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https://mathoverflow.net/questions/159688 | 1 | Maybe this is well known, but i could not find a pointer to some literature:
Let us assume $E$ is a rank n vector bundle on the Riemann sphere $\mathbb{C}\mathbb{P}^1$. We know that $E=\bigoplus\limits\_{i=1}^n L\_i$ for some line bundles $L\_i$.
Now if $E$ is also equipped with a Hermitian metric $h$, does this me... | https://mathoverflow.net/users/43247 | Do Hermitian metrics also split on the Riemann sphere? | No, such an hermitian metric does not necessarily split, even if $n=2$.
As a matter of fact, vector bundles are a quite rigid sort of objects, while metrics are pretty smooth and can be tweaked locally to have any prescribed look
and a given small enough open subset.
By partitions of unity, it is easy to define metri... | 2 | https://mathoverflow.net/users/10696 | 159692 | 84,133 |
https://mathoverflow.net/questions/156979 | 5 | This question is motivated by Joel David Hamkins' answer to [Gödel's Constructible Universe in Infinitary Logics (A Possible Approach to HOD Problem)](https://mathoverflow.net/questions/156940/godels-constructible-universe-in-infinitary-logics-a-possible-approach-to-hod), in which he shows that, if we replace first-ord... | https://mathoverflow.net/users/8133 | Climbing quickly up $L$ | In one sense I think you almost have the answer anyway. Assume $V=L$. For any cardinal $\beta$ let $C\_\beta = \{ \gamma\in (\beta,\beta^+)| \,L\_\gamma \prec L\_{\beta^+}\}$ ($\prec $ meaning First Order elementarily).
Let $\hat L\_0 =L\_\omega=HF$. Then $\hat L\_1 = L\_{\gamma\_1}$ where $\gamma\_1$ is the minimal... | 6 | https://mathoverflow.net/users/6942 | 159696 | 84,134 |
https://mathoverflow.net/questions/159693 | 1 | I have a covariance matrix C. I have then formulated an quadratic optimization problem that involves the following matrix in the quadratic form:
>
> [ C C ]
>
> [ C C ]
>
>
>
However, the quadratic solver complains that this matrix is not positive definite. I can also reformulate the optimization problem s... | https://mathoverflow.net/users/47932 | Block Covariance Matrix - Positive Definite? (Quadratic Optimization) | If $C$ is positive semidefinite, then so is $\begin{bmatrix} C & C\\ C & C\end{bmatrix}$ for the simple reason that it is nothing but the Kronecker product of $\begin{bmatrix} 1 & 1\\ 1 & 1\end{bmatrix}$ with $C$.
*EDIT*
For those who don't like Kronecker products, here is an alternative proof using block-matrices... | 3 | https://mathoverflow.net/users/8430 | 159700 | 84,137 |
https://mathoverflow.net/questions/159697 | 8 | Let $K$ be a field and $A$ the associative unital $K$-algebra of all $n\times n$ upper triangular matrices with entries in $K$. What is $\dim\_K$ of its hochschild homology $HH\_k(A;A)$?
Is there any software to compute Hochschild homology?
| https://mathoverflow.net/users/11317 | Hochschild homology of upper triangular matrix algebra? | Sasha's argument is pretty technological. You can really do this almost by hand, though.
Let $A$ be the algebra in question, let $r$ be its Jacobson radical (that is, the subspace of strictly upper triangular matrices), and let $E$ be the subalgebra of the diagonal matrices in $A$ (which is a complement to $r$) Notic... | 12 | https://mathoverflow.net/users/1409 | 159702 | 84,138 |
https://mathoverflow.net/questions/159604 | 1 | In the course of doing some calculations I have found myself wanting to answer the following question:
Let $D/\mathbb{Q}$ be a quaternion algebra ramified at a prime $p$ and at $\infty$ and let $\mathcal{O}$ be a maximal order.
Is it possible to find $\lambda,\mu\in\mathcal{O}$ such that:
$1. N(\lambda) = p-1$
... | https://mathoverflow.net/users/21698 | Integral elements of quaternion algebras with predescribed properties | First of all, it is not even true that every such order $\mathcal{O}$ contains an element $\mu \in \mathcal{O}$ with $\mathrm{nrd}(\mu)=p$. The first counterexample is $p=37$, and this corresponds to the fact that this is the first prime $p$ such that there is a supersingular elliptic curve with $j$-invariant $\in \mat... | 1 | https://mathoverflow.net/users/4433 | 159705 | 84,140 |
https://mathoverflow.net/questions/159529 | 17 | The category of representations $\text{Rep}(D(G))$ of the quantum double of a finite group is well-known to be a modular tensor category. Can these modular tensor categories also be obtained as representation categories of vertex operator algebras?
| https://mathoverflow.net/users/799 | $\text{Rep}(D(G))$ as representation category of a vertex operator algebra | This answer is related to my answer here: [Duality between orbifold and quasi-Hopf algebra (twisted quantum doubles)](https://mathoverflow.net/questions/153264/duality-between-orbifold-and-quasi-hopf-algebra-twisted-quantum-doubles/153270#153270) and the comment by Scott.
Every finite group $G$ can be embedded in som... | 7 | https://mathoverflow.net/users/10718 | 159714 | 84,144 |
https://mathoverflow.net/questions/159422 | 7 | Let $X$ be a smooth finite type variety over $\mathbb{C}$. Suppose that $\theta$ is a closed algebraic $1$-form whose cohomology class is weight $2$.
Can we always express $\theta$ as
$$\theta = \sum\_i c\_i \ d \log f\_i$$
where the $c\_i$ are constants and the $f\_i$ are nowhere vanishing algebraic functions?
As lo... | https://mathoverflow.net/users/297 | Log forms and Tate classes | What is true is that the exponential map gives surjection
$$H^0(\mathcal{O}\_X^\*)\to Hom\_{MHS}(\mathbb{Z},H^1(X,\mathbb{Z}(1))$$
Classes on the right can be interpreted as weight two classes represented by $1$-forms logarthmic at infinity with integral periods; the map can be identified with $d\log$. I'm not sure if ... | 5 | https://mathoverflow.net/users/4144 | 159740 | 84,149 |
https://mathoverflow.net/questions/159749 | 12 | Is there a proof of the consistency of Analysis (second order arithmetic), which is similar to Gentzen's proof of the consistency of arithmetic?
**Update:**
Which (different) methods can be used to prove the consistency of Analysis? and where can I find such proofs?
| https://mathoverflow.net/users/11115 | Consistency of Analysis (second order arithmetic) | I believe the answer is "no": certainly the proof-theoretic ordinal (the optimal object taking the role of "$\epsilon\_0$" in Gentzen's proof) is totally unknown, and my understanding is that there is no non-trivial upper bound on it, either. See also [Proof-Theoretic Ordinal of ZFC or Consistent ZFC Extensions?](https... | 14 | https://mathoverflow.net/users/8133 | 159752 | 84,153 |
https://mathoverflow.net/questions/159771 | 0 | If $X$ is a set of ordinals, define $f(X, \alpha)$ such that $f(X, 0) = X$, $f(X, \beta + 1) = f(X, \beta) \cap Lim(f(X, \beta))$, where $Lim(Y) = \{$limit points of $Y\}$, and $f(X, \gamma) = \bigcap \{f(X, \beta) : \beta < \gamma\}$ for limit $\gamma$.
Now let $\kappa$ be uncountable regular, and let $X \subset \ka... | https://mathoverflow.net/users/29231 | Is there such a sufficient condition for "$X$ is a stationary subset of uncountable regular $\kappa$" involving limit points? | The answer is no. Let $\langle \gamma\_\alpha \rangle\_{\alpha\lt\kappa}$ be a continuous sequence such that $\gamma\_{\alpha+1} \geq \gamma\_\alpha + \alpha$ for every $\alpha\lt\kappa$. Then $X = \kappa\setminus\{\gamma\_\alpha:\alpha\lt\kappa\}$ is nonstationary but $f(X,\alpha)$ is always unbounded in $\kappa$ sinc... | 1 | https://mathoverflow.net/users/2000 | 159772 | 84,162 |
https://mathoverflow.net/questions/159767 | 3 | Let $p\_1,...,p\_8\in\mathbb{P}^{3}$ be points in linear general position. Then there exists a unique elliptic curve $C$ of degree $4$ passing through $p\_1,...,p\_8$. I am interested in what happens for nodal rational curves of degree $4$.
Now, let us suppose we have seven points $p\_1,...,p\_7\in\mathbb{P}^{3}$ in ... | https://mathoverflow.net/users/14514 | Counting curves of degree 4 in $\mathbb{P}^{3}$ | Zero. Indeed, if the intersection $Q\_1 \cap Q\_2$ of two quadrics is singular at $p\_1$, then there is a quadric $Q$ in the pencil generated by $Q\_1$ and $Q\_2$ which is singular at $p\_1$. On the other hand, if a quadric cone with vertex at $p\_1$ passes through $p\_2,\dots,p\_7$, then the images of $p\_2,\dots,p\_7... | 10 | https://mathoverflow.net/users/4428 | 159773 | 84,163 |
https://mathoverflow.net/questions/159774 | 10 | Suppose $E$ is an elliptic curve defined over $\mathbb{Q}$ with good ordinary reduction at a prime $p$. Then one can define nonnegative integers $ \lambda\_{E}^{alg} $, $ \mu\_{E}^{alg} $, $ \lambda\_{E}^{an} $ and $ \mu\_{E}^{an} $.
The "algebraic" Iwasawa invariants $ \lambda\_{E}^{alg} $ and $ \mu\_{E}^{alg} $ ar... | https://mathoverflow.net/users/30999 | Main conjecture for elliptic curves | The main conjecture is a theorem if the image of the mod $p$ Galois representation of E is the whole of $GL\_2(\mathbf{F}\_p)$. The full statement of the conjecture, which implies what you wrote about lambda and mu invariants but is quite a bit stronger, is the claim that the p-adic L-function generates the characteris... | 10 | https://mathoverflow.net/users/2481 | 159791 | 84,170 |
https://mathoverflow.net/questions/159671 | 8 | Given a smooth hypersurface $H$ in $\mathbb{C}^n$, a theorem of Hironaka promises that one can find a strict normal crossings compactification $\bar{H}$ inside of a projective variety $X$. For me, this result is the definition of a "black box fact". What are some techniques for computing such a compactification in prac... | https://mathoverflow.net/users/36931 | Making Hironaka's theorem explicit for hypersurfaces | By now there are more tractable proofs of resolution of singularities than Hironaka's, so it no longer has to be a black box. A relatively elementary approach is described in Kollár's [Lectures on Resolution of Singularities](http://books.google.com/books/about/Lectures_on_Resolution_of_Singularities.html?id=Oygejj1QFh... | 5 | https://mathoverflow.net/users/10076 | 159793 | 84,172 |
https://mathoverflow.net/questions/159794 | 3 | If $M$ is a $2n\times 2n$ real symmetric matrix, I would like to ask what could be its canonical form under unitary congruence. We view a unitary $n\times n$ matrix $U$ as a real $2n\times 2n$ matrix, and try to see if we can ``diagonalize'' $M$ by
$$U^T M U.$$
In addition, what is the canonical form if we only requi... | https://mathoverflow.net/users/46433 | What is the canonical form of real symmetric 2n\times 2n matrix under unitary congruence? | The difficulty is that the space $S\_{2n}(\mathbb{R})$ of $2n$-by-$2n$ symmetric matrices with real entries is not irreducible under the action of $\mathrm{U}(n)$. In fact, there is a $\mathrm{U}(n)$-irreducible decomposition
$$
S\_{2n}(\mathbb{R}) = \mathbb{R}\cdot I\_{2n}\oplus A\_n\oplus B\_n
$$
where $A\_n$ is the ... | 10 | https://mathoverflow.net/users/13972 | 159801 | 84,176 |
https://mathoverflow.net/questions/159762 | 10 | Let me for simplicity start with affine case. If $X=\operatorname{Spec}(A)$ is an affine variety $Z \subset X$ is a closed affine subvariety $Z=\operatorname{Spec}(A/I)$. What conditions are sufficient to say that completion of the (total space of) normal bundle $N\_{Z/X}$ along zero section is isomorphic to the comple... | https://mathoverflow.net/users/21029 | Formal completion of the normal bundle | The formal completion of $X$ along $Z$ is always isomorphic to that of the normal bundle $N\_{Z \subset X}$ along its zero section if $X= \mathrm{Spec}(A)$ and $Z = \mathrm{Spec}(A/I)$ are smooth and affine over some base $k$. If the affineness or smoothness assumption is dropped, it is easy to give counterexamples.
... | 6 | https://mathoverflow.net/users/45704 | 159806 | 84,179 |
https://mathoverflow.net/questions/159803 | 16 | Motivated by [Consistency of Analysis (second order arithmetic)](https://mathoverflow.net/questions/159749/consistency-of-analysis-second-order-arithmetic) and [Proof-Theoretic Ordinal of ZFC or Consistent ZFC Extensions?](https://mathoverflow.net/questions/144041/proof-theoretic-ordinal-of-zfc-or-consistent-zfc-extens... | https://mathoverflow.net/users/8133 | Nontrivial upper bounds on proof-theoretic ordinals of strong theories: do we have any? | I think the short answer is no, and I'm not sure there really could be: I don't think we "know" any ordinals above the ordinal of $\Pi^1\_2-CA$ but below $\omega\_1^{CK}$. Theoretically someone could write down a large notation for, say, $\Pi^1\_3-CA$ and then discover it was too big, but in practice the hard part seem... | 14 | https://mathoverflow.net/users/8991 | 159810 | 84,181 |
https://mathoverflow.net/questions/159703 | 3 | There are a number of concepts in model theory - often situated around Hrushovski's amalgamation method (see for instance <http://math.univ-lyon1.fr/~wagner/nijmegen.pdf>) - which are colorfully named:
* Green fields (e.g. <http://arxiv.org/pdf/1003.5149.pdf>)
* Red fields, or red additive groups of fields (e.g., <ht... | https://mathoverflow.net/users/8133 | Colorful model theory | These colors were supplied by Poizat as an abbreviation. If there was any rationale for the choice of colors it escaped me.
| 4 | https://mathoverflow.net/users/31339 | 159815 | 84,183 |
https://mathoverflow.net/questions/159816 | 4 | Let $S$ be a finite set, and let $W$ be a nonempty set of subsets of $S$; we will identify every subset of $S$ with its characteristic function, a 0-1 vector in $\mathbb R^S$. The *combinatorial* dual $\hat W$ of $W$ is the set of all subsets of $S$ that have non-empty intersection with every subset of $W$. The *positi... | https://mathoverflow.net/users/8252 | combinatorial and linear duality | No, let $S=\{1,2,3\}$, let $W$ consist of the "large" subsets of $S$, i.e., the ones of cardinality 2 or 3. Then $\hat W=W$. Let $v=(1/3,2/3,2/3)$. Then $v\in W^\bot$.
But any vector in the convex production of $\hat W$ must lie above the half plane $x+y+z\ge 2$, which $v$ does not.
| 3 | https://mathoverflow.net/users/4600 | 159830 | 84,188 |
https://mathoverflow.net/questions/159820 | 3 | Let $F=\{z\in H: |z|\geq1, |Re(z)|\leq 1/2\}$. It is a fundamental domain of the modular group $\Gamma$ acting on the upper half plane $H$. (Strictly speaking, one should take part of the boundary from $F$ to make it a fundamental domain. For example, one can use $F\_1=F-\{z\in F:Re(z)=1/2\}-\{z\in F:Re(z)>0 \text{ and... | https://mathoverflow.net/users/45392 | the Action of $SL_2(\mathbb{R})$ on a fundamental domain of $\Gamma$ | As Misha says, this is false as stated. To see this: Apply the dilation $A \colon z \mapsto z/2$ to $F$. Note $i$ lies in the interior of $A(F)$. Since $i$ is fixed by a torsion element of $SL(2,\mathbb{Z})$, deduce that $A(F)$ is not a fundamental domain.
There are several different things you might be trying to as... | 4 | https://mathoverflow.net/users/1650 | 159838 | 84,193 |
https://mathoverflow.net/questions/159731 | 17 | Every year I teach an introductory class on the differential geometry of surfaces, including numerical aspects (e.g., how to solve PDEs on surfaces). Historically this class has included an introduction to exterior calculus, describing (for example) differential operators on curved surfaces in terms of the exterior der... | https://mathoverflow.net/users/1557 | How useful/pervasive are differential forms in surface theory? | Well, there's really not a whole lot more to say beyond what Deane already wrote. He certainly hit the main points, but maybe I can expand a bit on what he wrote and comment on my own experience over the years both learning and teaching differential geometry.
Maybe I should say a little bit about my own education: Or... | 18 | https://mathoverflow.net/users/13972 | 159848 | 84,197 |
https://mathoverflow.net/questions/159847 | 1 | Let $M$ be a Riemannian manifold and $\gamma$ be a non closed geodesic.
Is there an isometric embedding of $M$ into some $\mathbb{R}^{n}$ which send $\gamma$ into an straight line?
The second question: Assume $M$ is foliated with a family of non closed geodesics. Is there an isometric embedding of $M$ into $\mathbb{R}^... | https://mathoverflow.net/users/36688 | An special isometric embedding | Concerning the first question, what about an irrational geodesic in a torus? Since it has accumulation points, it cannot be mapped isometrically to a straight line.
| 4 | https://mathoverflow.net/users/44953 | 159849 | 84,198 |
https://mathoverflow.net/questions/159614 | 4 | Let $D$ be a $(v,k,\lambda)$-design. By the **domination number of $D$** I mean the domination number $\gamma(L(D))$ of the bipartite incidence graph of $D$.
>
> Is $\gamma(L(D))$ determined only by $v,k$, and $\lambda$,
> irrespective of the actual structure of $D$?
>
>
>
I can prove this for finite projecti... | https://mathoverflow.net/users/22051 | Is the domination number of a combinatorial design determined by the design parameters? | Well, interestingly enough, all the 80 Steiner triple systems on 15 points have minimum dominating sets of size 10 - there is more tradeoff between the points/blocks than I first recognised.
But if we go a bit bigger then we can find some variation. Here is a $2$-$(25,4,1)$ design with 50 blocks.
```
0 1 2 3; 0 4 ... | 7 | https://mathoverflow.net/users/1492 | 159851 | 84,199 |
https://mathoverflow.net/questions/159853 | 14 | Every version of Riesz's representation theorem (the one expressing linear functionals as integrals) that I have found so far assumes that the underlying topological space is locally-compact. (For instance, Rudin in "Real and Complex Analysis" uses this assumption in the proof of Urysohn's lemma, upon which he bases th... | https://mathoverflow.net/users/54780 | Riesz's representation theorem for non-locally compact spaces | The basic idea is that of the strict topology on $C^b(X)$. This locally convex topology was introduced in the case of a locally compact space by R.C. Buck in the fifties using weighted seminorms and generalised to the completely regular case by many authors in the sixties and seventies. It can be succinctly described a... | 10 | https://mathoverflow.net/users/47969 | 159857 | 84,201 |
https://mathoverflow.net/questions/159866 | 1 | Let $G$ be any algebraic subgroup of $\mathrm{GL}\_n$ over an algebraically closed field of any characteristic.
If $s$ is a semisimple element of $G$, can the $G$-conjugacy class of $s$ fail to be closed in $G$?
| https://mathoverflow.net/users/48010 | Is a semisimple conjugacy class closed? | No---it's always closed. See 18.2 in Humphreys, *Linear Algebraic Groups*.
| 4 | https://mathoverflow.net/users/430 | 159869 | 84,205 |
https://mathoverflow.net/questions/159761 | 1 | Let $M$ be a symplectic manifold, Then a subbundle $P\subset TM^{\mathbf{C}}$ of the complexified tangent bundle is called a complex polarization if
1. $P$ is Lagrangian
2. P involutive
3. dim$P\cap\bar P \cap TM$ is constant.
Let $P\_1, P\_2$ be two complex polarization, then when $\bar P\_1\cap P\_2$ is involuti... | https://mathoverflow.net/users/nan | A question about complex polarization | I may be missing something, but isn't it completely obvious that $\bar P\_1\cap P\_2$ is always involutive?
Indeed, if $\xi$, $\eta$ are two complex vector fields taking values in $\bar P\_1\cap P\_2$, then $[\xi,\eta]$ takes its values in $\bar P\_1$ because $\bar P\_1$ is involutive, and in $P\_2$ because $P\_2$ is... | 2 | https://mathoverflow.net/users/19276 | 159876 | 84,208 |
https://mathoverflow.net/questions/159873 | 2 | Denote by $H(X)$ the group of homeomorphisms of a topological space $X$. Assume further that $X$ is either compact or locally compact and locally connected. In both cases $H(X)$ becomes a topological group with respect to the compact open topology $\tau\_C$.
What can be said about the connectedness of $(H(X),\tau\_C)... | https://mathoverflow.net/users/31446 | Connectedness properties of groups of homeomorphisms | Regarding the last part of your question:
Let $f\_{0}$ and $f\_{1}$ be two orientation preserving homeomorphism in $H\_{+}(S^{1})$. Lift $f\_{0}$ and $f\_{1}$ to homeomorphisms $F\_{0}$ and $F\_{1}$ on $\mathbb{R}$, as in <http://en.wikipedia.org/wiki/Rotation_number>.
Then consider the quotient of the homotopy $F\... | 1 | https://mathoverflow.net/users/36688 | 159877 | 84,209 |
https://mathoverflow.net/questions/159874 | 5 | Let $G$ be a group (possibly infinite) and $k$ be a field. A module $M$ over $k[G]$ is said to be of type $\text{FP}\_{\infty}(k)$ if it has a projective resolution each of whose terms is finitely generated. We say that $G$ itself is of type $\text{FP}\_{\infty}(k)$ if the trivial $k[G]$-module $k$ is of type $\text{FP... | https://mathoverflow.net/users/48013 | Are finite-dimensional representations of groups of type $\text{FP}_{\infty}$? | **Cleaned up answer (6/11/18) after comments of Andy Putman.**
The answer is yes.
Thm 2 of <https://www.tandfonline.com/doi/abs/10.1080/00927870600796110> shows that if G is $FP\_\infty$ over $k$, then $kG$ has a free resolution as a bimodule by finitely generated free bimodules in each dimension.
If you tensor ... | 7 | https://mathoverflow.net/users/15934 | 159886 | 84,211 |
https://mathoverflow.net/questions/159094 | 10 | According to Lusin's theorem (and the slightly *weaker* converse of that result), measurable functions on locally compact topological spaces that are equipped with a regular measure may be characterized as follows:
**Definition.** Let $X$ be a locally compact space and $\mu$ be a regular measure on $X$. We call $f:X\... | https://mathoverflow.net/users/nan | Is the "continuous on compact subsets" characterization of measurable functions actually useful? | Tautological answer: the cases where you know the measure of compact sets (as in section 3 of chapter IX of Bourbaki), but not the class of all measurable sets (which you then can define using the above definition applied to the indicator function of the set). So the next question is: should one take the topology (or e... | 4 | https://mathoverflow.net/users/46855 | 159889 | 84,213 |
https://mathoverflow.net/questions/159891 | -4 | We know that if $ p$ is a prime number then $ O^p (G) $ is the smallest normal subgroup of $ G $ such that $ G/O^p (G) $ is a $ p $-group.
Now let $ G $ be a finite group of order $ p^aq^b $ where $ p $ and $ q $ are prime numbers. Is this true that $ O^p (G) \ne G$ and also $ O^q (G)\ne G $? We note that by the solv... | https://mathoverflow.net/users/31045 | on the solvable groups of order $p^aq^b$ | The alternating group $A\_4$ is a counterexample: It has order $2^2\cdot 3$, so $O^2(A\_4)$ will contain an order $3$ element. But any order $3$ element of $A\_4$ generates the whole group as a normal divisor, as is seen by playing around with permutations. So $O^2(A\_4) = A\_4$.
If you put some additional assumption... | 1 | https://mathoverflow.net/users/39747 | 159893 | 84,215 |
https://mathoverflow.net/questions/159895 | 9 | Let
$$
\begin{array}{rccccl}
A\_0&\to& B\_0&\to& C\_0&\to\\
\downarrow & &\downarrow&&\downarrow\\
A\_1&\to& B\_1&\to& C\_1&\to\\
\downarrow & &\downarrow&&\downarrow\\
\vdots & &\vdots&&\vdots\\
\end{array}
$$
be a commutative diagram in a triangulated category such that all the rows are exact triangles. One defines t... | https://mathoverflow.net/users/46913 | Is 'the' homotopy colimit of a sequence of exact triangles an exact triangle? | I'll assume that you at least want your triangulated category to have the property that a (countable) coproduct of exact triangles is an exact triangle?
Even then, I think this is probably not true in a general triangulated category, but it is in many (most?) familiar examples. For example, in the derived category (o... | 5 | https://mathoverflow.net/users/22989 | 159896 | 84,216 |
https://mathoverflow.net/questions/159899 | 9 | (Edited in accordance with suggestions in comments.)
I remember once I read a quote that sounded like "graph theory is the scum of topology" (please approximate). I can not find it on the web, and I can not remember who said this. Or did I dream it after a heavy meal?
| https://mathoverflow.net/users/nan | Source for derogatory quote about graph theory | *Graph theory is the slum of topology...*
You may have read it in the first paragraph of the very first **opinion** of Prof. [Zeilberger](https://mathoverflow.net/users/5822/doron-zeilberger):
[Topology: The slum of combinatorics OR "Don't show off too much, your specialty will soon be trivialized"](http://www.math... | 12 | https://mathoverflow.net/users/1593 | 159900 | 84,218 |
https://mathoverflow.net/questions/134430 | 20 | For $\epsilon > 0$ sufficiently small, can a regular hexagon with sides of length $1 + \epsilon$ be covered by seven equilateral triangles with sides of length $1$?
Motivation: Conway and Soifer [showed that](http://link.springer.com/chapter/10.1007%252F978-0-387-74652-4_15) an equilateral triangle with sides $n + \e... | https://mathoverflow.net/users/4558 | Covering a hexagon | The case of hexagon is exactly the problem that is Alexander and I posed in:
Karabash, D., and Soifer, A., On Covering of Trigons, Geombinatorics XV(1) (2005), 13–17.
Hexagon is type of 6-trigon (n-trigon is n connected triangles from triangulation); n-trigons for n<6 are trivial counting of vertecies and hence 6-tri... | 11 | https://mathoverflow.net/users/48020 | 159902 | 84,219 |
https://mathoverflow.net/questions/159856 | 1 | Recently I read about the dee dee bar lemma ($\partial\bar \partial$-lemma) in Gang Tian's *Canonical metrics in Kähler Geometry*. In the middle of Page 16, the author writes that: "The following equation can be solved for $u$,
$$\bar\partial^\*\theta=\bar\partial^\*\bar\partial u=-g^{i\bar j}u\_{\bar j i}$$
because $\... | https://mathoverflow.net/users/48006 | Solvable question of dee dee bar lemma | Firstly, I think you meant to write $$\bar\partial^\* \theta = \bar\partial^\*\partial u = -g^{i\bar j}u\_{i\bar j}$$
On Kahler manifolds the R.H.S is just the laplacian of $u$. So basically you want to solve $$-\Delta u = f$$ whenever $\int\limits\_{M}f\omega^n = 0$. Here $f = \bar\partial^\*\theta$ in your question... | 1 | https://mathoverflow.net/users/11945 | 159906 | 84,221 |
https://mathoverflow.net/questions/154953 | 5 | I've seen several citations of a letter from Weierstrass, talking about his dispute with Kronecker, in which he refers to Christoffel as a 'whimsical eccentric' (presumably the German original is even more flavorful). In what respect was he whimsical or eccentric?
One reference is Barrow, 'Pi in the Sky,' page 200, q... | https://mathoverflow.net/users/2937 | How was Christoffel a 'whimsical eccentric'? | (a) "Kauz" is literally a "brown (or tawny) owl," colloquially a "queer (or strange, familiarly a 'rum') fellow" (Muret-Sanders German-English Dictionary); certainly a more pungent expression than "whimsical eccentric," but not as dismissive as "crank" or "crackpot"; a rather precise English colloquial equivalent is "a... | 13 | https://mathoverflow.net/users/48029 | 159914 | 84,224 |
https://mathoverflow.net/questions/159945 | 12 | The Ground Axiom ($GA$) is the assertion that the universe of
sets ($V$) is not a forcing extension of any inner model $W$ by nontrivial forcing
$P\in W$.
Is $GA$ consistent with any possible behavior of continuum function $\kappa\mapsto 2^{\kappa}$?
It seems in models of $GA$ like $L$ and some other canonical mode... | https://mathoverflow.net/users/nan | Ground Axiom and behaviors of continuum function | In the paper [The ground axiom is consistent with $V\ne{\rm HOD}$](http://dx.doi.org/10.1090/S0002-9939-08-09285-X) (J. D. Hamkins, J. Reitz, W.H. Woodin, PAMS 136(8):2008), we prove that the ground axiom is consistent with $V\neq\text{HOD}$, and remark there that:
>
>
> >
> > The proof of Theorem 1 is
> > flexi... | 12 | https://mathoverflow.net/users/1946 | 159952 | 84,240 |
https://mathoverflow.net/questions/159957 | 1 | Suppose that $I\subseteq\mathbb{C}[x\_0,\ldots,x\_n]$ is a saturated homogeneous ideal. Let $\mathcal{I}\subseteq\mathcal{O}\_{\mathbb{P}^n}$ denote the corresponding coherent ideal sheaf, and then let $$\pi:Y=\mathrm{Bl}\_{\mathcal{I}}(\mathbb{P}^n)\longrightarrow\mathbb{P}^n$$ be the blow-up of $\mathbb{P}^n$ along $... | https://mathoverflow.net/users/nan | Direct image of an ideal sheaf along a blow-up | It sounds like you want to compute the pushforward of some sheaf on $Y$.
Macaulay2 can do this using the **BGG** package [HERE](http://www.math.uiuc.edu/Macaulay2/doc/Macaulay2-1.6.0.1-20131031-2/share/doc/Macaulay2/BGG/html/). In particular, use the **directImageComplex** function. I would apply the function **reesA... | 1 | https://mathoverflow.net/users/3521 | 159958 | 84,242 |
https://mathoverflow.net/questions/159907 | 9 | Let us assume that there are Turing machines of arbitrary cardinality, by that I mean they can have input tapes of any arbitrarily high cardinality and compute for a number of steps also of arbitrarily high cardinality. Those machines are, in principle, much more powerful than Hamkins infinite time Turing machines. Bas... | https://mathoverflow.net/users/27974 | What set theoretical questions could never be answered by Turing machines of arbitrary cardinality? | Peter Koepke and his numerous collaborators have studied the ordinal-length tape version of infinite time Turing machines, where one has a tape stretching the length of the ordinals, and one imagines that the device follows deterministic local rules for working on the tape, reaching the higher levels of the tape in tra... | 17 | https://mathoverflow.net/users/1946 | 159964 | 84,246 |
https://mathoverflow.net/questions/159030 | 5 | Any Lebesgue probability space is mod. 0 isomorphic to some Polish probability space (with $\sigma$-algebra the completion of Borel algebra, and some Borel probability). I would like to see an example of a Lebesgue probability space which is not isomorphic to such a Polish space.
In other word, I want a Lebesgue pro... | https://mathoverflow.net/users/46931 | Non-Polish Lebesgue probability space? | When requesting inner regularity, you hit the ZFC - undecidable Cantor's continuum problem.
First note that a Lebesgue space has cardinality at most continuum since a countable collection of subsets separates points. A uncountable polish space has exactly cardinality of the continuum since it has a perfect subset (Ca... | 4 | https://mathoverflow.net/users/46855 | 159971 | 84,248 |
https://mathoverflow.net/questions/159917 | 1 | It would be great if someone can help me do these integrals - using numerical integration on Mathematica it seems that these converge - in what follows $a \in \mathbb{R}$ and $q \in \mathbb{N}$ and $n \in \mathbb{Z}$
* $\int \_0 ^\infty dx\text{ } tanh (\pi \sqrt{x} )[ \frac{1}{x + a^2 + (\frac{n}{q})^2 } - \frac{1}{... | https://mathoverflow.net/users/36554 | Request for help with two integrals | Here is my attempt at the first one:
\begin{multline\*}
\int\limits\_0^\infty\tanh(\pi\sqrt x)\left(\frac1{x+c^2}-\frac1{x+d^2}\right)dx\\=2\int\limits\_0^\infty y\tanh(\pi y)\left(\frac1{y^2+c^2}-\frac1{y^2+d^2}\right)dy\\=\frac2\pi\int\limits\_0^\infty\sum\_{k\in\mathbb Z}\frac{y^2}{y^2+(k+\frac12)^2}\left(\frac1{y^2... | 3 | https://mathoverflow.net/users/41291 | 159972 | 84,249 |
https://mathoverflow.net/questions/159920 | 6 | An integral polytope $P$ is normal if all lattice points inside the integer dilation $kP$ can be expressed as $p\_1+p\_2+\dots+p\_k$, where $p\_i \in P$ are lattice points.
I am looking for an example $P$ for which the above is true for $k=2$,
but fails for higher $k$.
A related question: is there a number $M$, th... | https://mathoverflow.net/users/1056 | Normal polytopes - counterexample? | Examples of this sort (and worse) are given, for dimension 5 and up, in <http://scholar.google.com/scholar?cluster=14055290405510744870&hl=en&oi=scholarr>
David Handelman [me], *Effectiveness of an affine invariant for indecomposable integral polytopes*, J Pure and Applied Algebra 66 (1990) 165–184, section 3, pp 16ff.... | 12 | https://mathoverflow.net/users/42278 | 159974 | 84,250 |
https://mathoverflow.net/questions/159926 | 0 | I have been unable to find a good reference for a book that study in details ergodic theory on sub shifts of finite type in dimension $D>1.$ The only reference that I got was actually a book by Gerhard Keller, Ergodic States in Ergodic Theory, London mathematical society, Student Text 42. However, it is a bit difficult... | https://mathoverflow.net/users/39115 | The Book for ergodic theory on SFT in dimension $D>1.$ | I don't think there are many books that deal with this. Is there a particular topic or direction you want to go in? Some areas of papers:
Algebraic SFTs have been treated by Kitchens, Schmidt and others;
Decidability and aperiodicity issues have been treated by Berger, Robinson, Kari, Culik and others;
Entropy ... | 3 | https://mathoverflow.net/users/11054 | 159982 | 84,253 |
https://mathoverflow.net/questions/159979 | 1 | Let $G$ be a connected reductive group over a non archimedean local field $k$.
Let $P \subset G$ be a parabolic subgroup with Levi decomposition $P=MN$, $Z\_M \subset L$ be the center of $M$ and $S\_M \subset Z\_M$ be the maximal split torus.
It is commonly used that there exist element $s \in S\_M(k)$ whose action b... | https://mathoverflow.net/users/24114 | Strictly contracting elements in the center of a Levi subgroup | Let $G$ be a connected reductive group of an arbitrary field $k$, $P$ a parabolic $k$-subgroup, and $U = \mathscr{R}\_u(P)$ the unipotent radical of $P$ (so $U$ is what is unfortunately traditionally denoted as $N$).
For any Levi $k$-subgroup $M$ of $P$ there is a 1-parameter $k$-subgroup $\lambda:{\rm{GL}}\_1 \righ... | 1 | https://mathoverflow.net/users/43107 | 159993 | 84,257 |
https://mathoverflow.net/questions/159970 | 10 | It is known that the isomorphism class of a quaternion algebra $A=\binom{a,b}{K}$ over a number field $K$ is determined by the finite set of places $v$ of $K$ where $A\otimes\_K K\_v$ is a division algebra, equivalently at which the projective curve $ax^2+by^2-z^2$ fails to have a $K\_v$-rational point.
Given two qua... | https://mathoverflow.net/users/12419 | Explicit isomorphism for quaternion algebras over $\mathbb{Q}$? | Timo Hanke has written on this problem in the generality of cyclic algebras (<http://arxiv.org/abs/math/0702681>). He shows that it is equivalent to the solution of a norm equation, which has an algorithmic solution over global fields. In your case, this would in general involve the solution of a norm equation over a *... | 9 | https://mathoverflow.net/users/4433 | 159998 | 84,259 |
https://mathoverflow.net/questions/160007 | 7 | I have what should be a very simple questions for Brownian motion experts...
Let $[a,b]$ be a given time interval. Let $f(x)$ be the probability that a linear Brownian motion with initial value $x$ at time $t=0$ has a zero in the interval $[a,b]$. I want to argue that $f(x)$ is maximal for $x=0$. This seems intuitively... | https://mathoverflow.net/users/12126 | Probability of Brownian motion to have a zero in an interval | For any other initial x, construct a coupling between BMs started at x and 0, where the processes move in opposite directions until they meet (if they do), then they stick together afterwards. Then the answer is apparent.
| 9 | https://mathoverflow.net/users/46191 | 160020 | 84,266 |
https://mathoverflow.net/questions/160031 | 17 | The following questions came to my mind while preparing the notes for the first class of (my first) course on algebraic geometry.
Question 1: Is there any motivation for choosing the term "variety" for zeroes of polynomials? For example, I can sort of guess/understand the logic behind the term "manifold": 2-fold, 3-... | https://mathoverflow.net/users/1508 | Why the name "variety" and the notation "V" for zeroes of polynomials? | Also in Italian "varietà" is the term for both.
Starting from this, I looked at the [Italian Wikipedia webpage for varieties](http://it.wikipedia.org/wiki/Variet%C3%A0_%28geometria%29) which, at the end, has [this remark](http://it.wikipedia.org/wiki/Variet%C3%A0_%28geometria%29#Origine_del_termine) about the origin ... | 21 | https://mathoverflow.net/users/42995 | 160034 | 84,271 |
https://mathoverflow.net/questions/159991 | 14 | Sequence [A003238 of the OEIS](http://oeis.org/A003238) counts ``rooted trees with $n$ vertices in which vertices at the same level have the same degree.'' The sequence, $a$, begins
1, 1, 2, 3, 5, 6, 10, 11, 16, ...
and it is of course easy to generate as many terms as you like via recurrences (basically $a(n)$ is ... | https://mathoverflow.net/users/1907 | Are the asymptotics of A003238 known? | I'll show that
$$
\log a(n) \sim \frac{(\log n)^2}{\log 4} \approx 0.7213\ldots (\log n)^2.
$$
So the range for the constant given in the conjecture is false, but an asymptotic of that general shape
holds. One can obtain more precise asymptotics by working harder with the argument below.
Roughly speaking what t... | 22 | https://mathoverflow.net/users/38624 | 160037 | 84,272 |
https://mathoverflow.net/questions/160041 | 6 | Let $Q\cong\mathbb{P^{1}\_{1}}\times\mathbb{P^{1}\_{2}}\subset\mathbb{P}^{3}$ be a smooth quadric surface. We have the following two actions on $Q$:
$$S\_2\times Q\rightarrow Q,\; (\sigma,(x,y))\mapsto\sigma(x,y)$$
where $S\_2$ is the symmetric group of order $2$, and
$$Aut(\mathbb{P^{1}\_{1}})\times Aut(\mathbb{P^{1}\... | https://mathoverflow.net/users/nan | Automorphisms of a smooth quadric surface $Q\subset\mathbb{P}^{3}$ | Yes, it is true. This is easily seen by looking for example at the Picard group, generated by $C$ and $D$, the two fibres of the projections. Since $C^2=D^2=0$ and$C\cdot D=1$, the only curves of self intersection $0$ are multiple of $C$ or $D$, and the irreducible ones are equivalent to $C$ or $D$. Composing an automo... | 9 | https://mathoverflow.net/users/23758 | 160042 | 84,273 |
https://mathoverflow.net/questions/160036 | 5 | By a *binary tree*, I mean in this question a full rooted binary tree in which left and right child are labeled. A leaf of such a tree is a vertex of degree at most 1 (most references would probably consider that a leaf is a vertex of degree exactly 1) and an internal vertex is a vertex of degree at least 2. With these... | https://mathoverflow.net/users/2284 | Average number of distinguished leaves in a binary tree | A generating function approach gives an exact formula (extracting asymptotics is slightly more tedious): Let $d\_{n,k}$ denote the number of
binary trees with $n$ interior leaves and $k$ distinguished vertices. We consider the polynomials $D\_n=\sum\_{k=0}^{n+1}d\_{n,k}t^k\in\mathbb N[t]$. We have $D\_0=t$ and the rec... | 4 | https://mathoverflow.net/users/4556 | 160049 | 84,276 |
https://mathoverflow.net/questions/160033 | 1 | Does anybody know the answer to this or a good way to go about working this out? I have a list for $GL\_2(Z/pZ)$ and I am trying to lift it to this; I have mostly been using fairly elementary algebraic methods. Thank you.
| https://mathoverflow.net/users/48096 | List of Conjugacy Classes of the General Linear group over $\mathbb{Z}/p^2\mathbb{Z}$ | I don't have time to give a full answer now, but here's how I would try and answer this question:
You first need to establish what the normal subgroups of $GL\_2(\mathbb{Z}/p^2\mathbb{Z})$ are. This is done in a fair amount of detail at [this MO question](https://mathoverflow.net/questions/104262/normal-subgroup-of-c... | 2 | https://mathoverflow.net/users/801 | 160052 | 84,277 |
https://mathoverflow.net/questions/159989 | 35 | I am looking for a closed statement (i.e. not depending on any parameter objects) which is true in the internal logic of the topos of simplicial sets, but is not an intuitionistic tautology. Ideally, I would like it to be a simple universal statement of propositional logic (e.g. "for all propositions P, Q, and R, blah ... | https://mathoverflow.net/users/49 | Internal logic of the topos of simplicial sets | Short answer: the Kreisel-Putnam axiom $(\lnot p \to (q \lor r)) \to ((\lnot p \to q) \lor (\lnot p \to r))$ is not an intuitionistic tautology but it is valid for any subobjects of an object in the topos of simplicial sets.
The longer answer relies on an interesting characterization of the subobject classifier $\Ome... | 35 | https://mathoverflow.net/users/2000 | 160067 | 84,279 |
https://mathoverflow.net/questions/160029 | 3 | Consider the Laplace equation in a ball $B(r) \subset \mathbb{R}^n$ of radius $r$:
$$
\begin{cases}
-\Delta u &= 0, \quad \text {in} \quad B(r), \\
\ \ \ \ \ \, u&= g, \quad \text {in}\quad \partial B(r).
\end{cases}
$$
Suppose $g$ can be extended to a function $\overline g$ defined in $B((1+\sigma)r)$ such that $\over... | https://mathoverflow.net/users/46298 | Extending a harmonic function in a ball to subharmonic in a larger ball | One can build a distributional subsolution which is a Lipschitz extension of $u$ by making an extension with a positive jump in radial derivative across the boundary. Say $u$ is harmonic on $B\_1$ and we want to subharmonically extend to $B\_2$.
Let $v$ be a harmonic function on $B\_2-B\_1$ with boundary data $g$ on ... | 3 | https://mathoverflow.net/users/16659 | 160070 | 84,280 |
https://mathoverflow.net/questions/160051 | 2 | Let $(S,+)$ be a commutative semigroup. For $a,b\in S$ consider the equation $a=a+b$. Does such a relation between the given $a$ and $b$ have a name? I am currently using such equations quite often and I would like to find the appropriate references and basic facts. I have an impression seeing a similar subject in Math... | https://mathoverflow.net/users/18376 | Terminology for the equation $a=a+b$ in commutative semigroups | Tarski, cardinal algebras, pag. 10 writes:
Formulas of the type "$a+b=b$" [...] can be read "$a$ is absorbed by $b$" or "$b$ absorbs $a$". The relation of absorption plays an important role in the arithmetic of C.[ardinal]A.[lgebra]'s.
Wehrung, Injective positively ordered monoids I [ <http://www.sciencedirect.com/... | 4 | https://mathoverflow.net/users/46855 | 160074 | 84,282 |
https://mathoverflow.net/questions/160079 | 5 | Let $d(n)$ be the number of divisors of an integer $n$.
Does there exists a bound for $\sum\_{k\leq n}d^2(k)$?
I saw in a paper of Barry and Louboutin that the asymptotics is $\frac1{\pi^2}nlog^3n$ but there was no proof nor a reference or any indication that this is the best possible bound.
Any idea where can... | https://mathoverflow.net/users/16040 | Second moment for the number of divisors function | This can be proved by [Perron's formula](http://en.wikipedia.org/wiki/Perron%27s_formula), making use of the properties of the associated Dirichlet series
$$ g(s):=\sum\_{k=1}^\infty\frac{d^2(k)}{k^s},\quad\Re s>1. $$
We have an Euler product decomposition over the primes $p$,
$$ g(s)=\prod\_p\left(\sum\_{m=0}^\infty\f... | 14 | https://mathoverflow.net/users/11919 | 160084 | 84,284 |
https://mathoverflow.net/questions/160059 | 4 | I came across a poblem where they ask you to find an estimation of $\sum\_{n \leq x} \frac{k(n)}{n}$, with $k(n) = \prod\_{p \mid n} p$ the squarefree kernel of $n$, with an error term of $O(\sqrt{x})$.
I've worked on it for a while now, but I'm not able to find a solution. Does somebody know a solution?
| https://mathoverflow.net/users/48109 | Estimation of $\sum_{n \leq x} \frac{k(n)}{n}$ , with $k(n)$ the squarefree kernel | This is an exercise from Tenenbaum's 'Introduction to Analytic and Probablistic Number Theory' It is #11 from page 54. It has a hint which is a preceding exercise.
First consider a decomposition of power-full number($p|n\Rightarrow p^2|n$).
If $n$ is power-full, then $n$ can be written uniquely as $n=m^3d^2$ where ... | 10 | https://mathoverflow.net/users/21090 | 160085 | 84,285 |
https://mathoverflow.net/questions/160077 | 4 | Let $F: \mathbb{R} \to [0,1]$ be cumulative distribution function (cdf). The standard way to build a random variable $\tau$ on $([0,1],\mathcal{B},\text{Leb})$ with $F$ as its cdf is using the Skorokhod representation: For $x \in [0,1]$, let
$$
\tau(x) := \inf \{ z : F(z) > x \}
$$
be the generalized inverse of $F$.
... | https://mathoverflow.net/users/47510 | General version of Skorokhod representation of random variables | The statement is true for probability measures on a complete separable metric
space. I made a slight variation of the Skorokhod representation theorem
in my PhD thesis (way long time ago!) so I am more or less cutting and pasting,
sorry for the errors.
I got the original proof from Ikeda, Watanabe, *Stochastic differen... | 5 | https://mathoverflow.net/users/8966 | 160087 | 84,286 |
https://mathoverflow.net/questions/159406 | 6 | Let $\mathcal C$ be a strict, locally small 2-category.
Consider a subcategory $\mathcal L$ of $\mathcal C$ such that $\mathcal L$ has the same objects as $\mathcal C$, and the arrows of $\mathcal L$ are the left adjoints in the 2-category $\mathcal C$. Let $U:\mathcal L\to\mathcal C$ be the inclusion functor.
When ... | https://mathoverflow.net/users/4814 | Are left adjoints a left adjoint? | Your question is impossible to answer in full generality. There are, however, some interesting special cases. I shall focus on your motivating example (I think, I have written about this on many occasions, but cannot recall the exact sources at the moment).
If $\mathbb{C}$ is a (finitely complete, locally small) regu... | 3 | https://mathoverflow.net/users/13480 | 160093 | 84,287 |
https://mathoverflow.net/questions/160098 | 6 | Let $K$ be a field and $\Gamma$ a quiver (=multidigraph) and $K[\Gamma]$ its quiver algebra (free $K$-module on the set of all paths of length $\geq0$ where multiplication is concatenation if endpoints match and $0$ otherwise).
I'm looking for families of $\Gamma$ for which Hochschild homology $HH\_\ast(K[\Gamma])$ ... | https://mathoverflow.net/users/11317 | Hochschild homology of quiver algebras | The Hochschild homology of *all* path algebras $A=k\Gamma$ is known. I will assume the quiver is finite.
They are hereditary algebras, so $HH\_p(A)$ is zero as soon as $p>0$ *provided* $A$ is finite dimensional (which in this case is the same thing as $\Gamma$ being acyclic): this follows from the theorem of Keller t... | 8 | https://mathoverflow.net/users/1409 | 160109 | 84,295 |
https://mathoverflow.net/questions/160103 | 3 | Let $G$ be a finite group and $N\le G$ be a subgroup. Consider the group algebra $kN$ as a subalgebra of $kG$ over an algebraically closed field $k$ of positive characteristic.
What can we deduce from the simple relation $J(kN)=J(kG)$ (where $J()$ is the Jacobson radical)? Are there any conclusions concerning $G,N$? E... | https://mathoverflow.net/users/23104 | Jacobson radical and group rings/subalgebras | In the case that $J(kG) \neq \{0\},$ (which is equivalent to $p | |G|,$ where $p$ is the characteristic of $k ),$ we may conclude that $N = G$ under the assumptions of the question. This is because $J(kG)$ is a two-sided ideal of $kG.$ Otherwise, for any group element $t \in G \backslash N,$ we would have $tJ(kN) \subs... | 7 | https://mathoverflow.net/users/14450 | 160114 | 84,298 |
https://mathoverflow.net/questions/159645 | 2 | Let $X\_{ij}$ with $1\leq i<j\leq n$ (that are $X\_{12},\dots, X\_{1n},\dots,X\_{(n-1)n}$) be ${n \choose 2}$ identically normal distributed $N(0,\sigma^2)$ such that
$
\text{corr}(X\_{ij},X\_{rs})=\rho
$
if $|\{i,j\}\cap\{r,s\}|=1$ and $0$ if $|\{i,j\}\cap\{r,s\}|=0$.
I'd like to estimate $\sigma^2$ if in the large... | https://mathoverflow.net/users/45305 | estimating variance of dependent normal distributed data | Not an answer, but no comments for me.
I don't think the problem is fully specified. You've given marginal distributions and second moments, but no joint distribution. For example, if 0 < rho < 1/2, there is a multivariate normal with these properties; X\_ij = Y\_i + Y\_j + e\_ij, where the Y's and e's are mutually i... | 1 | https://mathoverflow.net/users/46191 | 160126 | 84,304 |
https://mathoverflow.net/questions/160127 | 0 | $k$ algebraically field, $A$ $k$ algebra and valuation ring of $K$ ($K$ field fraction of $A$) and we have the transcendence degree of $K$ over $k$ is one.
i want to ask if $A$ is noetherian ring?
| https://mathoverflow.net/users/46460 | question about valuation ring | The answer is no via the following easy counterexample. Let $k$ be an algebraically closed field, and $K=\overline{k(t)}$. Let $v$ be an extension of the $t$-adic valuation to $K$, and let $A$ be the valuation ring of $v$. Then $A$ is a valuation ring of $K$ and $\operatorname{tr.deg}\_k(K)=1$, but since the value grou... | 5 | https://mathoverflow.net/users/6856 | 160128 | 84,305 |
https://mathoverflow.net/questions/160112 | 2 | The standard proof of the completeness theorem in first-order Gödel logic is based
on a first-order countable language. I want to know that is there any proof of the completeness theorem in first-order Gödel logic for uncountable language, or is there an example that refute the completeness of first-order Gödel logic ... | https://mathoverflow.net/users/45990 | Compactness and completeness in Gödel logic | You didn’t specify the semantics you are interested in.
Gödel–Dummett logic in languages of arbitrary cardinality is complete with respect to safe models over linearly order Gödel algebras, and with respect to appropriate linearly ordered Kripke models.
For uncountable languages, the logic is not complete with resp... | 4 | https://mathoverflow.net/users/12705 | 160129 | 84,306 |
https://mathoverflow.net/questions/160009 | 3 | If one could prove that for every 4-chromatic planar graph $X$ every color identical pair in $X$ is separated by a cycle, would that be a proof of the 4-color theorem?
Explanation:
A pair of vertices $\{u, v\}$ in a $k$-chromatic graph $G$ is a **color identical pair** iff the color of $u$ equals the color of $v$ ... | https://mathoverflow.net/users/48080 | Color identical pairs and the 4-color theorem | The claim appears to be equivalent to the four colour theorem. In the following I'll assume that "$x$ and $y$ can be separated by a cycle" is equivalent to "adding the edge between $x$ and $y$ produces a non-planar graph". (Perhaps this is what your question is actually about: formally showing that a reasonable definit... | 3 | https://mathoverflow.net/users/25485 | 160139 | 84,307 |
https://mathoverflow.net/questions/160136 | 6 | Assume an arbitrary $x$ and let $z$ be smaller than $y$, where $y$ is the length of the interval $[x,x+y]$. What I would like to know is:
>
> Let $W(z)=\prod\_{p\leq z}\left(1-\frac{1}{p}\right)$. For what values of $z=z(y)$ is
> $$
> y W(z) \sim \textrm{e}^{-\gamma} \frac{y}{\log z}
> $$
> an accurate estimate ... | https://mathoverflow.net/users/45947 | When does Merten's product theorem accurately estimate the number of coprimes in an interval? | This is the type of problem that is solved by the ``fundamental lemma of sieve theory." Put $z=y^{1/u}$. Then from the fundamental lemma it follows that the number of integers in $[x,x+y]$ that are coprime to all primes below $z$ is
$$
\sim y \prod\_{p\le z} \Big(1-\frac 1p \Big) (1+O(u^{-u})).
$$
So as $u$ goes to ... | 10 | https://mathoverflow.net/users/38624 | 160145 | 84,310 |
https://mathoverflow.net/questions/160090 | 4 | This is related to the edited form of [this question.](https://mathoverflow.net/questions/136025/frobenius-density-theorem) Suppose that I know that some set of primes has a certain Dirichlet density. What is the optimum statement one can make generally (for example, if the Dirichlet density is non-zero, is it true tha... | https://mathoverflow.net/users/11142 | Dirichlet density vs natural density | Consider the set of integers $A$ which are in between $10^{n^2-n}$ and $10^{n^2}$ for some $n$. Then the upper natural density of $A$ is $1$, because among the $10^{n^2}$ first integers, at least $10^{n^2}-10^{n^2-n}$ are in $A$, so a proportion of $1-10^{-n}$ are in $A$. The lower density of $A$, on the other hand is ... | 8 | https://mathoverflow.net/users/9317 | 160152 | 84,313 |
https://mathoverflow.net/questions/160150 | 7 | Let $f\_1,f\_2,f\_3,f\_4,f\_5 \in \mathbb{Q}[x]$ be linear and
coprime and not all constant.
>
> Is it possible $ f\_1^2+f\_2^2+f\_3^2+f\_4^2=f\_5^2$?
>
>
>
I suppose the answer is negative.
If this is possible, solving $f\_5(x)=N$ would give
deterministic representation of $N$ as sum of four
squares (probab... | https://mathoverflow.net/users/12481 | Would such polynomial identity exist? (related to sum of four squares) | To see that $f\_{1}$, $f\_{2}$, $\ldots$, $f\_{k}$ must be scalar multiples of $f\_{k+1}$, plug in the root of $f\_{k+1}$ into both sides of the equation.
| 18 | https://mathoverflow.net/users/48142 | 160155 | 84,314 |
https://mathoverflow.net/questions/160131 | 9 | The following observation between the spaces of global sections of line bundles on the nilpotent cone and the Hall-Littlewood polynomials is made in a recent physics preprint [1403.0585](http://arxiv.org/abs/1403.0585). Is this a known mathematical fact?
Pick a simple Lie algebra $\mathfrak{g}$ over $\mathbb{C}$. Let... | https://mathoverflow.net/users/5420 | Hall-Littlewood functions and functions on the nilpotent cone | I took notes on a lecture that Mark Haiman gave explaining how to do a similar thing on the cotangent bundle on the flag variety: (see 2.1 of <http://math.berkeley.edu/~monks/seminars/Notes.pdf> )
This is close to what you are asking if we consider the Springer resolution of the cotangent bundle to the nilpotent cone... | 6 | https://mathoverflow.net/users/321 | 160156 | 84,315 |
https://mathoverflow.net/questions/160069 | 3 | Let $(E):\ \sum\_{k=0}^n P\_k(z)g(z+k)=0$ with the $P\_k\in\mathbb C[z]$ a finite differences equation in $\mathbb C$.
Is it true that every any entire solution $g$ of (E)
of exponential type $<\pi$ is an exponential polynomial?
Thanks in advance
| https://mathoverflow.net/users/33128 | Entire solutions of finite difference equations | The answer is no. Function $g(z)=(e^z-1)/z$ is entire, of exponential type $1$,
is not an exponential polynomial, but it satisfies the equation
$$(z+2)g(z+2)-(1+e)(z+1)g(z+1)+ezg(z)=0.$$
Ref. J-P Bezivin et F. Gramain, Solutiuons entieres d'un systeme
d'equations aux differences, Ann. Inst. Fourier, Grenoble, 43 (1993)... | 7 | https://mathoverflow.net/users/25510 | 160159 | 84,317 |
https://mathoverflow.net/questions/160160 | 4 | It is well know that (in the sense having the same image not parametrization) the extremal curves of the energy functional on a Riemanian manifold $(M,g)$:
$E[\gamma(t)] = \int g\_{\gamma(t)}(\dot{\gamma}(t), \dot{\gamma}(t)) dt$
are the same as the length functional:
$L[\gamma(t)] = \int \sqrt{g\_{\gamma(t)}(\dot{\gam... | https://mathoverflow.net/users/41654 | Lagrangians with the same extremal curves | If you have a nondegenerate Lagrangian $L:TM\to\mathbb{R}$ (such as the energy Lagrangian of a pseudo-Riemannian metric or the square of a Finsler metric, though these are not the only cases) with the property that $R(L)$ is a function of $L$ (where $R$ is the 'radial' vector field on $TM$, i.e., it is tangent to the f... | 4 | https://mathoverflow.net/users/13972 | 160171 | 84,320 |
https://mathoverflow.net/questions/160161 | 19 | Is there an unlabeled locally-finite graph which is a Cayley graph of an infinitely many non-isomorphic groups with respect to suitably chosen generating sets?
| https://mathoverflow.net/users/10443 | Infinitely many finitely generated groups having the same Cayley graph | Here I answer your additional question about finitely presented groups. The answer is then no.
**Affirmation.** Only finitely many finitely presentable groups may have the same given Cayley graph.
This works as follows. Let $X\_1$ be the Cayley graph of some f.p. group. By finite presentability, there exists $n\_0$... | 23 | https://mathoverflow.net/users/14094 | 160178 | 84,322 |
https://mathoverflow.net/questions/160130 | 0 | I asked this question on mathstack (long time ago), however I received no answers, so I'm trying it here. I don't know whether it's suitable for this site, anyway.
I was reading this article <http://ncatlab.org/nlab/files/cech.pdf> and I could not understand the construction of the left adjoint of the inclusion $\mat... | https://mathoverflow.net/users/40883 | Sheaffication using a $\lambda$-transfinite colimit | First about the two notions of sheaf: Dugger supposes that his Grothendieck topology is given by a [pretopology](http://ncatlab.org/nlab/show/Grothendieck+pretopology), i.e. you specify for each object $X$ of the site which families of morphisms $\{U\_i \to X\}$ are covering families - from such a covering family you g... | 3 | https://mathoverflow.net/users/733 | 160180 | 84,323 |
https://mathoverflow.net/questions/160173 | 5 | Let $X,Y$ be Banach spaces, and let $X\_0$ be a subspace of $X$ (by subspace I mean a closed linear set). Consider the set $Ext(X\_0,Y)$ of all bounded linear operators $A\_0:X\_0\to Y$ which have an extension to a bounded linear operator $A:X\to Y$
(not necessarily with the same norm).
Question: is $Ext(X\_0,Y)$ clos... | https://mathoverflow.net/users/48157 | Extension of bounded linear operators | The set $Ext(X\_0, Y)$ will be rarely closed, I believe. (Perhaps you want to use a different symbol for this set, as $Ext$ has its own meaning.)
Suppose that $X$ is a Banach space which is complemented in $X^{\*\*}$, $X$ is isomorphic to $X\oplus X$, $X$ has a Schauder basis and it contains an uncomplemented copy of... | 7 | https://mathoverflow.net/users/15129 | 160182 | 84,325 |
https://mathoverflow.net/questions/160124 | 6 | For any $r\in\mathbb{N}$, let $A\_r$ denote the set of all natural numbers that are potentially a side of a Pythagorean triple with hypotenuse $r$.
Given any $N\in\mathbb{N}$, does there exist $r,s$ such that $A\_r\cap A\_s$ contains at least $N$ elements?
I've been looking at the parameterization of Pythagorean tr... | https://mathoverflow.net/users/45842 | Intersecting Sets of Pythagorean Triples with Common Hypotenuses | Yes, given any $N \in \mathbb{N}$ there do exist $r$ and $s$ so that $A\_{r} \cap A\_{s}$ contains at least $N$ elements. In fact, I will show that one can take $s = 6r$. (The approach I suggest below is definitely not the most efficient, nor is it along the lines you were pursuing.)
We want to find values of $r$ so ... | 6 | https://mathoverflow.net/users/48142 | 160188 | 84,329 |
https://mathoverflow.net/questions/160198 | 2 | In time series analysis, a common assumption made is that the series is wide-sense stationary, ex. that it has time invariant mean and covariance. However, as this is often not the case in real life, a common approach is to take the difference of the time series:
D\_(i) = X\_(i) - X\_(i-1).
If it doesn't work, th... | https://mathoverflow.net/users/48168 | Why does differencing create wide-sense stationary time series? | I don't know why that is a common practice, but it makes sense in the case of a Brownian motion with drift $X\_t=\sigma W\_t+\nu t$. Then we have
$$
\mathbb E X\_{t+1}=\nu(t+1)
$$
which is not time invariant, but
$$
\mathbb E (X\_{t+1}-X\_t)=\nu
$$
which is time invariant. If the drift was of higher order, $\nu t^k$, t... | 1 | https://mathoverflow.net/users/4600 | 160199 | 84,333 |
https://mathoverflow.net/questions/160221 | 4 | Is there a Boolean lattice $(X,\le)$ and a nonempty collection $(a\_{ij})\_{i\in I,j\in J}\subseteq X$ such that
$$\bigvee\_{i\in I}\bigwedge\_{j\in J}{a\_{ij}}\ne \bigwedge\_{f:I\to J}\bigvee\_{i\in I}a\_{if(i)}\quad ?$$
where $\bigvee$ stands for $\sup$ and $\bigwedge\_{f:I\to J}\bigvee\_{i\in I}a\_{if(i)}$ is $\in... | https://mathoverflow.net/users/47958 | Existence of more distributive Boolean lattices | Yes as soon as $X$ is not atomic. See Handbook of Boolean algebras, Theorem 14.5 (page 215 in Volume 1)
| 4 | https://mathoverflow.net/users/41291 | 160222 | 84,343 |
https://mathoverflow.net/questions/160200 | 2 | suppose that $k$ is a finite field of characteristic 2 and $(V,q)$ a quadratic space, i.e., $V$ is a $k$-vector space and $q:V\to k$ quadratic form. Suppose that $\dim(V)\geq 4$ and that $q$ is non-defective, i.e., the $k$-bilinear $B(x,y)=q(x+y)-q(x)-q(y)$ is non-degenerated:
The question is:
Given a non-singular ... | https://mathoverflow.net/users/6517 | Stabilizer of a nonsingular vector in a quadratic space (char (k)=2) | A precise reference for this result is the book "The Subgroup Structure of the Finite Classical Groups" by P. Kleidman and M. Liebeck, Proposition 4.1.7.
The notation used for the various orthogonal groups varies a lot from book to book, so I am not completely sure what you mean by $O(n,q)$, but presumably $O'(n,q)$ ... | 1 | https://mathoverflow.net/users/35840 | 160223 | 84,344 |
https://mathoverflow.net/questions/159558 | 5 | A tournament is an orientation of a complete graph. A feedback arc set is a set of arcs in a digraph whose removal leave the digraph acyclic. The feedback arc set problem consists in finding a feedback arc set of minimum size. A polynomial time approximation scheme is an algorithm which takes an instance of an optimiza... | https://mathoverflow.net/users/nan | Is there a polynomial time approximation scheme for the feedback arc set problem for the class of tournaments? | Not only a PTAS is known for this problem.
It is also possible to compute a PTAS, even without seeing the entire adjacency matrix !
In 2011, Ailon [has showed](http://arxiv-web3.library.cornell.edu/pdf/1011.0108v3.pdf) that by a smart choice of queries you can compute a $(1+\epsilon)$-approximation while reading on... | 3 | https://mathoverflow.net/users/47499 | 160224 | 84,345 |
https://mathoverflow.net/questions/160232 | 4 | Let $h\_0, h\_1, \dots$ be the classical univariate Hermite polynomials, renormalized to have constant norm. Is
$$x\mapsto\prod\_{j=1}^n h\_{l\_j}(x\_j), \quad l\_j\in \mathbb N$$
a complete orthogonal system for $L^2(\mathbb R^n)$, as in the univariate case? The orthogonality is simple, but I still have some doubts on... | https://mathoverflow.net/users/28161 | Multivariate Hermite Polynomials | In one dimension, you have
$$
h\_k(t)e^{-π t^2}=e^{π t^2}(\frac{d}{dt})^k\bigl(e^{-2π t^2}\bigr),
$$
and $h\_k$ is easily proven to be with degree $k$.
The completeness question amounts to proving
$L^2=\overline{\text{span}\{t^ke^{-π t^2}\}}=E$.
Taking $u\in E^\perp$, you obtain
$$
\int u(t)\sum\_k\frac{t^ki^k\xi ^k}{k... | 5 | https://mathoverflow.net/users/21907 | 160240 | 84,348 |
https://mathoverflow.net/questions/160239 | 2 | (I posted this question on Mathstack but I haven't received any answers or comments so I thought I might as well try my luck here. I apologize if it is not an appropriate question.)
Theorem (Quillen) Let $F:\mathcal{C} \rightarrow \mathcal{D}$ suppose that $ N(F\downarrow d)$ is weakly equivalent to $\*$ for all $d \... | https://mathoverflow.net/users/38075 | A question about the proof of Quillen's Theorem A | If $S \colon \mathcal A \to \mathcal B$ is a functor, let $\newcommand{\op}{\mathrm{op}}S^\op \colon \mathcal A^\op \to \mathcal B^\op$ be its opposite. Then $(S \downarrow T)^\op \cong T^\op \downarrow S^\op$. Combine this with the equality $NF = NF^\op$ (if you identify $N\mathcal C = N\mathcal C^\op$, etc.)
| 7 | https://mathoverflow.net/users/1310 | 160243 | 84,351 |
https://mathoverflow.net/questions/159167 | 3 | The usual [second isomorphism theorem](https://en.wikipedia.org/wiki/Isomorphism_theorem#Second_isomorphism_theorem) for groups is: let $G$ be a group, $S$ and $N$ subgroups with $N$ normal, then $SN$ is a subgroup of $G$, $S\cap N$ is a normal subgroup of $S$ and $SN/N \simeq S /S \cap N$.
($N$ need not to be a nor... | https://mathoverflow.net/users/34538 | A second isomorphism theorem for the inclusions of groups | **No**, the following theorem gives counterexamples:
>
> **Theorem:** $\forall n \ge 3$, $A\_{n+1}$ and $S\_n$ are *normal intermediate subgroups* of the inclusion $(A\_n \subset S\_{n+1})$, but $(A\_n \subset S\_n \subset S\_{n+1} )$ and $(A\_n \subset A\_{n+1} \subset S\_{n+1} )$ are not equivalent.
>
>
>
*... | 2 | https://mathoverflow.net/users/34538 | 160253 | 84,353 |
https://mathoverflow.net/questions/160250 | 5 | There are many well-known equivalences in reverse mathematics between statements of the form "Every set is contained a countable coded $\omega$-model of $T$" and $S$, where $S, T$ are subsystems of second order arithmetic and $T < S$ in the usual hierarchies. For example:
* $\mathsf{ACA}\_0^+$ is equivalent over $\ma... | https://mathoverflow.net/users/10384 | Attribution of an equivalence of the existence of omega-models of RCA0 | There are a lot of subtleties here. For technical reasons, I'll use $\mathsf{WKL}\_0$ instead of $\mathsf{RCA}\_0$ to explain them.
The following two theorems are found in Simpson's book:
Theorem VIII.2.6. *The following is provable in $\mathsf{WKL}\_0$. For all $X \subseteq \mathbb{N}$, there exists a countable co... | 4 | https://mathoverflow.net/users/2000 | 160256 | 84,356 |
https://mathoverflow.net/questions/159880 | 0 | Let $X$ be the subset of the long line consist of rational points with the topology inherits from the long line.
Is $X$ a metrizable space?
| https://mathoverflow.net/users/36688 | Rational points in the Alexandroff line | As stated in the comments, one can show that $\omega\_{1}$ is not paracompact hence non-metrizable. However, one can also give a direct proof that $\omega\_{1}$ is not metrizable without resorting to paracompactness. Here is one way to prooe that $\omega\_{1}$ is not metrizable. Suppose to the contrary that $d$ is a me... | 2 | https://mathoverflow.net/users/22277 | 160266 | 84,359 |
https://mathoverflow.net/questions/160261 | 5 | Is there a Book or a bunch of exercises to get used to diffeological spaces from a **practical point of view?** It seems to me that papers on this topic are mostly concerned with their very good categorical properties, but do little care about how well theses spaces behave in actuall calculations.
On ordinary manifol... | https://mathoverflow.net/users/21965 | Exercises around Diffeological Spaces or a Diffeologic Atlas Theory | Have you seen the book of Patrick Iglesias-Zemmour "Diffeology"?
<http://math.huji.ac.il/~piz/Site/The%20Book.html>
It is very nice, with plenty of examples.
| 9 | https://mathoverflow.net/users/27816 | 160267 | 84,360 |
https://mathoverflow.net/questions/160092 | 6 | A finite matroid $M$ is a pair $(E,I)$ where $E$ is a finite set and $I$ is a family of independent set with the following properties:
1) There is at least an independent system
2) Every subset of an independent set is independent.
3) If $A$ and $B$ are two independent sets and $A$ has more elements than
$B$. Th... | https://mathoverflow.net/users/47659 | matroids axioms and independence system | The substantial question 2 being already answered, I will address the less substantial 1.
Even if not substantial, the formalization of 1 might not be immediate (and at the end all the question might be considered by someone only a logical sophism, but since questions of the kind "is a canonical construction availabl... | 3 | https://mathoverflow.net/users/46855 | 160283 | 84,367 |
https://mathoverflow.net/questions/160285 | 11 | Let $[X,S^3]$ be the set of homotopy classes of maps from the pointed CW complex X to the pointed 3-sphere. The group structure on S^3 endows this set with a group structure. Is this group ever noncommutative?
| https://mathoverflow.net/users/48226 | Can the group of homotopy classes of maps into S^3 be noncommutative? | Yes, by Yoneda's lemma. To generalize a bit, you have a group object $G$ in some category with finite products, and you ask whether the functor it represents is pointwise abelian. This is the same as asking whether two natural transformations $[-,G\times G]=[-,G]\times[-,G]\to[-,G]$ are equal (one being $(f,g)\mapsto f... | 10 | https://mathoverflow.net/users/75 | 160293 | 84,370 |
https://mathoverflow.net/questions/160297 | 21 | What is the group of pointed homotopy classes of maps from $S^3 \times S^3$ to $S^3$? The group structure induced by the group structure on the codomain. This question is a followup to Eric's answer to [another question](https://mathoverflow.net/questions/160285/can-the-group-of-homotopy-classes-of-maps-into-s3-be-nonc... | https://mathoverflow.net/users/48226 | What is the group of pointed homotopy classes of maps from $S^3 \times S^3$ to $S^3$? | We have that $S^3 \simeq \Omega \mathbb{H}P^{\infty}$, so by adjointness we can as well consider the group of maps $[\Sigma S^3 \times S^3, \mathbb{H}P^{\infty}]$. It is well-known that $[X,\Omega Y] \simeq [\Sigma X, Y]$ are isomorphic as groups, when you define the group structure on the first one by loop composition... | 32 | https://mathoverflow.net/users/39747 | 160307 | 84,377 |
https://mathoverflow.net/questions/160277 | 3 | Let $F$ be a non-Archimedean local field with ring of integers $O$, $\pi$ be a uniformizer. Let $\tilde{G}$ be a connected algebraic group over $F$ and splits over $F$, fix a split maximal torus $\tilde{T}$. Let, $$K=\tilde{G}(O)\subset G=\tilde{G}(F)$$ be a Gelfand pair of compact and locally compact groups.
We know... | https://mathoverflow.net/users/36735 | Gelfand pair and double coset decomposition | $K\pi^\lambda K$ has a transitive right action of $K$.
The stabilizer of $K\pi^\lambda$ for this action is $K\cap \pi^{-\lambda}K\pi^\lambda$.
Thus, $K\pi^\lambda K = \coprod\_x K\pi^\lambda x$ as $x$ runs over a set of representatives for $(K\cap \pi^{-\lambda}K\pi^\lambda)\backslash K$.
The subgroup $K\cap \pi^{-\l... | 2 | https://mathoverflow.net/users/9672 | 160310 | 84,378 |
https://mathoverflow.net/questions/160289 | 6 | If $\Pi\_n$ is the set of partitions of $n$, then for $\lambda, \mu\in \Pi\_n$ we say $\mu$ dominates $\lambda$ if $\sum\limits\_{i=1}^k \lambda\_i \leq \sum\limits\_{i=1}^k \mu\_i$ for all $k$. This gives a partial order on $\Pi\_n$ which is not a total order if $n>5$. For example, $(4,1,1)$ and $(3,3)$ are not compar... | https://mathoverflow.net/users/38495 | Length of the longest chain in dominance order | The longest (descending) chain starts obviously with the partition $n$ having
a unique part and ends with the partition $1+1+\dots+1$ into singletons.
One should move from $n$ to $1+\dots+1$ by moving dots (unities) down row by row
whenever possible along staircases: \begin{eqnarray\*}&&n,(n-1)+1,(n-2)+2,(n-2)+1+1,(n-3... | 7 | https://mathoverflow.net/users/4556 | 160313 | 84,380 |
https://mathoverflow.net/questions/160320 | 2 | Suppose $a\_n \to a$ in a unital C\*-algebra $A$. If $\lambda\_n \in \sigma(a\_n)$ and $\lambda\_n \to \lambda$, then $\lambda \in \sigma(a)$. Does the converse hold?
So if $\lambda \in \sigma(a)$, does there exist a sequence $\lambda\_n \in \sigma(a\_n)$ with $\lambda\_n \to \lambda$?
It is true if $A$ is commutat... | https://mathoverflow.net/users/46472 | Continuous depedence of the spectrum on elements | The result is not true.
The sequence of operators $U\_k$ on $\ell\_2(\mathbb{Z})$ defined by $U\_ke\_n = e\_{n-1}$ for $n\in\mathbb{Z}\setminus 0$ and $U\_ke\_0=k^{-1}e\_{-1}$ converges to $U\_\infty$ defined by $U\_\infty e\_n = e\_{n-1}$ for $n\in\mathbb{Z}\setminus 0$ and
$U\_\infty e\_0=0$.
The spectrum of $... | 2 | https://mathoverflow.net/users/39421 | 160325 | 84,383 |
https://mathoverflow.net/questions/160207 | 20 | A friend asked me the following problem:
>
> Is it true that for every $X\subset A\subset \mathbb{Z}$, where $A$ is finite and $X$ is non-empty, that $$\frac{|A+X|}{|X|}\geq \frac{|A+A|}{|A|}?$$
>
>
>
Here the notation $A+B$ denotes the set $\{ a+b\ : a\in A, b\in B\}$.
It follows from Rusza's triangle inequa... | https://mathoverflow.net/users/12176 | A sumset inequality | The proposed inequality is not true. I do not claim originality for this example: all I have done is take Seva's observation that the inequality would imply an improvement $|3A|\leq K^2|A|$ given $|2A|\leq K|A|$, find a standard counterexample to this hypothetical improvement, and then simplify it a bit for the present... | 11 | https://mathoverflow.net/users/20598 | 160327 | 84,384 |
https://mathoverflow.net/questions/160329 | 3 | I want to find some subset of R^2 which its intersection with every vertical line is measure zero if we see it as a subset of R and it is not measure zero in R^2?
| https://mathoverflow.net/users/48251 | measure zero in R but not in R^2 | It is a well known result of Sierpinski that there exists non-measurable subsets of $\mathbb{R}^{2}$ which intersect each line in at most two points. Furthermore, there exists a real-valued function whose graph is a non-measurable subset of $\mathbb{R}^{2}$.
1. W. Sierpi´nski: Sur un probl`eme concernant les ensemble... | 10 | https://mathoverflow.net/users/22277 | 160334 | 84,387 |
https://mathoverflow.net/questions/160279 | 3 | It is possible to define a weak solution of a parabolic PDE
$$u\_t - Au = f$$
$$u(0) = u\_0$$
as $u \in L^2(0,T;H^1)$ such that
$$-\int\_0^T\int\_\Omega u(t)\varphi'(t) + \int\_0^T\int\_\Omega Au(t)\varphi(t) = \int\_0^T\int\_\Omega f(t)\varphi(t) + \int\_{\Omega} u\_0\varphi(0)$$
holds for all smooth test functions $\... | https://mathoverflow.net/users/48222 | Checking initial data in parabolic PDE with no control on time derivative | The fundament solution approximates the $\delta$ function as $t\to0^+$. Thus if we represent the solution as an integral, it converges to the initial value as $t\to 0^+$ in $L^2$. This is similar to the situation of approximations to the identity using mollifier.
| 2 | https://mathoverflow.net/users/22238 | 160347 | 84,392 |
https://mathoverflow.net/questions/160355 | 2 | The modal logic S4.2 with the characteristic axioms
4: $\square \alpha \rightarrow \square \square \alpha$
and
.2: $\lozenge \square \alpha \rightarrow \square \lozenge \alpha$
and
T: $\square \alpha \rightarrow \alpha$
is sound and complete for transitive, reflexive and connected frames. Such frames valid... | https://mathoverflow.net/users/37385 | A question on the modal logic S4.2 | $\let\B\Box\let\D\Diamond$
\begin{align\*}
\D\B\alpha\land\D\B\beta&\to\D\B\B\alpha\land\D\B\B\beta\\
&\to\B\D\B\alpha\land\D\B\B\beta\\
&\to\D(\D\B\alpha\land\B\B\beta)\\
&\to\D\D(\B\alpha\land\B\beta)\\
&\to\D\D\B(\alpha\land\beta)\\
&\to\D\B(\alpha\land\beta)
\end{align\*}
using the K-provable principle $\B p\land\D... | 6 | https://mathoverflow.net/users/12705 | 160361 | 84,399 |
https://mathoverflow.net/questions/160356 | 2 | I wonder if the following is known:
let $n$ be a (square-free) positive integer. Is there ever/always a sequence of prime numbers $p$ that can be written in the form $$p = x^2 + ny^2,$$ where
$x, y$ were positive integers and the ratio $y/x$ would approach zero?
Thank you!
| https://mathoverflow.net/users/10591 | Principally split primes with factors in arbitrarily small angular sectors | Yes, this can be proved similarly as Dirichlet's theorem on arithmetic progressions. More precisely, using $L(1,\chi)\neq 0$ for all the nontrivial unramified Grössencharacters of $\mathbb{Q}(\sqrt{-n})$ it can be proved that the prime ideals of this number field are equidistributed in the class group and, within each ... | 4 | https://mathoverflow.net/users/11919 | 160363 | 84,401 |
https://mathoverflow.net/questions/160354 | 3 | Let $ f \in \mathbb{C}[z\_1, \ldots, z\_n]$ be a polynomial such that $f'(z) \neq 0$ if $z \neq 0$ ($f'$ means $\left( \frac{\partial f}{\partial z\_1}, \ldots, \frac{\partial f}{\partial z\_n}\right)$ ). Let $S = S^{2n-1} \subset \mathbb{C}^n$ be a sphere in $\mathbb{C}^n$.
Is the set $L = \{ z \in S: f(z) = 0 \}$ ... | https://mathoverflow.net/users/48261 | Zeroes of a complex polynomial on a sphere as a manifold | This is a well-known fact, and I'm not sure if this question is suitable for MO.
Anyway, I will give a short answer.
Let $X=V(f) \subset \mathbb{C}^{n}$ be an affine hypersurface with at most an isolated singularity at the origin and let $L= X \cap S\_{\varepsilon}$ be the intersection of $X$ with a sphere $S\_{\var... | 9 | https://mathoverflow.net/users/7460 | 160365 | 84,403 |
https://mathoverflow.net/questions/160350 | 1 | Consider solution of
$$x^3-y^2=1728 \text{ unit} \qquad (1)$$
in a number field.
This is related to the discriminant of elliptic curve
in terms of $c\_4,c\_6$.
Via elliptic curves it might have infinitely solutions
for fixed unit.
I am interested if $x,y$ are in the ring of integers
of the number field.
If on... | https://mathoverflow.net/users/12481 | On $x^3-y^2=1728 \text{ unit}$ in number fields | You can always absorb 6th powers into $x$ and $y$, so there is a finite set of units $u\_1,\ldots,u\_n$ (covering the congruence classes of units mod 6th powers), so that every solution to $x^3-y^2=1728u$ with $x,y\in\mathcal{O}\_K$ and $u\in\mathcal{O}\_K^\*$ yields a solution $(X,Y)\in\mathcal{O}\_K\times\mathcal{O}\... | 7 | https://mathoverflow.net/users/11926 | 160367 | 84,404 |
https://mathoverflow.net/questions/160353 | 3 | Is there a Karhunen-Loeve theorem for discrete-time process?
For example, let $\left\{X\_i\right\}$ be a sequence of independent random variable which are uniformly distributed on the set $\{-1,1\}$. Define
$$
S\_t = \sum\_{i=1}^t X\_i,
$$
with $t$ an integer between 1 and $n$.
I would like to write $S$ as
$$
S\_t = ... | https://mathoverflow.net/users/nan | Karhunen-Loeve expansion for discrete-time process | There does not seem to be a problem. I assume that you have a random field
$$S\_t: \{1,\dotsc, n\}\to \mathbb{Z}$$
The covariance kernel is $(t\leq \tau)$ $\newcommand{\bE}{\mathbb{E}}$
$$ \bE(S\_tS\_\tau)=\bE(S\_t^2)= \sum\_{i=1}^t \bE(X\_i^2)=t=\min(t,\tau). $$
Now all you need to do is find the eigenvalues ... | 1 | https://mathoverflow.net/users/20302 | 160370 | 84,405 |
https://mathoverflow.net/questions/160345 | 0 | Let me recall some quick definitions. A projective hyperkahler manifold is a simply connected smooth projective variety $M$ such that $H^0(M,\Omega\_M^2)=\mathbb C\sigma$, with $\sigma$ an everywhere non-degenerate holomorphic 2-form.
A normal projective variety $M$ is said to have $\textit{symplectic singularities}... | https://mathoverflow.net/users/13139 | Dominant map from hyperkahler manifolds to normal projective varieties with symplectic singularities | So apparently I was just being silly and it does indeed follow from the definitions. Let $f:M\dashrightarrow \overline{M}$ be our dominant rational map between a hyperkahler manifold $M$ and a projective normal variety $\overline{M}$ with symplectic singularities. Let $\pi:\tilde{M}\rightarrow \overline{M}$ be a resolu... | 0 | https://mathoverflow.net/users/13139 | 160381 | 84,407 |
https://mathoverflow.net/questions/160387 | 25 | Would anyone be able to recommend text books that give an introduction to Geometric Representation Theory and survey papers that give an outline of the work that has been done in the field? I'm looking for references that would be suitable as a follow up to James Humphrey's "Linear Algebraic Groups" and a first year gr... | https://mathoverflow.net/users/48273 | Introductory References for Geometric Representation Theory | I would encourage you to consider "Representation Theory and Complex Geometry" by Chriss and Ginzburg. In particular, I think you might enjoy the realization of irreducible representations of the Weyl group of a complex semisimple group $G$ on the Borel-Moore homology of the fibres of the Springer resolution of the nil... | 14 | https://mathoverflow.net/users/25358 | 160393 | 84,411 |
https://mathoverflow.net/questions/159727 | 2 | Let $q$ be a large prime and $\delta\in(0,1)$. Let $k$ be an integer which is not a multiple of $q$. Define $e(x)=e^{-2\pi ix}$. Can we get any non-trivial upper bounds for $$\sum\_{a=1}^{q^{1-\delta}}e(\frac{k}{q}a) $$?
A bound like $q^{1-\delta-\epsilon}$ would be good enough. I have searched literature in Weyl's sum... | https://mathoverflow.net/users/4519 | upper bound for an incomplete exponential sum | In general no non-trivial bound exists. Suppose e.g. that $a=1$. Then for all $k$ in the range of summation we have $\frac{ka}{q}\in[0, q^{-\delta}]$, thus $e(\frac{ka}{q})=1+\mathcal{O}(q^{-\delta})$.
If you have more information on $a$ you might use
$$
\sum\_{a=1}^{N}e(\frac{ak}{q}) = \frac{1-e(\frac{(N+1)a}{q})}{1... | 1 | https://mathoverflow.net/users/37555 | 160396 | 84,413 |
https://mathoverflow.net/questions/160421 | 4 | I read a [physics paper](https://arxiv.org/abs/1212.3324), of which the main idea is based on the topological classification of maps from 3-torus to the space of $N\times N$ unitary matrices. To quote their equation (4), which gives a winding number:
$$W(U)=\frac{1}{8\pi^2}\int\_{T^3} dtdk\_xdk\_y U^{-1}\partial\_t U\c... | https://mathoverflow.net/users/40893 | How to classify continuous/differentiable maps from $T^2$ to $U(N)$? | You are asking about the structure of the set $M=[T^2,U(N)]$ of homotopy classes of maps from $T^2$ to $U(N)$. The functor $H\_1$ gives a map $M\to\text{Hom}(H\_1(T^2),H\_1(U(n)))$, and standard calculations show that $H\_1(T^2)=\mathbb{Z}^2$ and $H\_1(U(N))=\mathbb{Z}$, so we end up with a map $\phi\colon M\to\mathbb{... | 8 | https://mathoverflow.net/users/10366 | 160425 | 84,424 |
https://mathoverflow.net/questions/160403 | 2 | It is possible that a more technical version of this question has been
asked and answered in the literature. If so, then a reference is much
appreciated. I will phrase it in terms of colored tapes placed on a number
line.
I have an unlimited supply of transparent but partially colored
arithmetic progression tapes,
o... | https://mathoverflow.net/users/3402 | Infinite play with tape, or covering the integers with prime arithmetic progressions | There is a map $\varphi: \mathbb{Z}\rightarrow \prod \mathbb{F}\_p$, given by reduction mod $p$ in each coordinate.
If you place the tapes centered at $0$ (to account for divisibility), you can read off the number of colors $n\in \mathbb{Z}$ gets by counting the zero-entries of $\varphi(n)$.
Now in your situation, yo... | 4 | https://mathoverflow.net/users/39747 | 160436 | 84,428 |
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