non-constant holomorphic map between compact riemann surfaces | This is quite standard and probably the question belongs to <http://math.stackexchange.com> rather than Mathoverflow, anyway let me give an answer.
The holomorphic map $t \colon X \to \mathbb{P}^1$ induces a natural map $dt$ between tangent bundles, hence a short exact sequence of coherent sheaves on $X$: $$0 \longri... | 4 | https://mathoverflow.net/users/7460 | 72667 | 44,319 |
https://mathoverflow.net/questions/72671 | 5 | Following is quoted from -Nakayama, On Weierstrass models-,
" Let $S$ be a complex surface, and $L$ a line bundle on it. consider $P=\mathbb{P}(\mathcal{O}\_S\oplus L^2 \oplus L^3)$. Let $a$ and $b$ be arbitrary sections of $L^{-4}$ and $L^{-6}$ and let $(x,y,z)$ be the canonical sections of $\mathcal{O}\_P(1)\otime... | https://mathoverflow.net/users/5259 | Question on a projective bundle associated to a vector bundle | There are two competing definitions for $\mathbb P(\mathcal E)$, one classifies subbundles of $\mathcal E$ of rank 1, the other classifies quotients. With the latter (as used in EGA or Hartshorne), you have a canonical quotient $p^\*\mathcal E\to\mathcal O(1)$ instead of a canonical subbundle $\mathcal O(-1)\to p^\*\ma... | 4 | https://mathoverflow.net/users/2035 | 72673 | 44,322 |
https://mathoverflow.net/questions/72631 | 3 | Let $E/k$ be an elliptic curve over a field of characteristic $\neq$ 2, 3. Then we have an isomorphism $ [ \ \ ] :\mu\_n \rightarrow\mathrm{Aut}\_{\overline{k}}(E)$, $[ \zeta ] : (x,y) \rightarrow (\zeta^2x, \zeta^3y) $, here $n=2, 4,6$, depending on the $j$-invariant $j(E) $. See Corollary 10.2 on Ch3 in "The arithmet... | https://mathoverflow.net/users/5482 | Automorphism groups of Elliptic curves as Galois module | It's as Matt indicates. You might find it easier to look at the commutative diagram, which is what it means for two maps to commute. Thus
$$
\begin{aligned}
E(\bar{k}) &\qquad\xrightarrow{[\zeta]} & E(\bar{k}) \cr
\downarrow\sigma && \downarrow\sigma \cr
E(\bar{k}) &\qquad\xrightarrow{[\zeta^\sigma]} & E(\bar{k}) \... | 4 | https://mathoverflow.net/users/11926 | 72676 | 44,324 |
https://mathoverflow.net/questions/72665 | 1 | [Sums of cubes and more](https://mathoverflow.net/questions/437/sums-of-cubes-and-more)
In the selected answer to the above question the writer states ""what is the least $G=G(k)$ such that for some $N$, every integer greater than $N$ can be represented as the sum of $G$ $k$-th powers"
For $G(k)$ that we have expli... | https://mathoverflow.net/users/16007 | Sum of higher powers (bound on $N$) | For the first few relevant cases even exact values are known or conjectured. Fairly recent work on this was done by Deshouillers and different groups of coauthors:
In particular, it is known that for $k=4$ that, recall $G(4)=16$, that $13792$ is the last number to requier $17$ fourth powers, and this is thus your (op... | 3 | https://mathoverflow.net/users/nan | 72685 | 44,329 |
https://mathoverflow.net/questions/72677 | 5 | Assume that the formal power series $a(z)=\sum\limits\_{n\geq 1} a\_n z^n$ satisfies an algebraic equation with polynomial coefficients (that is, there exists a nonzero polynomial $F(z,y)$ such that $F(z,a(z))=0$), and a finite number of terms of $a(z)$ determine uniquely the whole series as a solution to that equation... | https://mathoverflow.net/users/1306 | Asymptotics/growth for coefficients of algebraic power series | This should follow from Theorem VII.8 in Flajolet and Sedgewick's [*Analytic Combinatorics*](http://algo.inria.fr/flajolet/Publications/books.html) (freely available at the link). As I understand the argument, you can obtain a straightforward general form for asymptotics of coefficients of algebraic generating function... | 5 | https://mathoverflow.net/users/290 | 72687 | 44,330 |
https://mathoverflow.net/questions/72710 | 7 | Suppose that $X$ is a smooth threefold, and $C \subset X$ a smooth curve. Let $Y$ be the blowup of $X$ along $C$, with exceptional divisor $E$. What is the intersection number $E^3$ on $Y$? (in terms of the genus and normal bundle of $C$, etc)
I assume that I could extract the answer from Theorem 6.7 of Fulton's book... | https://mathoverflow.net/users/17092 | Self-intersection of exceptional divisor | The intersection number $E^3$ equals $-\deg N\_{C|X}$ the negative of the degree of the normal bundle of $C$. Here, as usual, $\deg N\_{C|X}=2g-2-K\_X.C$. This statement and the proof can be found in Griffiths-Harris and in Iskovskikh-Prokhorov Algebraic Geometry V III. $\S$ 2.3.
| 7 | https://mathoverflow.net/users/3996 | 72711 | 44,340 |
https://mathoverflow.net/questions/72714 | 1 | I'm working the proof of the Stone-Weierstrass Approximation theorem using probability theory from "A Second Course in Probability" by Ross and Pekoz. The statement of the theorem in the book omits UNIFORM from the limit, where as the statement seen in other books such as Rudin is UNIFOM. The proof below is slightly mo... | https://mathoverflow.net/users/17094 | Proving Uniform Convergence from AS Convergence | For uniformity you need a bit more than the SLLN, namely that there is some kind of uniformity in $t$. Typically this is handled by computing also the variance, see [this page](http://en.wikipedia.org/wiki/Bernstein_polynomial).
| 2 | https://mathoverflow.net/users/1044 | 72718 | 44,342 |
https://mathoverflow.net/questions/72727 | 0 | Let $\mathfrak{M}$ and $\mathfrak{N}$ be perfect Polish spaces, $P$ a nonempty perfect subset of $\mathfrak{M}$, and $f: \mathfrak{M} \rightarrow \mathfrak{N}$ a continuous surjection that's injective on $P$. Naively, the image $f[P]$ should be perfect. For if it has an isolated point $x$ with a neighborhood $X$ contai... | https://mathoverflow.net/users/8547 | When can the one-one continuous image of a perfect set fail to be perfect? | A perfect subset of a space $X$ is required not only to have no isolated points but also to be closed in $X$. Compactness of $P$ is used to ensure that $f[P]$ is (compact and therefore) closed.
| 4 | https://mathoverflow.net/users/6794 | 72733 | 44,351 |
https://mathoverflow.net/questions/72734 | 6 | Let $A$ be a free group and $G = A\*\_t$. When is $G$ also a free group?
Suppose $t y t^{-1} = z$ and there is a splitting $A = B\*C$ so that $y \in B$ and $z \in C$ and $z$ is a member of some basis of $C$ then clearly $G$ is free. Is this the only case that $G$ is free?
| https://mathoverflow.net/users/9472 | When is an HNN extension a free group? | Yes. This is a theorem of [Shenitzer](http://www.ams.org/mathscinet/search/publdoc.html?arg3=&co4=AND&co5=AND&co6=AND&co7=AND&dr=all&pg4=AUCN&pg5=RT&pg6=RT&pg7=ALLF&pg8=ET&review_format=html&s4=shenitzer&s5=&s6=&s7=&s8=All&vfpref=html&yearRangeFirst=&yearRangeSecond=&yrop=eq&r=13&mx-pid=69174). For a modern treatment s... | 10 | https://mathoverflow.net/users/1463 | 72740 | 44,352 |
https://mathoverflow.net/questions/72736 | 2 | Hi,
Is set membership *defined* in the signature of ZFC, or is it \*specified" in the signature of ZFC? The wikipedia article <http://en.wikipedia.org/wiki/Zermelo%E2%80%93Fraenkel_set_theory> says that the signature *has* set membership, but what does it mean by *has*?
If I understand correctly, the axioms specify... | https://mathoverflow.net/users/17106 | ZFC, set membership and FOL | A *specification* is a prescription of how things are supposed to be. A *model* (or *implementation* in computer-sciency talk) is an instance of something that satisfies a specification.
In first-order logic and model theory a specification is called a *theory*, and it consists of two parts:
* a *signature* which p... | 5 | https://mathoverflow.net/users/1176 | 72746 | 44,356 |
https://mathoverflow.net/questions/72593 | 0 | I know that minimal flows are actions for which no proper closed invariant subsets exist, but I am unclear how to understand this concept.
If a coset flow on a quotient space Gamma/S is ergodic, strongly mixing, and minimal, does minimal mean that the orbits are neither periodic nor have an infinite number of period... | https://mathoverflow.net/users/17069 | Basic question on minimal flows | Your question is about a theorem of Furstenberg.
About the definitions - obviously every periodic orbit is minimal, if exists, hence in the case the action is minimal, you won't have any periodic orbits.
Notice that in the case of homogeneous flows, "usually" (due to measure classification, which is known for many ... | 2 | https://mathoverflow.net/users/8857 | 72748 | 44,357 |
https://mathoverflow.net/questions/72747 | 4 | Let $(X\_N)\_N$ be a sequence of trace class operators acting on, say, $L^2(\mathbb{R})$. What are the minimal assumptions in order to have the convergence of their Fredholm determinant
$$
\lim\_N\det(I+X\_N) ?
$$
I know $X\mapsto \det(I+X\_N)$ is continuous for the trace class norm topology (once restricted to trace ... | https://mathoverflow.net/users/15517 | Convergence of Fredholm determinants | The Fredholm determinant is not sequentially continuous in the strong topology. Take the sequence of 1-dimensional projectors $X\_n=\langle e\_n,\cdot\rangle e\_n$ with the $e\_n$ forming an ON basis. You then have $X\_n \to 0$ strongly, but $0=\det(I-X\_n)$ does not converge to $1=\det(I)$. Or what did you have in min... | 4 | https://mathoverflow.net/users/13034 | 72754 | 44,358 |
https://mathoverflow.net/questions/72690 | 14 | The category of line bundles (possibly with connection)
on a smooth manifold M can be defined in two different ways:
The first definition uses transition functions that satisfy a cocycle condition
(possibly with additional data of a 1-form that defines a connection),
and the second definition uses invertible vector bun... | https://mathoverflow.net/users/402 | Are bundle gerbes bundles of algebras? | There is a bicategory of Dixmier-Douady bundles of algebras which is equivalent to the bicategory of bundle gerbes. In particular, sections into these bundles form algebras.
The price you pay is that the bundles are infinite-dimensional; for that reson I am not sure if that picture persists in a setting "with connec... | 6 | https://mathoverflow.net/users/3473 | 72756 | 44,359 |
https://mathoverflow.net/questions/72761 | 1 | I'd like to sample the elements of a symmetric square matrix uniformly. For example, for a $N\times N$ matrix, I'd like to only keep $\alpha$% of the matrix elements to build a sparse matrix, while keeping the symmetry property.
A simple method would be to scan the upper right part of the matrix, generating a random ... | https://mathoverflow.net/users/8646 | Random sampling a symmetric matrix | write down the $N^2$ pairs of integers $(n,m)$ with both $n$ and $m$ ranging from $1$ to $N$ ; randomly select a% of them; these are the nonzero elements $M\_{nm}$ of your sparse matrix; then evaluate the row with the largest number of nonzero elements, reflect in the diagonal to get the corresponding column, continue ... | 0 | https://mathoverflow.net/users/11260 | 72763 | 44,363 |
https://mathoverflow.net/questions/72735 | 3 | I'd like to compute
$\max\_{x,t} t$ such that $\forall i$, $t < a\_i + |x - b\_i|$.
where $a\_i,\ldots, a\_n$ and $b\_1,\ldots,b\_n$ are fixed and $x \in [0,1]$.
Can this be solved with a linear program? I'm familiar with a technique to minimize the maximum of absolute values, by doubling the number of constraint... | https://mathoverflow.net/users/17073 | Linear program to maximize the minimum absolute value of linear functions ? | Unfortunately, this problem can't be represented by an LP, since your feasible region is in general nonconvex, and the feasible region of an LP (being the intersection of a bunch of half spaces) is always convex.
To be more specific, consider the problem
$\max \;t $
$ t \leq | x- 1/2 | $
$ t \leq | x- 3/4 | $
... | 4 | https://mathoverflow.net/users/9022 | 72768 | 44,364 |
https://mathoverflow.net/questions/72567 | 2 | I would like to know if there are some central-simple algebras $D\_1$, $D\_2$ and $D\_3$ over a field $k$ satisfying the following properties :
* $ind(D\_1)=exp(D\_1)=4$ ($ind$ is the Schur index and $exp$ the exponent);
* $D\_2$ and $D\_3$ are two non-isomorphic quaternion algebras;
* $ind(D\_1^{\otimes2} \otimes D\... | https://mathoverflow.net/users/17057 | Central division and quaternion algebras | You can find such division algebras over $\mathbb{Q}(x\_1,x\_2,x\_3)$ or $k(x\_1,x\_2,x\_3,x\_4)$ where $k$ is any field using the results of "Nakayama, T. Über die direkte Zerlegung einer Divisionsalgebra. Japanese J. of Mathematics 12 (1935), 65–70". A simplified proof is given in the book "Associative algebras" by P... | 6 | https://mathoverflow.net/users/519 | 72769 | 44,365 |
https://mathoverflow.net/questions/72102 | 0 | [Picard's Big Theorem](http://en.wikipedia.org/wiki/Picard_theorem) says that if a function $f(z)$ has an isolated essential singularity at a point $w$, then in every neighborhood of $w$, $f(z)$ hits every complex number infinitely many times, with perhaps at most one exception.
Is there a version of Picard's theorem... | https://mathoverflow.net/users/5534 | Behavior of essential singularities in an 'open cone' | Maybe it's not exactly what you are asking for (and maybe you know it already), but a related concept to what you are asking is that of *Julia line*.
For sake of simplicity, consider an entire function $f$ with an essential singularity at $\infty$; let
$S(\phi,\epsilon)=\{z\ :\ |\mathrm{arg}(z)-\phi|<\epsilon\}$ ... | 4 | https://mathoverflow.net/users/17111 | 72774 | 44,368 |
https://mathoverflow.net/questions/72779 | 7 | Let $\mathcal{M}\_{0,n}$ be the complement of the boundary of the Mumford-Knudsen compactification of the moduli space of genus zero, n-pointed curves.
Is $Pic(\mathcal{M}\_{0,n})$ trivial?
| https://mathoverflow.net/users/4096 | Picard group of $\mathcal{M}_{0,n}$ | Yes. By fixing the three points $\{0,1,\infty\}$ one sees that $M\_{0,n}$ is isomorphic to an open subscheme of $\mathbb{A}^{n-3}$ which has trivial Picard group. Since it is smooth, the Picard group of any open subscheme is also trivial.
| 12 | https://mathoverflow.net/users/519 | 72781 | 44,369 |
https://mathoverflow.net/questions/72784 | 6 | Let G be a finitely generated group. Suppose we have two families F1 and F2 of finite index subgroups of G, and each family has trivial intersection and is filtered from below (i.e. for any two elements in the family their intersection contains some third element).
These families generate two profinite topologies on ... | https://mathoverflow.net/users/7307 | Profinite topologies on a group generated by different families of subgroups | No. $\widehat{\mathbb{Z}} \times \mathbb{Z}\_p$ is isomorphic to $\mathbb{Z}\_p \times \widehat{\mathbb{Z}}$, and the two topologies on $\mathbb{Z} \times \mathbb{Z}$ are different, though homeomorphic.
Am I interpreting your question too narrowly?
Also, in the case of a subfamily, you have a map $G'' \to G'$ from ... | 10 | https://mathoverflow.net/users/1335 | 72790 | 44,372 |
https://mathoverflow.net/questions/72785 | 2 | I'd like to compute
$\max\_{x,t} t$ such that $\forall i$, $t < a\_i + \|x - b\_i\|\_\infty$.
where $a\_i,\ldots, a\_n \in \mathbb R$ and $b\_1,\ldots,b\_n \in [0,1]^{21}$ are fixed, $x \in [0,1]^{21}$, $\|\cdot\|\_\infty$ is $\sup$-norm, and $n$ is roughly $1000$.
I understand this problem cannot be solved with ... | https://mathoverflow.net/users/17073 | Maximizing the minimum of piecewise linear functions in high dimensional space | As in your previous question, this is a nonconvex optimization problem, so it won't be LP, SOCP, or SDP representable.
You've only got a 21 dimensional problem, and the constraint functions have easy Lipschitz constants. If you've got the time (hours or days of computation) and really need a fairly accurate solution... | 3 | https://mathoverflow.net/users/9022 | 72798 | 44,375 |
https://mathoverflow.net/questions/72548 | 4 | I'm reading through Mukai's excellent book "Introduction to Invariants and Moduli", and am stuck on a proof in Chapter 4. He's proving that $G = SL\_n$ over a field $k$ of characteristic $0$ is linearly reductive, i.e. for every epimorphism $V \rightarrow W$ of representations of $G$, the induced map on invariants $V^G... | https://mathoverflow.net/users/2698 | Linear reductivity of $SL_n$ in char $0$: proof in Mukai's book | In Mukai's approach to this proof (in particular his Prop 4.49), care is needed typographically, all squaring must come before taking the trace. Note that having $\mathrm{tr}(\tilde{\rho}(h)) = 0$ for all h does not imply $\tilde\rho$ is trivial (eg 2 dim rep of SL\_2), the condition we need is $\mathrm{tr}(\tilde{\rho... | 4 | https://mathoverflow.net/users/425 | 72799 | 44,376 |
https://mathoverflow.net/questions/72601 | 2 | In our application we wish to estimate the actual path of objects. We have a set of samples of object locations $(t\_i, x\_i, y\_i, P\_i)$ where $t\_i$ is the sample time, $(x\_i, y\_i)$ is the 2D location, and $P\_i$ is the error covariance matrix for measurement $i$.
We estimate the path using polynomial regression... | https://mathoverflow.net/users/17071 | Point-wise error estimate in polynomial regression | This is just weighted least squares and here is how I would approach it. To keep my notation simple I'll just have polynomials of order 1. It's trivial to extend the approach to polynomials of any order. Let $a\_x$, $b\_x$ and $a\_y$, $b\_y$ be the 'true' coefficients describing the path. So the path is
$$ (x(t), y(t))... | 1 | https://mathoverflow.net/users/5378 | 72803 | 44,377 |
https://mathoverflow.net/questions/72800 | 57 | The title says it all: if $f\colon \mathbb{R} \to \mathbb{R}$ is any real function, there exists a dense subset $D$ of $\mathbb{R}$ such that $f|\_D$ is continuous.
Or so I'm told, but this leaves me stumped. Apart from the rather trivial fact that one can find a dense $D$ such that the graph of $f|\_D$ has no isolat... | https://mathoverflow.net/users/17064 | Every real function has a dense set on which its restriction is continuous | It is a theorem due to Blumberg (*New Properties of All Real Functions* - Trans. AMS (1922)) and a topological space $X$ such that every real valued function admits a dense set on which it is continuous is sometimes called a Blumberg space.
Moreover, in Bredford & Goffman, *Metric Spaces in which Blumberg's Theorem H... | 49 | https://mathoverflow.net/users/17111 | 72804 | 44,378 |
https://mathoverflow.net/questions/72807 | 5 | Fix a natural number, $n \geq 1$. Consider the class, M, of all sets hereditarily ordinal-definable using some $\Sigma\_n$ formula. Since there is a universal $\Sigma\_n$ formula, M is definable. Is M necessarily a model of ZF? It seems to me that it is closed under Godel operations and almost universal for the same re... | https://mathoverflow.net/users/3183 | $\Sigma_n$ version of HOD | It follows from the [Reflection Principle](http://en.wikipedia.org/wiki/Reflection_principle) that every ordinal definable set is ordinal definable by a $\Sigma\_2$-formula in the language of set theory. Indeed, if $A = \{x : \phi(x,\bar\alpha)\}$, then $$\exists\gamma(\gamma \in \mathrm{Ord} \land \bar\alpha \in \gamm... | 6 | https://mathoverflow.net/users/2000 | 72809 | 44,382 |
https://mathoverflow.net/questions/72810 | 25 | I was recently trying to learn a little bit about group cohomology, but one point has been confusing me. According to wikipedia (<https://en.wikipedia.org/wiki/Group_cohomology> and some other sources on the internet), given a (topological) group $G$, we have that the group cohomology $H^n(G)$ is the same as the singul... | https://mathoverflow.net/users/17121 | The relationship between group cohomology and topological cohomology theories | Short answer: you don't want to consider group cohomology as defined for finite groups for Lie groups like $U(1)$, or indeed topological groups in general. There are other cohomology theories (not Stasheff's) that are the 'right' cohomology groups, in that there are the right isomorphisms in low dimensions with various... | 22 | https://mathoverflow.net/users/4177 | 72813 | 44,384 |
https://mathoverflow.net/questions/70339 | 2 | This is a question on Fourier series convergence. The problem is, in the applications of the Stone Weierstrass approximation theorem on [wikipedia](http://en.wikipedia.org/wiki/Stone%E2%80%93Weierstrass_theorem#Applications_2), there's stated that as a consequence of the theorem the space of trigonometrical polynomials... | https://mathoverflow.net/users/13108 | Stone-Weierstrass theorem applied to Fourier series | The density of trigonometric polynomials in $C[0,1]$ with respect to the sup norm does not imply that the Fourier Series of some $f\in{}C[0,1]$ must converge pointwise.
Let $e\_k:=e^{2\pi{}ikx}$ for $k\in{}\mathbb{Z}$. Then it can be shown that
1. $\{e\_k|k\in\mathbb{Z}\}$ is an orthonormal basis for $L^2[0,1]$ *... | 5 | https://mathoverflow.net/users/17100 | 72814 | 44,385 |
https://mathoverflow.net/questions/72520 | 5 | Can a representation of a Banach algebra be unbounded?
To clarify, I'm not asking about a representation as unbounded operators, but
rather a homomorphism $\pi: A \to B(H)$ for some Hilbert space $H$, with the property
that $\sup\_{a \neq 0} \frac{\|\pi(a)\|}{\|a\|} = \infty$.
This question is inspired by chapter 2... | https://mathoverflow.net/users/17050 | Unbounded representations of Banach algebras | As Dima Shlyakhtenko indicated, there are examples, obtained for example by modifying your zero product example to $x\mapsto\begin{bmatrix}0&f(x)\\\ 0&0\end{bmatrix}$ for some unbounded linear functional $f$.
A generalization of the result you mentioned is found in Theorem 4.1.20 of Rickart's *General theory of Bana... | 4 | https://mathoverflow.net/users/1119 | 72819 | 44,388 |
https://mathoverflow.net/questions/64898 | 9 | *(This Question was taken from [MSE](https://math.stackexchange.com/questions/38731/values-of-the-riemann-zeta-function-and-the-ramanujan-summation-how-strong-is-t). As Eric Naslund pointed out there, this [question](https://mathoverflow.net/questions/3204/does-any-method-of-summing-divergent-series-work-on-the-harmoni... | https://mathoverflow.net/users/93724 | Values of the Riemann zeta function and the Ramanujan summation - How strong is the connection? | The answer can be obtained with the [following interpretation](http://en.wikipedia.org/wiki/Ramanujan_summation) of the Ramanujan summation:
>
> More recently, the use of $C(1)$ has been proposed as Ramanujan's summation, since then it can be assured that one series admits one and only one Ramanujan's summation, de... | 7 | https://mathoverflow.net/users/14551 | 72823 | 44,391 |
https://mathoverflow.net/questions/72750 | 3 | Assume that $X,Y$ are infinite dimensional Banach spaces. Is it true that if density character of $X$ is less then or equal to density character of $Y$ then $card X \leq card (Y)$ ?
| https://mathoverflow.net/users/17110 | Density character and cardinality | Let $\aleph$ be the density character of the Banach space $X$ and let's compute the cardinality of $X$ using AC but not much set theory, since these days cardinal arithmetic is generally given short shrift in real analysis courses.
Take a dense set $D=\{x\_a: a<\aleph\}$ in $X$ and observe that all tails of this set ... | 3 | https://mathoverflow.net/users/2554 | 72824 | 44,392 |
https://mathoverflow.net/questions/72773 | 2 | Let $H\_{\mathbf{Q}}$ and $H\_{\mathbf{Q}}'$ be two pure Hodge structures of weight $n$ and
$n'$ respectively. How do you prove the following simple fact:
**fact:** If $n>n'$ and $f:H\_{\mathbf{Q}}\rightarrow H\_{\mathbf{Q}}'$ is a morphism which respects the filtrations over $\mathbf{C}$, then $f=0$.
I don't quite... | https://mathoverflow.net/users/11765 | On morphisms of pure Hodge structures of decreasing weight | Here is a proof of the Scholie:
For a pure Hodge structure $H$ of weight $n$ we have $H\_{\mathbb{C}} = \oplus\_{p+q=n} H^{p,q}$ where we have $H^{p,q} := F^p \cap \bar{F}^q$. The key point is that $H^{p,q}$ as defined above is $0$ if $p+q > n$ (but not if $p+q < n$). This is because for a pure Hodge structure of wei... | 2 | https://mathoverflow.net/users/519 | 72825 | 44,393 |
https://mathoverflow.net/questions/72663 | 2 | Let $\mathcal{A}$ be a collection of $n$ lines. Assume that $\mathcal{A}$ is not a pencil. It is known (see <http://www.springerlink.com/content/320p742475v6q746/>) that if all lines are in $\mathbb{RP}^2$, then there are at least $6n/13$ nodes.
What is the minimal number of nodes, if all lines are in $\mathbb{CP}^2... | https://mathoverflow.net/users/2348 | Minimal number of nodes in a complex line arrangement. | In a paper by Hirzebruch "Singularities of algebraic surfaces and characteristic numbers", Cont. Math. vol. 58, part I, 1986, there is an inequality for the number of double and triple points of an arrangement of $n$ complex lines.
Denote by $t\_k$ the number of points lying on precisely $k$ lines; if $t\_n=t\_{n-1}=t\... | 3 | https://mathoverflow.net/users/10610 | 72827 | 44,394 |
https://mathoverflow.net/questions/72826 | 0 | i have the following problem. Let $M$ be a complex n-dim manifold and $X \subset M$ be a n-dim real analytic submanifold. Consider $d\_{X}(z)$ be the squared distance from $z \in M$ to $X$. For $z$ sufficiently near $X$ this function is smooth. My quaestion is: Is (with respect to complex coordinates $z\_{1}, ..., z\_{... | https://mathoverflow.net/users/17125 | is the differential of the distance function holomorphic? | Take the most simple example $M=\mathbb C$ and $X$ the unit circle. Then $d\_X(z)=|z-\frac{z}{|z|}|^2$ is not holomorphic (so as its Laplacian $i\partial \bar \partial( d\_X)$), as you can easily see by expanding the expression.
| 4 | https://mathoverflow.net/users/5659 | 72828 | 44,395 |
https://mathoverflow.net/questions/72834 | 23 | Let $X$ be a variety over $\mathbb{C}$, say separated. According to Deligne's results, there is a "mixed Hodge structure" on the total cohomology $H^\bullet(X(\mathbb{C}), \mathbb{Z})$. One component of this is a "weight filtration" on $H^\bullet(X(\mathbb{C}), \mathbb{Q})$. I haven't read Deligne's "Theorie de Hodge" ... | https://mathoverflow.net/users/344 | What exactly does the weight filtration in Hodge theory have to do with the Weil conjectures? | Dear Akhil,
This is a big topic, although one that has been discussed at various times here, e.g.
[In what setting does one usually define mixed sheaves and weights for them?](https://mathoverflow.net/questions/31223/in-what-setting-does-one-usually-define-mixed-sheaves-and-weights-for-them/31239#31239)
The idea ... | 15 | https://mathoverflow.net/users/4144 | 72836 | 44,397 |
https://mathoverflow.net/questions/72839 | 15 | Let $X$ be a contractible 2-dimensional simplicial complex. Are there nice necessary and sufficient conditions for $X$ to be embeddable in $\mathbb R^3$? Clearly it is necessary that the link of every vertex be a planar graph. Is this sufficient?
| https://mathoverflow.net/users/35353 | When can a contractible 2-complex be embedded in R^3? | If your complex is finite, then figure out the possible ways of
thickening it to a 3-manifold. The possible thickenings are
determined by the various embeddings of the links of the vertices
into $S^2$, then seeing if these induce compatible thickenings
over the edges (determined by the same cyclic ordering over
the li... | 14 | https://mathoverflow.net/users/1345 | 72847 | 44,402 |
https://mathoverflow.net/questions/72862 | 0 | Let there be > included in AxB as a binary relation.
What does (x)>^2(y) mean? What is the meaning of an order relation raised to a power?
My first tought was that >^2 = >x> which is a cartesian product of > included in AxB.
I have a solution in a Linear Algebra book : (x)>^2(y) = x+2 >= y , wich is a bit confusing; I ... | https://mathoverflow.net/users/17137 | Power of an order relation | If $R$ and $S$ are binary relations, then the composition relation $R\circ S$ is usually defined by $a\mathrel{R\circ S} c$ if and only if there is $b$ such that $a\mathrel{R}b$ and $b\mathrel{S}c$.
In the special case where the relations are the (graphs) of functions $f$ and $g$, this produces the (graph) of the us... | 2 | https://mathoverflow.net/users/1946 | 72864 | 44,411 |
https://mathoverflow.net/questions/72887 | 0 | In the real field. Given a diagonal matrix $D$ and a symmetric matrix $A$. For every skew symmetric matrix $S$, is there always a symmetric matrix $H$ such that $-\operatorname{trace}(DSAS)=\operatorname{trace}(DHAH)$?
If $A$ is also diagonal, this can be easily seen true.
| https://mathoverflow.net/users/6858 | Existence of a symmetric matrix. | Maybe I'm missing something, but let $f(H)=\mathrm{tr}(DHAH)$, then
$$f(\alpha H)=\alpha^2f(H)$$
for every real $\alpha$. Obviously $f(0)=0$. So the problem boils down to find an $H$ such that $f(H)>0$ and another one such that $f(H)<0$.
Now, denoting by $h\_k$ the $k-$th column of $H$,
$$(HAH)\_{ij}=(h\_i)... | 4 | https://mathoverflow.net/users/17111 | 72895 | 44,426 |
https://mathoverflow.net/questions/72886 | 19 | Allow me to quote a definition from Gelbart in "*Modular Forms and Fermat's Last Theorem*":
**Definition.** Let $E/\mathbb{Q}$ be an elliptic curve. We say that $E$ is *modular* if there is some normalised eigenform
$$ f(z) = \sum\_{i=1}^{\infty} \ a\_ne^{2\pi inz} \in S\_2(\Gamma\_0(N),\epsilon), $$
for some lev... | https://mathoverflow.net/users/5744 | Why does the definition of modularity demand weight 2? | $\newcommand\Q{\mathbf{Q}}$
$\newcommand\Qbar{\overline{\Q}}$
$\newcommand\Gal{\mathrm{Gal}}$
$\newcommand\C{\mathbf{C}}$
$\newcommand\Sym{\mathrm{Sym}}$
$\newcommand\E{\mathcal{E}}$
$\newcommand\Betti{\mathrm{Betti}}$
$\newcommand\Z{\mathbf{Z}}$
$\newcommand\Hom{\mathrm{Hom}}$
$\newcommand\T{\mathbf{T}}$
To answer thi... | 27 | https://mathoverflow.net/users/2024 | 72898 | 44,428 |
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