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https://mathoverflow.net/questions/252640 | 6 | I'm trying to read bits and pieces of W. Lawvere's [Categories of Space and of Quantity](https://github.com/mattearnshaw/lawvere/blob/master/pdfs/1992-categories-of-space-and-quantity.pdf), and, as usual have lots of questions. Page numbers will refer to the number printed on the corners of the pages.
The article is ... | https://mathoverflow.net/users/69037 | Lawvere's 'Categories of space and of quantity" - the projection formula | One way to interpret the projection formula as a "homogeneity" condition is by thinking of it as saying that the push-forward is a module map, i.e. it preserves an action. I don't know if this is what is meant by Lawvere, however.
I will be vague and general. We start with some category of things (maybe spaces or gro... | 3 | https://mathoverflow.net/users/458 | 252662 | 114,516 |
https://mathoverflow.net/questions/252645 | 5 | Let $R$ be a Noetherian commutative ring and $M$ a finitely generated $R$-module.
>
> What is known about the following subset of $ \mathrm {Spec}(R)$:
>
>
> $$\mathrm{supp}\_{fl}(M)=\{P\in \mathrm {Spec}(R):\ \mathrm{Tor}^R\_1(M, R/P)= 0\}?$$
>
>
>
In particular,
>
> Is $\mathrm{supp}\_{fl}(M)$ open in ... | https://mathoverflow.net/users/10482 | The flat support of a module | I suspect that Graham Denham's argument only shows that the weaker condition $\mathrm{Tor}\_1(M,\kappa(P))=0$ is open, where $\kappa(P)$ is the residue field of $P$. With the problem as stated, here is a counterexample: for a field $k$, take $R=k[x,y]$ and $M=R/(x,y)$. Then it is easy to check that $\mathrm{supp}\_{fl}... | 7 | https://mathoverflow.net/users/7666 | 252665 | 114,517 |
https://mathoverflow.net/questions/252544 | 1 | I am thinking of the following situation:
On a probability space $\left( \Omega, \mathscr{F}, \cal{P} \right)$ with **arbitrary** structure, suppose we are given a random function (as it is called in the Stochastic Programming literature) $g: \Omega \times \mathbb{R}^N \rightarrow \mathbb{R}$, which is additionally j... | https://mathoverflow.net/users/99108 | Conditional Expectation Relative to "Random Time" - Consistency of the Substitution Rule | **Please, Your Attention:**
I have taken my time to work out a proof based on Dynkin's Multiplicative System Theorem, in the fashion on Exercise 14.7 in [these notes](http://www.math.ucsd.edu/~bdriver/280_09-10/Lecture_Notes/2009-2010-Probability%20Lecture%20Notes.pdf). From the discussion that follows, I think that,... | 0 | https://mathoverflow.net/users/99108 | 252669 | 114,519 |
https://mathoverflow.net/questions/252670 | 2 | I performed some computations in [wolfram alpha](https://www.wolframalpha.com/input/?i=%7Czeta%5Be%5E(174521i))%5D%7C)%5Cleq+log(174521)) looking at the behavior of the values of $|\zeta(e^{ni})|$ trying to predict a lower bound. I have got the following result:
For $n > 19 :|\zeta(e^{ni})|\leq \log(n)$
**My questi... | https://mathoverflow.net/users/51189 | Is $|\zeta(e^{ni})|\leq \log(n)$ true for $n > 19$ and how do i can show it if it is? | The minimal counterexample is $n=25$ (see the [numerical verification](https://www.wolframalpha.com/input/?i=%7Czeta%5Be%5E(25i))%5D%7C)%5Cleq+log(25))).
Another small counterexample is $n=44$, which as suggested by Gerry Myerson comes from the approximation $2\pi \approx 44/7$.
| 4 | https://mathoverflow.net/users/43108 | 252684 | 114,526 |
https://mathoverflow.net/questions/250484 | 2 | For a fixed set $X$ and a finite collection $E\_1,E\_2,\ldots,E\_k\subseteq X$, define the binary relation *adjacency* as follows: $E\_i,E\_j$ are adjacent
if their intersection is nonempty.
We term the transitive closure of this relation by
*transitive adjacency*
and define the *adjacent union* by
$$
\tilde\cup(E\_1,\... | https://mathoverflow.net/users/12518 | Is there a standard term for this graph/set theoretic concept? | You can think in terms of [intersection graphs](https://en.wikipedia.org/wiki/Intersection_graph). The transitive adjacency tells you when two vertices are adjacent. The adjacency union is then empty if and only if two of $E\_1,\ldots,E\_k$ are in distinct connected components.
| 4 | https://mathoverflow.net/users/8193 | 252686 | 114,527 |
https://mathoverflow.net/questions/248689 | 0 | Given two orientations $Q, Q'$ of a Dyinkin diagram. Is it always true that after a sequence of mutations, $Q$ becomes $Q'$? Are the some references about this? Thank you very much.
| https://mathoverflow.net/users/11877 | Mutation equivalence of quivers | Yes, it's true for any two acyclic quivers with the same underlying diagram having different orientations. See [lectures notes on cluster algebras](http://bookzz.org/book/2324891/6e1fab) by Robert Marsh.
| 3 | https://mathoverflow.net/users/100019 | 252692 | 114,531 |
https://mathoverflow.net/questions/252646 | 1 | I was wondering if a Smith number can be a Lucas Carmichael number.
Is there a proof that there is no such number?
If there is such a number, can you tell me it and how you got it?
I have written a brute force program and so far I have found no such number between 1 and 1,000,000.
Thanks, Remy
| https://mathoverflow.net/users/100006 | Can a Lucas Carmichael number also be a Smith number? | Here are all numbers below $10^9$ that are both Smith and Lucas-Carmichael:
8164079, 8421335, 21408695, 30071327, 47324639, 77350559, 103727519, 121538879, 134151479, 202767551, 239875559, 287432495, 306871487, 466861199, 560974259, 566019167, 574342145, 592557119, 594633599, 602758079, 677913599, 832477799
UPDATE.... | 3 | https://mathoverflow.net/users/7076 | 252697 | 114,533 |
https://mathoverflow.net/questions/252695 | 2 |
>
> **QUESTIONS.**
>
>
> (a) What is the formal system $(\Pi^1\_1$-CA)${}+{}$BI?
>
>
> (b) And what is ID${}\_\omega$, the formal theory of $\omega$-times iterated inductive definitions?
>
>
>
They are both mentioned in the following paper without any further explanations:
[W. Buchholz, An independence res... | https://mathoverflow.net/users/27742 | What is ($\Pi^1_1$-CA)${}+{}$BI? And what is ID${}_\omega$? | BI = Bar Induction.
See <https://www.jstor.org/stable/2270902> for the abbreviation
and
<https://en.wikipedia.org/wiki/Bar_induction> for the definition.
| 2 | https://mathoverflow.net/users/4600 | 252705 | 114,534 |
https://mathoverflow.net/questions/252076 | 2 | Let $A$ be a linear second-order difference operator acting on the space of complex sequences as
$$(Af)\_{n}=f\_{n-1}+a\_{n}f\_{n}+f\_{n+1}, \quad n\in\mathbb{Z},$$
where $a\_{n}\in\mathbb{C}$. Further, let for fixed $k\in\mathbb{N}\_{0}$, $f^{(0)},f^{(1)},\dots,f^{(k)}$ are nonzero sequences satisfying
$$ Af^{(0)}=0 \... | https://mathoverflow.net/users/56553 | A problem from linear algebra and difference equations | I'm going to use the notation defined above here. When a letter is used alone, it denotes the entire sequence.
Claim: If $C^{[j]}(f, g) = 0$ for all $j \leq k$, then $g^{[k]}$ is a linear combination of the $f^{[i]}, i \leq k$.
We work by induction. This is obvious for $k = 0$. Assume the statement is true up to $... | 3 | https://mathoverflow.net/users/44191 | 252706 | 114,535 |
https://mathoverflow.net/questions/252649 | 7 | $\require{AMScd}$I've been told that the Gray tensor product of two 2-categories can be obtained with a Kan extension (forgive me for my carelessness of foundational issues): the product $\boldsymbol\otimes \colon 2\textbf{-Cat} \times 2\textbf{-Cat} \to 2\textbf{-Cat}$ can be obtained with a Kan extension of $\Phi$ al... | https://mathoverflow.net/users/7952 | The Gray tensor product as a Kan extension | This is the subject of an [unpublished manuscript of Ross Street](http://maths.mq.edu.au/~street/GrayTensor.pdf). Street applies the technique for left Kan extending a monoidal structure along a dense functor from Brian Day's PhD thesis to construct the lax Gray tensor product of 2-categories. He defines a monoidal str... | 11 | https://mathoverflow.net/users/57405 | 252708 | 114,537 |
https://mathoverflow.net/questions/252703 | 24 | I ran into the following sanity check. Is the following statement true?
>
> Every smooth fiber bundle (with compact fiber) over $S^2$ can be extended to a smooth fiber bundle over $\mathbb{C}P^\infty$ (or just $\mathbb{C}P^n$).
>
>
>
For the case of $\mathbb{C}P^2$, if the bundle is a principal bundle, then it... | https://mathoverflow.net/users/40517 | All fiber bundles over $S^2$ extendable to $\mathbb{C}P^\infty$? | No it isn't, but I had to dig quite deep to get a counterexample. Let us look at smooth $(D^7, \partial D^7)$-bundles over $S^2$, i.e $D^7 \to E \overset{\pi} \to S^2$ with an identification $\partial E \cong S^2 \times \partial D^7$. These are classified by a map
$$f : S^2 \to BDiff\_\partial(D^7)$$
to the classifying... | 37 | https://mathoverflow.net/users/318 | 252715 | 114,540 |
https://mathoverflow.net/questions/252728 | 18 | Turing proved that not all real numbers are effectively computable. In the sense that no algorithm exists to compute some real numbers.
Here is Turing's definition: *A real number is computable if its digit sequence can be produced by some algorithm or Turing machine. The algorithm takes an integer $ n \geq 1$ as in... | https://mathoverflow.net/users/8784 | Hard-to-compute real numbers | EDIT: This was in a comment below, but I now think it should be part of the main answer:
There are two different ways to ask the question in the OP:
* Is there a real number $r$ such that no polytime algorithm computes *all* the bits of $r$?
* Is there a real number $r$ such that no *individual* bit of $r$ can be c... | 23 | https://mathoverflow.net/users/8133 | 252737 | 114,545 |
https://mathoverflow.net/questions/252713 | 6 | Currently I'm working on the following version of the AHSS $$ E^2\_{pq}\cong H\_p(M\eta; MSpin\_q(\ast))\Rightarrow MSpin\_{p+q}(M\eta)$$
where $\eta \colon B \to BSO$ is a stable vector bundle, and $M\eta$ denotes its Thom Spectrum.
My experience with the AHSS is "graduate level", therefore most properties and, let'... | https://mathoverflow.net/users/93538 | References for properties of Atiyah-Hirzebruch Spectral Sequence for a spectrum $X$ and generalised homology theory $MSpin_*$ | Answer to Q2: By the Thom isomorphism, $E\_{n,0}=H\_n(B;\mathbb Z)$. The bordism group you are interested in is isomorphic to a twisted spin bordism group, i.e. elements are represented by $f:M^n\to B$ together with a spin structure on $TM\oplus f^\*\eta$, up to bordism. This follows by a Pontryagin-Thom construction. ... | 7 | https://mathoverflow.net/users/95545 | 252742 | 114,548 |
https://mathoverflow.net/questions/252671 | 60 | I am looking for examples of Markov Chains which are surprising in the following sense: a stochastic process $X\_1,X\_2,...$ which is "natural" but for which the Markov property is not obvious at first glance. For example, it could be that the natural definition of the process is in terms of some process $Y\_1,Y\_2,...... | https://mathoverflow.net/users/83481 | "Surprising" examples of Markov chains | I believe that if $(X\_n)$ is a biased simple random walk on $[-N,N]$, then $|X\_n|$ is a Markov chain.
| 30 | https://mathoverflow.net/users/11054 | 252752 | 114,556 |
https://mathoverflow.net/questions/252700 | 4 | The problem is:
$$\min\_{\alpha}\frac{\alpha^T A \alpha}{\alpha^T\alpha}\frac{ \alpha^T B \alpha}{\alpha^T\alpha}$$
where $A$ and $B$ are symmetric and positive definite matrix.
I think the explicit solution may be hard to find. If anyone have some reference?
| https://mathoverflow.net/users/100022 | minimizing the product of rayleigh quotient | Here is a paper on [minimizing products of positive definite forms](http://epubs.siam.org/doi/pdf/10.1137/110821196) that you should find very useful --- it gives necessary and sufficient conditions for the products of positive definite quadratic forms to be convex, which will help do the optimization.
| 3 | https://mathoverflow.net/users/8430 | 252759 | 114,559 |
https://mathoverflow.net/questions/252761 | 6 | Let $k$ be an algebraically closed field and let $X$ be a (not necessarily affine) variety over $k$. Is the coordinate ring of $X$ ($k[X]$ or $O\_{X}(X)$) always a finitely generated $k$-algebra?
| https://mathoverflow.net/users/100065 | Is the coordinate ring of a variety a finitely generated algebra? | Feed google with "variety whose ring of global sections is not finitely generated". One gets [An example of a nice variety whose ring of global sections is not finitely generated](http://math.stanford.edu/~vakil/files/nonfg.pdf) by Ravi Vakil.
| 12 | https://mathoverflow.net/users/98306 | 252762 | 114,561 |
https://mathoverflow.net/questions/252756 | 1 | By an admissable subcategory $A$ in a triangulated category $B$, I mean a triangulated subcategory that has $A \oplus B \in A$, then $A$, $B \in A$, and so that there is either a right or left adjoint to the inclusion of $A$ into $B$.
Then, if $X$ is a projective variety (possibly with mild assumptions on the singula... | https://mathoverflow.net/users/41873 | Let $X$ be a projective variety. Is the bounded derived category of perfect complexes admissable in $D^b(X)$? | $D^b\_{perf}(X)$ is not admissible in $D^b(X)$, unless $X$ is smooth (in which ase $D^b\_{perf}(X) = D^b(X)$). Still one can consider the quotient category $D^b(X)/D^b\_{perf}(X)$. It is called triangulated category of singularities, and was much studied.
| 3 | https://mathoverflow.net/users/4428 | 252773 | 114,563 |
https://mathoverflow.net/questions/252755 | 6 | I was wondering if anybody could provide a sketch of Weil's proof of the Riemann hypothesis for curves that uses the Jacobian $\text{Pic}^0(X)$ and a bit of the intersection theory on $X \times X$ and the underlying intuitions.
The previous version of this question was asking for a "proof without words", which I real... | https://mathoverflow.net/users/nan | Sketch of Weil's proof of the Riemann hypothesis for curves | There is an REU paper from Chicago which seems to cover it <https://math.uchicago.edu/~may/VIGRE/VIGRE2007/REUPapers/FINALFULL/Raskin.pdf>
| 7 | https://mathoverflow.net/users/6084 | 252774 | 114,564 |
https://mathoverflow.net/questions/252778 | 0 | I am considering a smooth-enough real-valued function $ f: (0,1) \to (0,\infty) $ such that
1. $ f $ is decreasing,
2. $\lim\_{x\rightarrow0^{+}}f(x)=\infty $,
3. $ x \mapsto x^{2} f'(x) $ is decreasing,
4. $\lim\_{x\rightarrow0^{+}}x^{2} f'(x)=0 $.
>
> **QUESTION.** Under these constraints, does the limit
> $$ ... | https://mathoverflow.net/users/100069 | On the existence of $ \lim_{x \to 0^{+}} \frac{\log(f(x))}{\log(x)} $ under some constraints | Here's one way to get a counterexample.
Let $f(x) = g(1/x)$, so $g'(t) = -(1/t^2) f'(1/t)$.
Your conditions say that as $t \to +\infty$, we have $g$ positive and increasing to $\infty$, with $g'$ decreasing to $0$.
We can choose sequences $t\_n$ and $y\_n$ such that
1. $y\_n = t\_n^{1/4}$ if $n$ is even, $t\_n^{1/... | 2 | https://mathoverflow.net/users/13650 | 252779 | 114,565 |
https://mathoverflow.net/questions/252769 | 1 | Given smooth functions $Q\_1(t), Q\_2(s,y)$ which are both bounded above and below, consider the following variational problem for $T,t>0$ and $T>t$:
$\int\_0^T \delta w(s) Q\_1(s) \ ds + \int\_0^T \int\_0^t \delta w(s) Q(s,t) \ ds \ dt = 0$, for all smooth and bounded variations $\delta w$ which need not be zero at ... | https://mathoverflow.net/users/41654 | strong form of variational problem in control theory | Note that the second integral is the integral over the triangle $\newcommand{\bR}{\mathbb{R}}$
$$ \Delta\_T=\big\{ (s,t)\in\bR^2;\;\;0\leq s\leq t\leq T\big\}. $$
Fubini's theorem shows that for any integrable function $f: \Delta\_T\to\bR$ we have
$$\int\_0^T \left(\int\_0^t f(s,t) ds\right)dt=\int\_{\Delta\_T} f(s... | 1 | https://mathoverflow.net/users/20302 | 252780 | 114,566 |
https://mathoverflow.net/questions/252808 | 3 | Let $A,B$ be two K-algebras over a field K.
1. $A$ and $B$ are said to be $Morita $ $equivalent$ if the category $Mod A$ and $Mod B$ are equivalent.
2. $A$ and $B$ are said to be $derived$ $equivalent$ if $\mathcal{D}^b(Mod A)$ and $\mathcal{D}^b(Mod B)$ are equivalent as triangulated categories.
3. Given a minimal ... | https://mathoverflow.net/users/83554 | Whether Morita equivalence holds the following properties? | 1) Yes, Morita equivalence trivially implies derived equivalence. Note that two algebras over an algebraically closed field are Morita equivalent iff their quiver algebras are isomorphic. So compared to derived equivalence, being Morita equivalent is rather easy to check and can be reduced to calculating quivers and ch... | 3 | https://mathoverflow.net/users/61949 | 252809 | 114,575 |
https://mathoverflow.net/questions/252608 | 13 | Suppose $\kappa$ is a measurable cardinal and $j:V\to M$ is the ultrapower by a normal measure on $\kappa$. Let's say, for instance, that $2^\kappa=\kappa^{++}$ (note that this assumption has consistency strength greater than just a measurable cardinal). It is easy to see that $j(\kappa)$ has size $\kappa^{++}$ in $V$.... | https://mathoverflow.net/users/5984 | Cofinality of $j(\kappa)$ for a measurability embedding $j:V\to M$ with critical point $\kappa$ | As Mohammad Golshani remarked, it is possible to control the cofinality of $j(\kappa)$ by iterating the forcing that adds a function $f\colon \kappa \to \kappa$ which is eventually larger than any ground model function.
The conditions of the forcing notion are pairs of the form $(s, g)$ where $s{\in} ^{<\kappa}\kapp... | 10 | https://mathoverflow.net/users/41953 | 252827 | 114,580 |
https://mathoverflow.net/questions/252831 | 6 | Let $f : \mathbb{R} \to \mathbb{R}$ such that the $k^{\rm th}$ derivative of $f$ is strictly positive for every $x \in \mathbb{R}$. Define the forward difference operator to be:
$$\Delta(g,h) = g(x+h) - g(x),$$
and for $h\_1, \ldots , h\_k > 0$,
$$\Delta(g, h\_1, \ldots , h\_k) = \Delta( \Delta(g , h\_1 , \ldots... | https://mathoverflow.net/users/50426 | kth finite difference always positive when kth derivative is? | Yes, this is true. For the function $\Delta(f,h)$ its $(k-1)$-st derivative is strictly positive, since by Lagrange theorem it equals $$f^{(k-1)}(x+h)-f^{(k-1)}(x)=hf^{(k)}(x+\theta h)>0,$$ for some $\theta\in (0,1)$. Then induct on $k$.
| 6 | https://mathoverflow.net/users/4312 | 252832 | 114,582 |
https://mathoverflow.net/questions/252838 | 2 | I look for a simple algorithm for *parametric minimum spanning tree* where the weight of the edge e is $a\_e + \lambda b\_e $. Can we simplify the algorithm in case $b\_e=1$ for all edges?
| https://mathoverflow.net/users/100012 | Parametric minimum spanning tree | I don't know of any "simple" algorithms, but algorithms are known. See for example [Using Sparsification for Parametric Minimum Spanning Tree Problems](http://web.cs.iastate.edu/~fernande/PAPERS/sparsif.pdf) or [Parametric and Kinetic Minimum Spanning Trees](https://www.ics.uci.edu/~eppstein/pubs/AgaEppGui-FOCS-98.pdf)... | 1 | https://mathoverflow.net/users/51668 | 252841 | 114,585 |
https://mathoverflow.net/questions/252815 | 4 | I'm reviewing some notes of mine on regular categories to try and get a better feel for regular, strong, and extremal epimorphisms, and this leads me to ask: is a category $\mathsf C$ regular if and only if $(\mathrm{ExtEpi},\mathrm{Mono})$ is a strong/orthogonal factorization system?
If anyone cares, here's my motiv... | https://mathoverflow.net/users/69037 | Is a category regular iff extremal epis and monomorphisms are a strong factorization system? | It is an old result of Joyal that a category is regular if and only if it is finitely complete and if the classes (strong epi, mono) form a **stable** (orthogonal) factorisation system, that is, if every morphism can be factorised as a strong epi followed by a mono, and the pullback of a strong epi is a strong epi. For... | 8 | https://mathoverflow.net/users/57405 | 252842 | 114,586 |
https://mathoverflow.net/questions/252848 | 5 | If $X$ is a space, we can form $QX=\varinjlim \Omega^n\Sigma^nX$ which is an infinite loop space with homotopy groups $\pi\_i(QX)=\pi^{s}\_i(X)$ the stable homotopy groups of $X.$ But these are the *unstable* homotopy groups of $QX.$
**Q:** Is there any similar expression for the *stable* homotopy groups of $QX$?
J... | https://mathoverflow.net/users/39713 | Stable homotopy groups of $QX$ | The "Snaith splitting" gives the following spectrum level statement: for a pointed connected space $X$, there is a weak equivalence:
$$
\Sigma^\infty\_+ (\Omega^\infty \Sigma^\infty X) \simeq \bigvee\_{n \geq 0} \Sigma^\infty (X^{\wedge n})\_{h\Sigma\_n}
$$
There is also an unbased version:
$$
\Sigma^\infty (\Omega^\in... | 10 | https://mathoverflow.net/users/360 | 252850 | 114,588 |
https://mathoverflow.net/questions/252846 | 2 | Define $\phi(n,x)= \sum\_{m\leq x,\gcd(m,n)=1} 1$, the number of elements in the interval $[1,x]$ that is relatively prime to $n$. $\omega(n)$ is the number of distinct prime factors of $n$.
It's not difficult to show that $\phi(n,x) \geq x \phi(n)/n - 2^{\omega(n)}$ (see [this question](https://mathoverflow.net/ques... | https://mathoverflow.net/users/6886 | Bounds for relative totient function for small values | The fundamental lemma of sieve theory will give you good results here. Put $n\_0 = \prod\_{p|n, p< (\log n)^2} p$, so that $n\_0$ is the product of the small prime factors of $n$. The fundamental lemma gives
$$
\Big| \sum\_{\substack{m \le x\\ (m,n\_0)=1}} 1 - \frac{\phi(n\_0)}{n\_0} x \Big| \ll u^{-u(1+o(1))} x \fra... | 5 | https://mathoverflow.net/users/38624 | 252852 | 114,589 |
https://mathoverflow.net/questions/252849 | 3 | Let $(L,\vee,\wedge,0,1)$ be a lattice with unique least and greatest elements $0$ and $1$, respectively. I'll say that an *antichain* $A$ in $L$ is a subset of $L\setminus\{0\}$ such that for every $a,b\in A$ distinct, $a\wedge b=0$. (This is a set-theorist's antichain.)
I want to consider the following "finite-join... | https://mathoverflow.net/users/16107 | Finite-join antichains in lattices | I will add a few comments to Bjørn Kjos-Hanssen's answer.
>
> 1. Does property (A) have a name in the literature? Is it a studied notion?
>
>
>
The property is a kind of independence property. A set-theorist's antichain $S$ of a lattice $L$ is called **weakly independent** if
whenever $a\_1,\ldots,a\_{n+1}\in ... | 4 | https://mathoverflow.net/users/75735 | 252856 | 114,591 |
https://mathoverflow.net/questions/252813 | 11 | I am interested in learning differential cohomology and differential characters, and am currently studying these [lecture notes](https://arxiv.org/abs/1208.3961) on the subject. I sometimes feel it would be great if I could keep some more good references besides the lecture notes, as it could greatly help me speed up w... | https://mathoverflow.net/users/40386 | References for differential cohomology and differential characters | Bunke's notes are indeed a great source for this material! However, in order to get to the main definitions and properties, he does breeze through a lot of the fundamental prerequisites. I will briefly add to the great list Omar has provided above.
For a classical, non sheaf theoretic approach: Jeff Cheeger, James Si... | 10 | https://mathoverflow.net/users/43687 | 252862 | 114,595 |
https://mathoverflow.net/questions/252517 | 4 | Consider $(\mathbb{C}^2, \omega)$ where $\omega$ is a non-degenerate complex skew-symmetric bilinear form on $\mathbb{C}^2$. Let us write
$V = (\mathbb{C}^2, \omega)$
There are many spaces one can construct from $V$. For instance $Sym^n(V)$ is a vector space endowed with a non-degenerate complex bilinear form (resp... | https://mathoverflow.net/users/81645 | A question on complex semisimple Lie groups and $(\mathbb{C}^2, \omega)$ | I think that the kind of question you are asking is one that was treated by Dynkin back in the 1950s (see Semisimple subalgebras of semisimple Lie algebras. (Russian) Mat. Sbornik N.S. 30(72), (1952), though it is available in the AMS Translation series, and, if I understand correctly, there are more recent articles in... | 6 | https://mathoverflow.net/users/13972 | 252864 | 114,597 |
https://mathoverflow.net/questions/252869 | 7 | Let $S\_t = M\_t + D\_t$ be the sum of a martingale $\left(M\_t\right)\_{t=1,2,\ldots}$ and a predictable process $(D\_t)\_{t=1,2,\ldots}$ such that the variance of the increments of $M$ is uniformly bounded $$\mathbb{E}\left[\left(M\_{t+1}-M\_t\right)^2 | \mathcal{F}\_t\right] \leq v$$
and the increments of $D$ are un... | https://mathoverflow.net/users/82510 | Large deviation/concentration inequality for submartingale | For convenience, suppose that $D\_0 = 0$ and $M\_0=0$. The lower bound $D\_{t+1} - D\_t \ge \Delta$ implies that $D\_{t} \ge \Delta t$ a.s., i.e., $D\_t$ grows at least linearly with $t$. Thus, for any $t \in \mathbb{N}$ we have that
$$
\{ S\_t = M\_t + D\_t \le \alpha t \} \subset \{ M\_t + \Delta t \le \alpha t \} \t... | 3 | https://mathoverflow.net/users/64449 | 252877 | 114,600 |
https://mathoverflow.net/questions/252879 | 11 | We have an elementary sharp lower bound for the regulator of a real quadratic field as a function of the discriminant
$$R\geq \tfrac{1}{2}(\sqrt{d-4}+\sqrt{d})$$
It is sharp because the equality holds infinitely often for $d=x^2+4$.
The problem of finding a good upper seems much more complicated, but there's stil... | https://mathoverflow.net/users/43108 | Upper bounds for regulators of real quadratic fields | [Stephane Louboutin](http://iml.univ-mrs.fr/~loubouti/) has several papers on getting explicit bounds for $L(1,\chi)$, for $\chi$ a character $\pmod q$. They're all of the strength of $1/2 \log q + $ an explicit constant. Some of his results include information on $\chi(2)$ which is sometimes helpful.
The best theor... | 11 | https://mathoverflow.net/users/38624 | 252882 | 114,602 |
https://mathoverflow.net/questions/252698 | 4 | Does the bicategory $\bf Prof$ of categories, profunctors and natural transformations admit all pseudolimits?
By [Kel89, Prop. 5.1] it is enough to show that $\bf Prof$ admits products, cotensors, iso-inserters and equifiers. It's easy to show that products and cotensors exist, but what about inserters and equifiers?... | https://mathoverflow.net/users/7952 | Does $\bf Prof$ admit all pseudolimits? | There is another simple kind of peusdo-limits called inverters, in which one universally inverts a 2-morphism between two parallel 1-morphisms (such pseudo-limits can also be constructed in a simple way from iso-inserters and equifiers). We now claim that ${\bf Prof}$ does **not** have all inverters.
To construct a ... | 3 | https://mathoverflow.net/users/51164 | 252886 | 114,603 |
https://mathoverflow.net/questions/252884 | 7 | I am interested in an explicit description of the principal homomorphism from $SL(2,\mathbb{C})$ to $G$, for each complex semisimple Lie group $G$. Does any one have specific references please? Kostant's original paper is of course great, and contains a lot, but I do not think it contains explicit descriptions for each... | https://mathoverflow.net/users/81645 | Where can I find explicit descriptions of principal $SL(2,\mathbb{C})$s? | You may try exploring the reference
"Lie Algebras, Geometry, and Toda-Type Systems"
by Alexander V. Razumov, Mikhail V. Saveliev, Cambridge University Press.
| 2 | https://mathoverflow.net/users/66131 | 252890 | 114,605 |
https://mathoverflow.net/questions/252878 | 2 | $\newcommand{\sig}{\sigma}$
$\newcommand{\tr}{\operatorname{tr}\_{\eta}}$
$\newcommand{\al}{\alpha}$
$\newcommand{\be}{\beta}$
$\newcommand{\til}{\tilde}$
Let $E$ be a smooth vector bundle over a manifold $M$. Suppose $E$ is equipped with a metric $\eta$ and with a compatible metric connection $\nabla$. Note that $\n... | https://mathoverflow.net/users/46290 | Commuting of exterior derivative and contraction (vector-valued forms) | The general version you are looking for is the following: Suppose that$\Phi:V\to W$ is a vector bundle map between two vector bundles $V$ and $W$ over $M$. Suppose that we have connections $\nabla^V$ and $\nabla^W$, which are compatible with $\Phi$ in the sense that $\nabla^V\_\xi \Phi(s)=\Phi(\nabla^W s)$ for all $s\i... | 6 | https://mathoverflow.net/users/64141 | 252920 | 114,613 |
https://mathoverflow.net/questions/252918 | 2 | Let $ G $ be a locally compact abelian group and $ \omega: G \times G \to \mathbb{T} $ a continuous multiplier on $ G $, i.e.,
\begin{align}
\forall r,s,t \in G: \qquad
\omega(s,t) ~ \omega(r,s + t) & = \omega(r,s) ~ \omega(r + s,t), \\
\omega(0\_{G},r) & = 1 = \omega(r,0\_{G}).
\end{align}
Let $ G^{\omega} $ be the gr... | https://mathoverflow.net/users/50614 | Constructing an explicit extension of a continuous character on a closed subgroup of a certain locally compact abelian group | It is a known fact (e.g. [Baggett & Kleppner 1973](http://www.ams.org/mathscinet-getitem?mr=364537), p. 308) that a (continuous) multiplier $\omega$ on a locally compact abelian group is symmetric iff it is trivial, i.e. $\omega(r,s) = \xi(r)\xi(s)\xi(r+s)^{-1}$ for some (continuous) $\xi:G\to\mathbf T$. Then one check... | 1 | https://mathoverflow.net/users/19276 | 252927 | 114,616 |
https://mathoverflow.net/questions/252943 | 10 | Suppose $f$ is a diffeomorphism from the standard two sphere $S^2$ to itself.
Given any three points $a,b,c$ on a geodesic curve on the sphere $S^2$
and $b$ is the middle points of the geodesic arc $\widehat{ac}$,
$f$ satisfies the following two conditions:
(1) $f(a), f(b), f(c)$ are also on a geodesic.
(2) $... | https://mathoverflow.net/users/42816 | Is this diffeomorphism on standard two sphere an isometry? | Yes, $f$ is an isometry; you only need to know that $f$ is a homeomorpphism, diffeomorphism is not needed. You first show that (by continuity of $f$) that $f$ sends geodesics to geodesics. From this, you conclude that $f$ is the lift of a projective transformation $g: RP^2\to RP^2$. ($f$ preserves anipodality since ant... | 11 | https://mathoverflow.net/users/21684 | 252947 | 114,622 |
https://mathoverflow.net/questions/252897 | 24 | I am currently reading Sullivan's Geometric Topology: Localization, Periodicity, and Galois Symmetry, on page 34 Sullivan claims that the degree 2 map $2:S^4 \to S^4$ induces the map $\left(\begin{smallmatrix} 0 & 1 \\ 0 & 0 \end{smallmatrix}\right)$ on $\pi\_8(S^4) \cong \mathbb{Z}/2 \oplus \mathbb{Z}/2$. I don't see ... | https://mathoverflow.net/users/81484 | Action of the degree 2 map on $\pi_8(S^4)$ | Use Theorem 8.9 from Whitehead's "Elements of Homotopy Theory" on page 537. Take $k=2$, $\beta=\iota\_4$ and $\alpha\in\pi\_8S^4$. Use $[\iota\_4,\iota\_4]=\Sigma\nu'-2\nu\_4$, $h\_0(\nu\_4\circ\eta\_7)=h\_0(\nu\_4)\circ\eta\_8=\iota\_7\circ\eta\_8=\eta\_8$ and $h\_0(\Sigma\nu'\circ\eta\_7)=h\_0(\Sigma(\nu\circ\eta\_6)... | 18 | https://mathoverflow.net/users/54788 | 252951 | 114,624 |
https://mathoverflow.net/questions/252959 | 6 | The irrational rotation on the circle is both a homeomorphism and minimal but is not topologically mixing. The argument-doubling transformation on the circle is topologically mixing but is neither a homeomorphism nor is it minimal.
Is there a topologically mixing and minimal homeomorphism on the circle (or on $\mathb... | https://mathoverflow.net/users/100195 | Is there a topologically mixing and minimal homeomorphism on the circle (or on $\mathbb S^2$)? | There's no topologically mixing self-homeomorphism of the circle. Indeed, pick 3 points, so that the complement of these 3 points consists of 3 intervals $A,B,C$. If $g$ is a self-homeomorphism such that $g(A)$ meets all of $A,B,C$, then it has to contain entirely one of $A,B,C$.
Therefore it is not possible that all... | 11 | https://mathoverflow.net/users/14094 | 252965 | 114,630 |
https://mathoverflow.net/questions/252963 | 8 | In my research on optimization research, I thought about attempts to bridge gradient descent (deterministic standard) via a dynamical system of sorts, meaning if I look at the iterations of Gradient Descent as the discrete orbit of some dynamical system in the phase space (maybe a Poincare map here...) So I was wonderi... | https://mathoverflow.net/users/69446 | Steepest descent/gradient descent as dynamical system | This topic has long history. Here are some references:
1. Bloch, Anthony M. "Steepest descent, linear programming and Hamiltonian flows." Contemp. Math. AMS 114 (1990): 77-88.
2. Brockett, Roger W. [Dynamical systems that sort lists, diagonalize matrices and solve linear programming problems](https://doi.org/10.1109/... | 14 | https://mathoverflow.net/users/30684 | 252971 | 114,633 |
https://mathoverflow.net/questions/252969 | 2 | What is the geometric meaning, for a metric in function of the time that is a solution of the Ricci flow ($g'(t)=-2Ric(t)$), compared to one that is not?
EXPLANATION
I'm interested to understand, being that not all metrics satisfy the equation, $g'(t)=-2Ric(t)$, what differences there are, from the geometrical point ... | https://mathoverflow.net/users/90594 | Geometric meaning of Ricci flow | If $g'(t) \neq -2Ric(t)$, it can be anything else. Ricci flow is in some sense a heat kernel applied to the Riemannian metric tensor, uniformizing it over time; if you start modifying the metric tensor according to whatever you see fit, you will end up with any space you like. You may want to restrict the types of thin... | 6 | https://mathoverflow.net/users/97686 | 252978 | 114,635 |
https://mathoverflow.net/questions/252885 | 9 | I am interested in type theory and proof theory. I have read a lot of papers and books that use the term "computational content" (For example: <https://scholar.google.com/scholar?hl=en&q=%22computational+content+of%22&btnG=&as_sdt=1%2C33&as_sdtp=>) and I have developed an intuitive sense of the meaning, but I have neve... | https://mathoverflow.net/users/2377 | What is the definition of computational content? | Let me respond to your comments to Bjorn and Matt's answers:
Let's say I have a sentence of the form $\forall x\exists y\theta(x, y)$, where "$\theta(x, y)$" is "simple" (say, only bounded quantifiers) - I'm not interested in how hard it is to evaluate $\theta(a, b)$ for a given $a, b$. Then a computational interpret... | 6 | https://mathoverflow.net/users/8133 | 252980 | 114,636 |
https://mathoverflow.net/questions/252974 | 4 | Jacobian varieties of Shimura curves are very interesting objects. For one thing they provide a geometric relation between elliptic curves and modular forms of weight 2 (say we are over $\mathbb{Q}$). So I was wondering what happens when one consider other Shimura varieties, and to start I would be happy to understand ... | https://mathoverflow.net/users/69558 | Albanese of Siegel modular variety $\mathcal{A}_2$ | Take a look at Sankaran, *Fundamental group of locally symmetric varieties,* Manuscripta (1995), and references therein. I think that the toroidal compactification of $\mathcal{A}\_2$ and related spaces have finite fundamental group, and therefore trivial Albanese.
| 5 | https://mathoverflow.net/users/4144 | 252982 | 114,637 |
https://mathoverflow.net/questions/252865 | 4 | Consider the matrix algebra $\mathcal{M}\_n(\mathbb{C})$ (acting on $n$ dimensional space $V$) and let $R$ be subring of matrices of $\mathcal{M}\_n(\mathbb{C})$.
>
> Suppose that any two elements of $R$ have a common eigenvector. Does it follow that there is a common eigenvector for all $R$?
>
>
>
---
No... | https://mathoverflow.net/users/100140 | Subring of matrix algebra with common eigenvectors | In an attempt to prove this, I finally got the following counterexample. Fix $2\le n<k$ (for instance $(n,k)=(2,3)$, in size 5). Consider the subalgebra $R$ of $(n+k)$-square matrices of the form
$$M(A,B,t)=\begin{pmatrix}A & B\\ 0 & tI\_k\end{pmatrix},\quad A\in M\_n(\mathbf{C}),\quad B\in M\_{n,k}(\mathbf{C}).$$
For... | 2 | https://mathoverflow.net/users/14094 | 252983 | 114,638 |
https://mathoverflow.net/questions/252954 | 10 | Take $F$ a number field, $\pi$ a cuspidal automorphic representation of $GL(3, \mathbb{A}\_F).$ Suppose $\pi \cong \pi\otimes \chi.$ Comparing central characters we see that $\chi$ must be cubic.
Now suppose $\chi$ is a cubic character of $\mathbb{A}\_F^\times$. Is it known how to construct $\pi$ such that $\pi \con... | https://mathoverflow.net/users/21252 | Is there a known construction of Cuspidal representations of GL(3) isomorphic to their own twist? | Let $E / F$ be the cyclic cubic extension corresponding to $\chi$ by class field theory. Let $\sigma$ be a generator of $\operatorname{Gal}(E / F)$, and let $\psi$ be a character of $E^\times \backslash \mathbb{A}\_E^\times$ such that $\psi \ne \psi^\sigma$.
Then we can consider the automorphic induction of $\psi$ to... | 10 | https://mathoverflow.net/users/2481 | 252985 | 114,639 |
https://mathoverflow.net/questions/252979 | 2 | Let $F : C \to E$ and $G : D \to E$ be functors. Consider the comma category $(F \downarrow G)$ with its projections $\pi\_1 : (F \downarrow G) \to C$ and $\pi\_2 : (F \downarrow G) \to D$. Using the elementary construction of a comma category, it is easy to prove the following equality: $$F \; \circ \; \pi\_1 \;\; = \... | https://mathoverflow.net/users/55915 | Comma category as weighted limit | The equality you state is not true in general. An object $(c,f,d)$ of the comma category consists of an object $c$ of $C$, an object $d$ of $D$, and a morphism $f \colon Fc \longrightarrow Gd$ in $E$. The projection functors $\pi\_1$ and $\pi\_2$ send this object to $\pi\_1(c,f,d) = c$ and $\pi\_2(c,f,d) = d$, so we ha... | 8 | https://mathoverflow.net/users/57405 | 252991 | 114,641 |
https://mathoverflow.net/questions/252981 | 3 | I believe that this question is an extension of [this other one](https://mathoverflow.net/questions/14964/estimate-population-size-based-on-repeated-observation), and is vaguely reminscent of the German tank problem.
Let us assume I am dealing with an organisation, and on each encouter I deal with a random person. Af... | https://mathoverflow.net/users/100207 | Estimating population size by number of repeats | Here is an answer via [maximum likelihood](https://en.m.wikipedia.org/wiki/Maximum_likelihood_estimation): choose the population size for which the observed distribution would be most likely.
Let $n$ be the population size. There are 77 people encountered, whom we identify as: person 1, the person encountered four t... | 3 | https://mathoverflow.net/users/nan | 253002 | 114,644 |
https://mathoverflow.net/questions/253030 | 3 | Is any known method capable of bounding (either definitely or as an estimate) the maximum number of unknowns $x$, $y\_1$, .. $y\_n$
for which there is guaranteed to be a rational solution of $x^4 \pm y\_1^4 \pm y\_2^4 .. \pm y\_n^4 = k$ for any given positive integer $k$, where the sign options are independent?
The 4... | https://mathoverflow.net/users/10454 | Rational solutions of $x^4 \pm y_1^4 \pm .. y_n^4 = |k|$ for given rational $k$ | (Not a complete answer.)
Rational solutions with negative signs allowed may be found by polynomial formulae, like $$x^4+(2x+17)^4-(x+16)^4-(2x+15)^4=-4080(x+8),$$
this may be an arbitrary rational number, thus 4 fourth powers are enough.
| 7 | https://mathoverflow.net/users/4312 | 253034 | 114,649 |
https://mathoverflow.net/questions/252156 | 7 | Let $(q;q)\_n=(1-q)(1-q^2)\cdots(1-q^n)$ with $(q;q)\_0:=1$. Define a $q$-exponential by
$$e(z;q)=\sum\_{n\geq0}\frac{z^n}{(q;q)\_n}.$$
There is a notion of $q$-Eulerian polynomials, see [the reference](https://arxiv.org/pdf/1201.4941.pdf). I like to introduce **$q$-Eulerian polynomial of type B** via the generating f... | https://mathoverflow.net/users/66131 | $q$-Eulerian type B enjoy symmetry | $\bf{Step~1}.$ $B\_{n,a}(q)=B\_{n,n-a}(q)$.
$\it{Proof}$. Write
$$
\sum\_{n\geq1}\dfrac{B\_n(t,q)}{t^{n/2}}\frac{z^n}{(q;q)\_n}=\frac{e(z/\sqrt{t};q)-e(z\sqrt{t};q)}{\dfrac{e(2z\sqrt{t};q)}{\sqrt{t}}-\sqrt{t}~e(2z/\sqrt{t};q)}\left(\dfrac{e(z\sqrt{t};q)}{\sqrt{t}}+\sqrt{t}~e(z/\sqrt{t};q)\right).
$$
The rhs is symmet... | 3 | https://mathoverflow.net/users/82588 | 253037 | 114,650 |
https://mathoverflow.net/questions/253032 | -2 | Several years ago, I was a trainee in a physics lab where I was supposed to study atomisation in sprays (ensemble of liquid drops). As we did observe that the drops tended to adopt a spherical shape over time, I thought that maybe some kind of curvature diffusion was occurring during this process. What I would like to ... | https://mathoverflow.net/users/13625 | Ricci flow and evolution of the shape of drops in spray | For shapes of liquid drops, it is probably not driven by Ricci flow.
Fluid interfaces with surface tension is better modeled by *mean curvature*, going back to [Young and Laplace](https://en.wikipedia.org/wiki/Young%E2%80%93Laplace_equation); and there is a lot of current work on structure of interfaces when coupled... | 5 | https://mathoverflow.net/users/3948 | 253039 | 114,652 |
https://mathoverflow.net/questions/253035 | 5 | If $G$ is a graph and $G-v$ is linkless for some vertex $v$, is $G$ necessarily knotless?
Of course, one can assume that $v$ is adjacent to every vertex in $G-v$.
Here, a graph is *linkless* if it has an embedding in 3-space with no two linked cycles. And a graph is *knotless* if it has an embedding in 3-space su... | https://mathoverflow.net/users/25980 | Vertex deletion in knotless and linkless graphs | **No**, this is false.
Your question was posed by Adams in 1994, and was disproved by [Foisy](http://onlinelibrary.wiley.com/doi/10.1002/jgt.10114/abstract) in 2003.
| 4 | https://mathoverflow.net/users/2233 | 253048 | 114,657 |
https://mathoverflow.net/questions/253010 | 35 | The classical lore is that $H^1(X,\mathcal F)$ is obstruction to lifting local data to global data. However I don't understand why one would want to compute $H^3(X,\mathcal F), H^4(X,\mathcal F), \cdots$.
---
For complex manifold $X$, $H^1(X,\mathcal O),H^{0,1}(X)$ both represent obstruction to local-to-global li... | https://mathoverflow.net/users/68215 | Why study Higher Sheaf Cohomology? | I think these kinds of lifting-based statements are the wrong place to start. For me these arguments about lifting and such are most convincing as answers to questions like "What are cohomology groups?". For instance $H^1(X,\mathcal O\_X)$ is the tangent space to the moduli space of line bundles on $X$. But that don't ... | 40 | https://mathoverflow.net/users/18060 | 253051 | 114,659 |
https://mathoverflow.net/questions/253042 | 0 | How to calculate $G^{-1}$ efficiently when $G$ is a large matrix knowing that:
\begin{eqnarray}
G=I⊗A + A⊗I
\end{eqnarray}
Or since i'm using $G^{-1}$ to multiply by some other matrix, how to find $X$ from $G.X = B$.
Of course we can just solve the standard equation $A.x = b$ what I was wondering is if there is ... | https://mathoverflow.net/users/100249 | How to calculate the inverse of the sum of kronecker products with the identity matrix | Thanks for the help, indeed it is similar to the Lyapunov equation.
Although for my case, the dimensions of $X$ and $B$ can be different from those of $G$ (that is: not square matrices, but of compatible dimensions).
But I can just solve the Lyapunov equation for each column of $B$ at a time and build my solution in ... | 1 | https://mathoverflow.net/users/100249 | 253055 | 114,661 |
https://mathoverflow.net/questions/252910 | 11 | Suppose that $A\_i$, $i=1,\ldots,m$ are positive semidefinite Hermitian $n\times n$ matrices, with $a\_i^{(j)}$ being the $j$-th eigenvalue of $A\_i$. Let $A\_0=I$.
>
> **QUESTION.** Can we extend the result of M. Fiedler for $m=2$ (*Bounds for the Determinant of the Sum of Hermitian Matrices*, Proc. Amer. Math. S... | https://mathoverflow.net/users/100164 | Determinant of arbitrary sum of positive semidefinite hermitian matrices | The previous answer was incorrect due to an incorrect use of majorization. I decided to replace it by a correct version below for completeness, and to remedy the embarrassing error (though now the answer becomes essentially equivalent to the OPs own answer, so feel free to ignore!).
$\newcommand{\da}{\downarrow} \newco... | 8 | https://mathoverflow.net/users/8430 | 253056 | 114,662 |
https://mathoverflow.net/questions/253064 | 1 | Let $f:X\dashrightarrow Y$ be a birational map between projective varieties not contracting any divisor. Assume that $X$ is smooth, and that $Y$ has at most ordinary singularities at finitely many points.
Is it true then that $K\_X = f^{\*}K\_Y$ ?
| https://mathoverflow.net/users/nan | Pull-back of the canonical divisor via a rational map | There are several issues with this question.
One issue is that if $f$ is a rational map and not a morphism, then you have to say what you mean by $f^\*$. Another issue is that if $K\_Y$ is not at least $\mathbb Q$-Cartier, then $f^\*K\_Y$ doesn't make sense even if $f$ is a morphism. (More precisely, there is no good... | 7 | https://mathoverflow.net/users/10076 | 253066 | 114,665 |
https://mathoverflow.net/questions/253076 | 1 | If $F$ is a non-Archimedean local field, then any inner form of $GL\_n(F)$ is isomorphic to $GL\_m(D)$, for a central $F$-division algebra of dimension $d^2, md=n$. Why is this true? I looked in Platonov-Rapinchuk and couldn't find the answer.
| https://mathoverflow.net/users/64244 | Inner forms of $GL_n(F)$ | This is Proposition 2.17 on page 87 of Platonov-Rapinchuk (for $SL\_n(F)$, instead of $GL\_n(F)$). There is no need to restrict to local fields.
| 5 | https://mathoverflow.net/users/68305 | 253084 | 114,668 |
https://mathoverflow.net/questions/253057 | 4 | If $G$ is a $k-$ group scheme (seeing as a functor) exist a good definition of what is the derived group scheme? (Or a good reference for a good definition). Where derived I'm talking in the sense of group theory.
PS:We may assume that $G$ is of finite type and smooth.
| https://mathoverflow.net/users/47300 | Derived Group Scheme | The following quote, from Pseudo-reductive groups by Conrad, Gabber and Prasad (Definition A.1.14, lightly edited) should answer your question.
>
> The derived group D(G) of a smooth group G of finite type
> over a field k is the unique smooth closed k-subgroup such that (D(G))(K) is
> the commutator subgroup of ... | 6 | https://mathoverflow.net/users/425 | 253089 | 114,670 |
https://mathoverflow.net/questions/252999 | 0 | The classic example of the sum of product binomial coefficients is Vandermonde's Identity:
$$
\sum\_{a\_{1,1}+a\_{1,2}+a\_{1,3}=x}\binom{n}{a\_{1,1}}\binom{n}{a\_{1,2}}\binom{n}{a\_{1,3}}=\binom{3n}{x}
$$
However, I'm interested in the following twist. To motivate the question, consider the simple case in which I'... | https://mathoverflow.net/users/65953 | Constrained sum of product binomial coefficients | I believe I have a working solution that works for general $L>1$:
```
binom.coeff <- function(x,l,size,cvec){
sum(apply(valid.counts(l=l,cvec=cvec,x=x),1,function(j) prod(choose(size,j))))
}
valid.counts <- function(l,cvec,x){
vec = seq(from=0,to=cvec[1])
lst = lapply(numeric(l), function(i) vec)
mat = as.mat... | 0 | https://mathoverflow.net/users/65953 | 253093 | 114,673 |
https://mathoverflow.net/questions/135902 | 17 | NICOLAS-SERRE THEORY
Let $F \in Z/2[[x]]$ be $x+x^9+x^{25}+...$, the exponents being the odd squares, and $V$ be the space spanned by the $F^k$ with $k$ odd. Nicolas and Serre define formal Hecke operators $T\_n$, $n$ odd, on $Z/2[[x]]$ and show:
* 1) $V$ is stable under the $T\_n$. If $n$ is not a square, then $T\... | https://mathoverflow.net/users/6214 | Higher level analogs of Nicolas-Serre theory | Working on this topic for some time,I've come to understand the level 3 and 5 analogues to the characteristic 2 level 1 theory of modular forms developed by Nicolas and Serre. My first results were presented in three arXiv articles: Variations on a Lemma of Nicolas and Serre (1604.02622 [math.NT]), A Hecke algebra atta... | 0 | https://mathoverflow.net/users/6214 | 253094 | 114,674 |
https://mathoverflow.net/questions/247212 | 4 | In eq. (10.35) of his book "Symmetric functions and Hall polynomials" I.G.Macdonald gives the following scalar product, under which Jack polynomials with different partitions $\mu\neq\lambda$ are orthogonal
$$\langle J^\alpha\_\lambda(z\_1,z\_2),J^\alpha\_\mu(z\_1,z\_2)\rangle'\_2=\frac{1}{2}\int\_T J^\alpha\_\lambda... | https://mathoverflow.net/users/37039 | Normalization of Jack polynomial integral-scalar product? | Yes, my friend. Take $J\_\lambda^{(\alpha)}$ in the J-normalization. Let $n$ be the number of variables (which for you is $2$). Let $\lambda'$ denote the conjugate partition to $\lambda$.
Define
$$
C\_\lambda^{(\alpha)}=\prod\_{(i,j) \in \lambda}(\alpha(\lambda\_i-j)+\lambda\_j'-i+1)(\alpha(\lambda\_i-j)+\lambda\_j'-i+... | 5 | https://mathoverflow.net/users/100284 | 253105 | 114,676 |
https://mathoverflow.net/questions/253103 | 2 | I have a question about an estimate of the surface area of a set.
Let $B(r)$ denotes the open ball of $\mathbb{R}^{d}$ centered at origin with radius $r>0$. Let $F:\mathbb{R}^{d} \to \mathbb{R}^{d}$ be a Lipschitz continuous function and define
\begin{equation}
\text{Lip}(F)=\inf\{L>0:|F(x)-F(y)| \le L|x-y|,x,y \in ... | https://mathoverflow.net/users/68463 | On the surface area of a set | There is no reason to assume that the boundary of the image is contained in the image of the boundary, even in the case $n=1$; consider for example the function $y=x^2$ on $[-1,1]$.
| 1 | https://mathoverflow.net/users/28128 | 253114 | 114,677 |
https://mathoverflow.net/questions/253070 | 4 | I'm interested in knowing more about the question if $f(\pi)$ is rational or not, where $f$ is some well-known function. For example, $\cos(\pi) =-1$ is rational, while ${e}^{\pi}$ is irrational as shown [here](http://www.wolframalpha.com/input/?i=e%5E%28pi%29++is+irrational+number) by WolframAlpha.
**My question her... | https://mathoverflow.net/users/51189 | Is it possible to know if $\log(\pi)$ is irrational or not since the $\log$ function is the inverse of the $\exp$ function? | The irrationality of $\log \pi$ is an open problem (see for example [this](https://arxiv.org/abs/0908.3253) recent paper).
It is expected to be transcendental (page 34 of [this slides](https://webusers.imj-prg.fr/~michel.waldschmidt/articles/pdf/TNTOpenPbs.pdf) by Michel Waldschmidt), and in fact this follows from Sc... | 15 | https://mathoverflow.net/users/43108 | 253127 | 114,680 |
https://mathoverflow.net/questions/253130 | 2 | This is a continuation of the question about [Minimal $T\_0$-spaces](https://mathoverflow.net/questions/202149/minimal-t-0-spaces) .
Let $X\neq \emptyset$ be a set and let $\text{Top}(X)$ denote the lattice of all topologies on $X$ and let $\tau\in\text{Top}(X)$.
Do we have $$\tau = \bigcap\{\sigma \in \text{Top}(X... | https://mathoverflow.net/users/8628 | Is every topology the intersection of the $T_0$-topologies containing it? | I've answered the respective full question around 1982 or even many years earlier. Possibly, the result was well known "always", before me.
The answer is contained in one of my OM Answers from a different thread:
**THEOREM**
* Every topology is an intersection of singular spaces.
* Every topology $\ T\ $ is the i... | 3 | https://mathoverflow.net/users/8385 | 253143 | 114,687 |
https://mathoverflow.net/questions/253075 | 2 | Let $k$ be an integer and disjoint closed sets $E,F\subset\mathbb{R}^2$. Consider the *Zolotarev number*
$$Z\_k(E,F):= \inf\_{r\in\mathcal{R}\_{k,k}}\frac{\sup\_{z\in E}|r(z)|}{\inf\_{z\in F}|r(z)|},$$
where $\mathcal{R}\_{k,k}$ is the space of degree $(k,k)$ rational functions.
In 1969, Gonchar showed that ([p... | https://mathoverflow.net/users/2011 | An upper bound on Zolotarev numbers | T. Ganelius has shown inequalities of the type
$$
Z\_k(E,F)\leq const. e^{-k/C(E,F)},\qquad k=1,2,\ldots,\qquad (\*)
$$
under additional assumptions on $E$ and $F$, see
[1] T. Ganelius, Some extremal functions and approximation. Fourier analysis and approximation theory (Proc. Colloq., Budapest, 1976), Vol. I, pp. 37... | 1 | https://mathoverflow.net/users/89429 | 253144 | 114,688 |
https://mathoverflow.net/questions/253136 | 0 | Let $A, B, C, D \in \mathbb{R^\*\_+}$.
Is it possible to solve
$$
\max\_{ \substack{0 \leq x\leq A \\ 0\leq y\leq B}} \frac{1+x+y}{(1+Cx)(1+Dy)}
$$
The KKT conditions give for an extrema $(x^\*,y^\*)$
If $\frac{1}{C} > (1+y^\*)$ then $x^\*=A$ otherwise $x^\*=0$,
If $\frac{1}{D} > (1+x^\*)$ then $y^\*=B$ otherwi... | https://mathoverflow.net/users/nan | Optimization function of two variables | The gradient is $0$ iff $[x,y] = [1/D-1, 1/C-1]$, at which point the objective value is $1/(C+D-CD)$. If this is in the feasible region $[0, A] \times [0,B]$, it may be optimal. Compare to the optimal values on the edges $\{0\} \times [0,B]$, $\{A\} \times [0,B]$, $[0,A] \times \{0\}$, $[0,A] \times \{B\}$.
| 0 | https://mathoverflow.net/users/13650 | 253160 | 114,693 |
https://mathoverflow.net/questions/250368 | 5 | Let $G$ be a reductive group over $\mathbb{Q}\_p$. Let $X\_G$ be a locally symmetric space associated to the group $G$, and let $\partial X\_G$ be the Borel-Serre boundary of $X\_G$. The space $\partial X\_G$ has a stratification
by locally symmetric spaces $X\_P$ associated to parabolic subgroups of $X\_G$.
Assuming I... | https://mathoverflow.net/users/93798 | Eisenstein cohomology - explicit computation and relation to Franke's trace formula | After rereading section 4 of *On the Eisenstein Cohomology of Arithmetric
Groups* by Schwermer and Li, I think I understand this now.
If $\lambda$ is sufficiently regular, then there is an isomorphism
$$H^i(X\_P,V\_{\lambda}) \cong \bigoplus\_{w \in W^M} H^{i+\dim \mathfrak{n}-\ell(w) }(X\_M,V\_{w(\lambda+\rho)-\rho}... | 3 | https://mathoverflow.net/users/93798 | 253164 | 114,696 |
https://mathoverflow.net/questions/253119 | 5 | Is there a fiber bundle $(E,B, \mathbb{R}^{n})$, with typical fiber $\mathbb{R}^{n}$, such that there is no any $n$-dimensional vector bundle structure on the pair $(E,B)$? That is there is no a continuous map $p:E \to B$ such that the triple $(E,B,P)$ would be a $n$ dimensional vector bundle.
| https://mathoverflow.net/users/36688 | A $\mathbb{R}^{n}$ -fiber bundle which do not admit a n-dimensional vector bundle structure | Such a fibre bundle does not exists if you suppose that it is endowed with a differentiable structure. Stewart has shown that the group of diffeomorphisms of $R^n$ retract to $O(n)$. So every $Diff(R^n)$-bundle has an $O(n)$-reduction.
Stewart, T. E. (1960). On groups of diffeomorphisms. Proceedings of the American M... | 10 | https://mathoverflow.net/users/80891 | 253167 | 114,698 |
https://mathoverflow.net/questions/253168 | 5 | Let $K/\mathbb{Q}$ be a degree $4$ number field. Is it known how to determine whether the Galois closure of $K$, say $K'$, contains the 4-th roots of unity?
In the cubic case, there is a succinct answer: the Galois closure $E'$ of a cubic field $E/\mathbb{Q}$ contains the third roots of unity if and only if $E$ is a ... | https://mathoverflow.net/users/10898 | Which quartic fields contain the 4th roots of unity in their Galois closure? | As explained in the comments, the only non-trivial case is where $K/\mathbb{Q}$ has Galois closure $K'/\mathbb{Q}$ with $\operatorname{Gal}(K'/\mathbb{Q})$ isomorphic to $D\_4$. I will do this case here since it is too long for a comment.
In that case, since all proper subgroups of $D\_4$ are contained in a subgroup ... | 7 | https://mathoverflow.net/users/17907 | 253170 | 114,699 |
https://mathoverflow.net/questions/253141 | 7 | Every reference I've seen on modular forms seems to jump from the general definition of a modular form for congruence subgroups to studying modular forms just for $\Gamma\_0(N)$.
Once upon a time, I saw something like
$$M\_k(\Gamma(N)) \cong\bigoplus\_{\chi\mod N} M\_k(N^2,\chi)$$
where $M\_k(\Gamma(N))$ is the v... | https://mathoverflow.net/users/88840 | Why does it suffice to study modular forms for $\Gamma_0(N)$? | Conjugating $\Gamma(N)$ by $\scriptstyle\begin{bmatrix} 1 & 0 \\ 0 & N \end{bmatrix}$
$$\scriptstyle\begin{bmatrix} 1 & 0 \\ 0 & N \end{bmatrix} \displaystyle\Gamma(N)\scriptstyle\begin{bmatrix} 1 & 0 \\ 0 & 1/N \end{bmatrix}
\scriptstyle \ \ = \ \ \underbrace{\left\{\ \scriptstyle\begin{bmatrix} aN+1 & b \\ cN^2 & d... | 5 | https://mathoverflow.net/users/84768 | 253173 | 114,701 |
https://mathoverflow.net/questions/252629 | 3 | I am interested in knowing if there are any types of graphs that are very sparse, perhaps consisting of just connections between paths and cycles, and for which $k$-coloring is $\mathsf{NP}$-hard for some $k \in \mathbb{N}$.
I know a single cycle and path are very easy to color. Are there types of graphs that, when c... | https://mathoverflow.net/users/82949 | Sparse graphs that are hard to color | If we consider very sparse graphs to be subcubic graphs (i.e. graphs with maximum degree at most 3) as suggested in the comments, the answer is that vertex coloring is never hard. Coloring of subcubic graphs can be done in polynomial time.
Let $G$ be a connected subcubic graph. By [Brooks' Theorem](https://en.wikiped... | 2 | https://mathoverflow.net/users/51668 | 253178 | 114,703 |
https://mathoverflow.net/questions/253175 | 2 | Let $p\ge 5$ be prime. For every $j\in\mathbb{F}\_p$, there are at most 6 twists of any elliptic curve over $\mathbb{F}\_p$ with $j$-invariant $j$, and in general only two twists.
Is there a formula which gives the point counts $\#E(\mathbb{F}\_p)$ (equivalently, the traces of Frobenius $a\_p(E)$) of the finitely man... | https://mathoverflow.net/users/88840 | twists of elliptic curves over finite fields | If $p>2$ and $j \neq 0,1728$ then the two twists are of the form $E: y^2=f(x)$ and
$E\_d: d{y}^2=f(x)$ with $d \notin {\mathbb{F}\_p^{\*}}^{2}$, so $a\_{E\_d} = -a\_E$, as follows: Denote $\chi(0) = 0$, $\chi(x) = 1$ if $x \in {\mathbb{F}\_p^{\*}}^{2}$, $\chi(x) = -1$ otherwise. Then $\#{E(\mathbb{F}\_p)} = 1 + \Sigma\... | 3 | https://mathoverflow.net/users/59248 | 253179 | 114,704 |
https://mathoverflow.net/questions/253090 | 28 | I am supposed to give a talk about the Riemann-Roch theorem to a seminar of first and second year graduate students. I want to do Riemann-Roch for compact Riemann surfaces, but I am open to perhaps doing the version for projective curves.
I assume the crowd knows little algebraic geometry. I assume however, knowledge... | https://mathoverflow.net/users/nan | Elementary Proof of Riemann-Roch for Compact Riemann Surfaces | RRT
There is a big difference in difficulty between the compact Riemann surface case and the projective curve case, for reasons already mentioned. Namely a projective curve comes equipped with a large supply of meromorphic functions, but the proof that they exist for a compact Riemann surface is a major step.
I sugge... | 38 | https://mathoverflow.net/users/9449 | 253187 | 114,707 |
https://mathoverflow.net/questions/253193 | 39 | I'd like to formulate an abstract definition of convex sets: a set $K$ is *convex* if it is endowed with a ternary operation $K\times[0,1]\times K\to K$, written $(x:t:y)$, satisfying axioms
* $(x:0:y)=(x:t:x)=x$
* $(x:t:y)=(y:1-t:x)$
* $(x:t:(y:\frac ut:z))=((x:\frac{t-u}{1-u}:y):u:z)$
The axioms imply that, for e... | https://mathoverflow.net/users/10481 | Abstract definition of convex set | There has been a bunch of work along these lines, and I think the idea has been rediscovered several times. I suggest looking at the papers of [Anna Romanowska](http://www.mini.pw.edu.pl/~aroman/www/?Publikacje), who refers to them as "barycentric algebras", to get an idea of what's known. Her book with Smith, "Modes",... | 25 | https://mathoverflow.net/users/3711 | 253194 | 114,709 |
https://mathoverflow.net/questions/253186 | 2 | A calculation of the dark matter density profile in a dissipative dark matter model leads to the integral
$$f(x,\theta)=\int\limits\_0^\infty\frac{y\,e^{-y}\,dy}{\sqrt{x^4+y^4+2x^2y^2\cos{2\theta}}}.$$
Is it possible to calculate this integral in a closed form? What is its limit in the $x\ll 1$ case?
| https://mathoverflow.net/users/32389 | Closed form of an dark matter related Integral | Denote $y=tx$, we get a Laplace transform $$f(x,\theta)=\int\_0^{\infty}\frac{t}{\sqrt{t^4+2t^2\cos 2\theta+1}}e^{-xt}dt.$$
Integral over $[0,1]$ is bounded uniformly in $x$, and for $[1,\infty)$ we have
$$
\int\_1^{\infty}\frac{t}{\sqrt{t^4+2t^2\cos 2\theta+1}}e^{-xt}dt=
\int\_{1}^\infty \frac1t e^{-xt}dt+\int\_1^{\in... | 4 | https://mathoverflow.net/users/4312 | 253196 | 114,710 |
https://mathoverflow.net/questions/252552 | 2 | Let
* $T>0$
* $(\Omega,\mathcal A,\operatorname P)$ be a probability space
* $(\mathcal F\_t)\_{t\in[0,\:T]}$ be a right-continuous filtration on $(\Omega,\mathcal A)$
* $B$ be a Brownian motion on $(\Omega,\mathcal A,\mathcal F,\operatorname P)$
Note that $$\mathcal E\_0:=\left\{\Phi\in\mathcal L^2(\operatorname P... | https://mathoverflow.net/users/91890 | How can we define the Stratonovich integral rigorously? | I think there might be a typo in your write up. Take a look at either chapter 8 from the book <http://press.princeton.edu/titles/7566.html> (<http://www-math.mit.edu/~dws/ito/ito8.pdf>). I will try to give a brief explanation, hope it helps.
Essentially the reason for defining Ito's integral of the process adapted p... | 1 | https://mathoverflow.net/users/57648 | 253201 | 114,711 |
https://mathoverflow.net/questions/253195 | 12 | There are several formulations and consequences of the Riemann Hypothesis over Function Fields (RH, from now on). I am interested in the logical implications between those, and in proofs\references for those implications (which are surely known to the experts).
Personally, I would be most satisfied with proofs which... | https://mathoverflow.net/users/31469 | Related Forms for the Riemann Hypothesis over Function Fields | 1 is not just morally equivalent with 2 - the equivalence is really easy. As you say, there is a correspondence between smooth, projective, geometrically connected algebraic curves over $\mathbb F\_q$ and function fields with constant field $\mathbb F\_q$. The equivalence sends the set effective divisors on the curve (... | 11 | https://mathoverflow.net/users/18060 | 253221 | 114,716 |
https://mathoverflow.net/questions/253233 | 3 | Let $G$ be a connected reductive group defined over a perfect field $F$. The *split component* $A$ of $G$ is the unique maximal $F$-split subtorus of the radical of $G$. For an algebraic group $H$ over $F$, let $X(H)$ denote the abelian group of rational characters $H \rightarrow \mathbf{G}\_m$, and let $X(H)\_F$ be th... | https://mathoverflow.net/users/38145 | Rational Characters of a reductive group have the same rank as split component | This is much easier than it looks. The point is that any reductive group $G$ is isogenous to the product of its radical, which is its centre $Z(G)$, and its commutator subgroup, which is a semisimple group. Since Hom into $\mathbb{R}$ will kill all torsion, it is sufficient to prove the statement when $G$ is either sem... | 6 | https://mathoverflow.net/users/2481 | 253237 | 114,718 |
https://mathoverflow.net/questions/253184 | 1 | Let $X,Y$ be continuous random variables with $X$ defined over $\mathcal{A}$, and let $f: \mathcal{A} \to \mathcal{A}$, $g: \mathcal{A} \to \mathcal{A}$ be any functions. Is it true that
$$
I(Y;f \circ g(X) ) \leq I(Y; f(X))
$$
where $I(\cdot\,; \cdot)$ denotes mutual information. Note that
data processing inequali... | https://mathoverflow.net/users/nan | Data processing inequality | There is no reason for that. For instance, one can choose the functions $f$ and $g$ in such a way that $g$ is a bijection on the range of $X$, whereas $f$ is constant on the range of $X$ and is a bijection on the range of $g(X)$. Then $I(Y,f(X))=0$, whereas $I(Y,f\circ g(X))=I(Y,X)$.
| 2 | https://mathoverflow.net/users/8588 | 253238 | 114,719 |
https://mathoverflow.net/questions/253248 | 6 | With $\Bbb K$ a commutative rings there a way to characterize $A,B\in\Bbb K^{n\times n}$ with $$Per(AB)=Per(A)Per(B)?$$
John provides a reasonable concept coverage multiplicative property.
How about if I seek $Per(A+B)=Per(A)+Per(B)$ instead of $Per(AB)=Per(A)Per(B)$?
| https://mathoverflow.net/users/nan | On additive/multiplicative property of permanent | Searching the literature brings up a concept known as permanent groups. A *permanent group* is a group of nonsingular matrices on which the permanent is multiplicative. In was conjectured by M. Marcus in [Permanents](https://www.maa.org/sites/default/files/pdf/upload_library/22/Ford/MarvinMarcusHenrykMinc.pdf) that the... | 9 | https://mathoverflow.net/users/51668 | 253255 | 114,727 |
https://mathoverflow.net/questions/253223 | 0 | When reading the paper
E. Carlen, J. Geronimo & M. Loss: SIAM J. MATH. ANAL., vol. 40, no. 1, 327-374
I found an argument like the following.
Given an bounded and self-adjoint linear operator $K: L^2(B) \rightarrow L^2(B)$, where $B$ is the unit ball in $\mathbb R^3$ equipped with the Lebesgue measure. Suppose ... | https://mathoverflow.net/users/44590 | The eigenfunctions of an operator commuting with all rotations. | In the paper you mention at page 345 it is shown that the operator $K$ is selfadjoint and, for every $N\geq 0$, the operator $K$ preserves the space $\newcommand{\eP}{\mathscr{P}}$ $\eP\_N$ of polynomials of degree $\leq N$. This space has an $SO(3)$-invariant orthogonal decomposition
$$\eP\_N= \bigoplus\_{k+2\ell\le... | 1 | https://mathoverflow.net/users/20302 | 253257 | 114,728 |
https://mathoverflow.net/questions/253176 | 9 | Fix $0<h\_1<h\_2<h\_3<1$ reals. All matrices below are $3\times3$ real.
Suppose the sequence of matrices $M(n)$ are symmetric positive definite and these converge (point-wise) to a symmetric positive definite matrix $M$ (point-wise). Assume that the eigenvalues of $M(n)$ converge to that of $M$.
>
> **QUESTION.*... | https://mathoverflow.net/users/66131 | convergence of 2nd eigenvalue | Let $a = n^{h\_3 - h\_1} \to + \infty$ and $b = n^{h\_1 - h\_2} \to 0$.
Note that $ab = n^{h\_3 - h\_2} \to +\infty$, but slower than $a$.
If $$M(n) = \pmatrix{m\_{11}(n) & m\_{12}(n) & m\_{13}(n)\cr m\_{12}(n) & m\_{22}(n) & m\_{23}(n)\cr
m\_{13}(n) & m\_{23}(n) & m\_{33}(n)}$$ the characteristic polynomial of $A(n... | 5 | https://mathoverflow.net/users/13650 | 253258 | 114,729 |
https://mathoverflow.net/questions/253249 | 1 | Consider two matrices $A, B\in\mathcal{M}\_n(\mathbb{C})$, such that $A, B$ has no common eigenvectors. Is it true that for some nonzero $t\in\mathbb{C}$, matrix $A+tB$ is similar to diagonal matrix $\text{diag}(a\_1, a\_2, \ldots, a\_n)$, where $a\_i\not = a\_j, \forall i\not= j$?
| https://mathoverflow.net/users/100140 | Pairs of matrices | It's false for every $n\ge 3$.
Consider $A\_n,B\_n$ $n\times n$ square matrices with no common eigenvalue (these exist for $n=0$ and $n\ge 2$). Then
$$A=\begin{pmatrix} A\_2 & 0 & 0\\ 0 & A\_2 & 0\\ 0 & 0 & A\_{n-4}\end{pmatrix},\qquad B=\begin{pmatrix} B\_2 & 0 & 0\\ 0 & B\_2 & 0\\ 0 & 0 & B\_{n-4}\end{pmatrix}$$
w... | 4 | https://mathoverflow.net/users/14094 | 253260 | 114,730 |
https://mathoverflow.net/questions/253246 | 5 | In analytic number theory, for example as in ternary Goldbach problem via circle method, when one has to deal with exponential sums over primes often people use von Mangoldt function or log weight. In other words, instead of
$$
\sum\_{p < X}e^{2 \pi i f(p)}
$$
one considers
$$
\sum\_{n < X} \Lambda(n) e^{2 \pi i f(n)... | https://mathoverflow.net/users/84272 | Reason for putting log weight for exponential sums over primes? | Ultimately, it is because of the fundamental theorem of arithmetic, which expresses each natural number $n$ as a product of the primes dividing it:
$$ n = p\_1^{a\_1} \dots p\_k^{a\_k}.$$
Taking logarithms to make this multiplicative formula additive, we conclude that
$$ \log n = \sum\_{p,j: p^j|n} \log p $$
or in term... | 10 | https://mathoverflow.net/users/766 | 253262 | 114,732 |
https://mathoverflow.net/questions/253259 | 4 | Consider the set of coprime integer pairs $\mathcal{C} \subset \mathbb{Z}^2$ and the circle of radius $r$ centered at the origin. The function
$$f(r) = \min\_{(m,n) \in \mathcal{C}} \bigl| \sqrt{m^2 + n^2} - r \bigr|$$
represents the minimum Euclidean distance between $\mathcal{C}$ and the circle of radius $r$.
I am ... | https://mathoverflow.net/users/100355 | Upper-bounding the min-distance between a circle and the set of coprime integer pairs | Here's an easy way to show that $f(r)$ goes to zero. Indeed it shows that $f(r) \le C r^{-1/2+\epsilon}$ for some constant $C$ and any $\epsilon > 0$.
We start with the observation that given any number $n$,
$$
\sum\_{\substack{k\le x \\ (k,n) =1}} 1 = \frac{\phi(n)}{n} x + O(2^{\omega(n)}),
$$
which follows from... | 6 | https://mathoverflow.net/users/38624 | 253281 | 114,735 |
https://mathoverflow.net/questions/252067 | 7 | Let $\mathcal M\_X(r,0,K\_X)$ be tha moduli space of semistable Higgs bundles over a smooth irreducible algebraic curve over $\mathbb C$. And let $E$ be a stable vector bundle of rank $r$ and degree $0$. Then clearly we have an injection $$H^0(X,E\otimes E^\*\otimes K\_X)\rightarrow \mathcal M\_X(r,0,K\_X)$$
>
> Is... | https://mathoverflow.net/users/66528 | Higgs bundles and stable vector bundle | This is a partial answer to my question!
We can deduce easly that if this map is closed immersion, then $E$ is very stable vector bundle (because in this case, the Hitchin map is finite). Hence if $E$ is stable non very stable vector bundle (there exist of course such vector bundles) then we deduce that the above map... | 3 | https://mathoverflow.net/users/66528 | 253302 | 114,740 |
https://mathoverflow.net/questions/253303 | 0 | For an $n$-dimensional space $V$ with a positive metric $g$, we can construct the Clifford algebra $Cl(V)$ and its representation space $S$, i.e.
$$c(V):S\to S,~\forall v\in V.$$
**Question**: Under what condition, we can find an another Clifford structure $\tilde{Cl}(V)$ acting on $S$, such that
these two Clifford ... | https://mathoverflow.net/users/95296 | New Clifford structure | I'm not sure what you're asking. You might be asking:
**Potential interpretation 1:** Can we find a second embedding $\tilde c: V \to \mathrm{Cliff}(V)$ so that the images of $c$ and $\tilde c$ commute with each other?
**Answer 1:** No (unless $\dim V = 0$). The graded center of $\mathrm{Cliff}(V)$ is just the scal... | 4 | https://mathoverflow.net/users/78 | 253308 | 114,742 |
https://mathoverflow.net/questions/253313 | 10 | By a kummer surface, I mean a quotient of two-dimensional complex torus by multiplication by $-1$. It is a K3 surface and known to be simply connected.
But it is not clear to me why it is simply-connected from its construction, which is a quotient of a surface with infinite fundamental group.
Is there a direct explan... | https://mathoverflow.net/users/38823 | Why is a Kummer surface simply-connected? | In the paper "The homology of Kummer manifolds" E. Spanier proves among other things that the fundental group of the resolution $K$ of singularities of the quotient
$$(S^1\times S^1)^{\times n}/(\mathbb{Z}/2)$$
is simply connected (theorem 1). The proof is elementary and can be easily adapted to the singular quotient i... | 13 | https://mathoverflow.net/users/27816 | 253317 | 114,744 |
https://mathoverflow.net/questions/253306 | 6 | Let $\beta: H^n(X, \mathbb{Z}\_2)\to H^{n+1}(X, \mathbb{Z})$ be the Bockstein homomorphism. Is it possible to define a cohomology operation $f: H^{n+1}(X, \mathbb{Z})\to H^{n+k+1}(X, \mathbb{Z})$ such that
\begin{eqnarray\*}\beta \text{Sq}^k=f\beta\end{eqnarray\*}
where $\text{Sq}^k: H^n(X, \mathbb{Z}\_2)\to H^{n+k}(X... | https://mathoverflow.net/users/85722 | An integral cohomology operation related to Steenrod square? | From the Adem relations we get $Sq^1Sq^{k-1}=(k-2)Sq^k$. It follows that for $k$ odd we have $Sq^k=Sq^1Sq^{k-1}$. Note that we have $Sq^1=\rho\beta$, where $\rho$ is mod 2 reduction, so if $k$ is odd $\beta Sq^k=\beta Sq^1Sq^{k-1}=\beta\rho\beta Sq^{k-1}=0$ since $\beta\rho\simeq \ast$. Therefore we are forced to take ... | 8 | https://mathoverflow.net/users/54788 | 253318 | 114,745 |
https://mathoverflow.net/questions/253269 | 5 | The binomial Sheffer sequence of Bell / Touchard / exponential polynomials $\phi\_n (x) $, whose coefficients are the Stirling numbers of the second kind, have the representation
$(RL)^n=\phi\_n (:RL:) $
where $R $ and $L $ are the raising and lowering operators of any sequence of Sheffer polynomials and $:RL:^n=R... | https://mathoverflow.net/users/12178 | A uniqueness of the Stirling numbers? | Well, upon further reflection, the answer is that any pair of Sheffer sequences that are an umbral inverse pair suffice to construct such an arrangement. Let $u\_n (x)$ and $v\_n (x)$ be such a pair, i.e.,
$u\_n (v.(x)) = x^n=v\_n(u.(x))$.
Then
$K^n = u\_n(v.(K))$ and $B\_n(K) = v\_n (K) $.
So, the Stirling pa... | 0 | https://mathoverflow.net/users/12178 | 253343 | 114,753 |
https://mathoverflow.net/questions/253309 | 7 | I need to use the following result (that I'm pretty sure is true):
>
> **Theorem.** Let $Y$ be a compact complex manifold and $B \subset Y$ be a connected submanifold of codimension one. Then isomorphism classes of connected analytic covers of degree $n$ $$f \colon X \longrightarrow Y,$$ branched at most over $B$, ... | https://mathoverflow.net/users/7460 | Monodromy representations and branched covers | For a modern treatment of the Grauert Remmert argument see Chapter 4 in the book Several Complex Variables vol 7 by Dethloff and Grauert . For an alternate proof using resolution of singularities see SGA 1 page 255 in the arxiv version . The idea of all these proofs is to restrict to the local case . In the case you ar... | 5 | https://mathoverflow.net/users/4696 | 253344 | 114,754 |
https://mathoverflow.net/questions/253335 | 3 | Let $c\_i$ be the number of complete subgraphs of size $i$ in a graph $G$. I learned the following result:
If $G$ has $n$ vertices and is connected and chordal, that is, $G$ does not have induced cycles of length at least 4, then
$$
\sum\_{i=1}^n (-1)^{i+1} c\_i = 1
$$
See for example Thm 4.1 (together with the fa... | https://mathoverflow.net/users/3736 | Clique numbers of chordal graphs | Here is another argument. A chordal graph has a *perfect elimination
ordering* $v\_1,\dots,v\_n$ of its vertices. This means that the
neighbors among $v\_1,\dots,v\_{i-1}$ of $v\_i$ form a clique $C\_i$ . See
for instance <https://en.wikipedia.org/wiki/Chordal_graph>. If $C\_i$ has
$p\_i$ vertices, then we obtain ${p\_... | 3 | https://mathoverflow.net/users/2807 | 253346 | 114,756 |
https://mathoverflow.net/questions/253349 | 6 | I would like to generate random graphs that might be geometric graphs in some
(unknown) dimension. So I would like every triangle in the graph to satisfy the
triangle inequality on its (random) edge lengths/weights.
I need something akin to the Erdős/Rényi model such as,
"[The weighted random graph model](https://arxi... | https://mathoverflow.net/users/6094 | Generate random graphs that satisfy the triangle inequality | I am not sure I understand the issues: First you generate an ER (or your favorite model) random graph. The constraints that the edge lengths are in $[0, 1]$ and satisfy all possible triangle inequalities defines a polytope in $\mathbb{R}^E,$ and you are just trying to find a uniform random point in the polytope, which ... | 7 | https://mathoverflow.net/users/11142 | 253351 | 114,760 |
https://mathoverflow.net/questions/253342 | 2 | Consider a generic diffusion of the form
$$dX\_t=f(t,X\_t)dt+dB\_t,$$
where $f$ is some *nice* function and $B\_t$ is a standard Brownian motion.
The marginal distributions of the integrals
$$I:=\int\_0^TB\_t~dt\qquad J:=\int\_0^TX\_t~dt$$
can in principle be computed with fairly straightforward methods:
$I$ is Gauss... | https://mathoverflow.net/users/50406 | Joint distribution of integrals of diffusion and driving noise | Consider the SDE:
$$
\begin{cases}
d X\_t = f(t,X\_t) dt + d B\_t \\
d Y\_t = d B\_t
\end{cases}
$$
The infinitesimal generator of the process $(X\_t,Y\_t)$ is given by:
$$
L\_t g(x,y) = f(t,x) \partial\_x g(x,y) + \frac{1}{2} \partial\_{xx} g(x,y) + \partial\_{xy} g(x,y) + \frac{1}{2} \partial\_{yy} g(x,y)
$$ Let $t\... | 3 | https://mathoverflow.net/users/64449 | 253355 | 114,761 |
https://mathoverflow.net/questions/252784 | 5 | For a fixed $k \geq 2$, are there infinitely many *non-trivial* coprime integer pairs $(x,y)$ for which $xy$ divides $(x+y)^k-1$? By *trivial* I mean parametrized pairs $(x,y)$ of the form $(-1,-1),(x,1),(1,y),(x,1-x),(x,1 - x^n),(1-y^n,y),(x,p(x)),(p(y),y)$, where $p(x)$ is a polynomial factor of $\frac{x^k-1}{x-1}$. ... | https://mathoverflow.net/users/22733 | On the divisibility of $(x+y)^k - 1$ by $xy$ | After doing some research online I found out that there's a lot of theory behind this subject and it was studied by several authors (see the references below).
As it was pointed out by @Geoff Robinson, the question essentially reduces to finding all coprime integer pairs $(x,y)$ such that $x \mid 1-y^k$ and $y \mid 1... | 3 | https://mathoverflow.net/users/22733 | 253362 | 114,765 |
https://mathoverflow.net/questions/253316 | 1 | Let $V$ be a separable Hilbert space and define $X=L^2(0,T;V)$. Then $u\_m\to u$ weakly in $X$ means
>
> for every $v\in X'=L^2(0,T;V')$
> $$
> \int\_0^T\langle v(t),u\_m(t)\rangle\ dt\to\int\_0^T\langle v(t),u(t)\rangle\ dt\tag{1}
> $$
> where $\langle\rangle$ is the dual pair $(V',V)$.
>
>
>
If one assumes... | https://mathoverflow.net/users/nan | Weak convergence in vector-valued Hilbert space | A subset $A\subset X'$ is called *total* if it is not contained in any proper closed subspace.
Note that $A$ is total in $X'$ iff its anihilator in $X$ is trivial.
Your space $X$ is a separable Hilbert space, thus closed bounded sets are compact and metrizable for the weak topology.
In particular, every bounded seque... | 1 | https://mathoverflow.net/users/89334 | 253367 | 114,766 |
https://mathoverflow.net/questions/253370 | 1 | I'm planning to write a paper about the possibility of describing modal logic and the multiple world aspect of it with techniques of automata theory. To not duplicate my work does anyone have more explicit links to done research in this area. My university doesn't seem to show anything specific in their database. Thank... | https://mathoverflow.net/users/97562 | Modal logic in combination with automata theory | What sources are you using for modal logic? What aspects of modal logic are you considering?
In for instance, the book: Tools and Techniques in Modal Logic, by Kracht, the section on Dynamic Logic includes a brief discussion of finite automata as it is relevant to that logic. Another perhaps more useful source to exa... | 1 | https://mathoverflow.net/users/3502 | 253376 | 114,770 |
https://mathoverflow.net/questions/251831 | 8 | I read this as a conjecture in the paper by Ballmann-Brin-Burns, titled "On Surfaces with No Conjugate points" JDG 25(249-273), 1987.
What is current status of this conjecture?
| https://mathoverflow.net/users/47336 | Does a compact nonflat surface without conjugate points have ergodic geodesic flow? | This is an open question. Actually it is still not known if a compact non-flat surface with non-positive curvature has an ergodic geodesic flow (with respect to the Liouville measure).
| 4 | https://mathoverflow.net/users/6129 | 253377 | 114,771 |
https://mathoverflow.net/questions/253379 | 18 | According to remark 6.14 in Shigeru Mukai's *An introduction to invariants and moduli* (unfortunately, the page is not available on Google Books, so I explain it below), the GIT-quotient of an affine variety $X$ by a reductive group $G$ with respect to a nontrivial character $\chi: G \to \mathbb G\_m$ may be considered... | https://mathoverflow.net/users/43639 | Why is Mumford's GIT-quotient so effective? | By my understanding, your question is not "Why is Mumford's construction better than the affine quotient". As you note, Proj is better than Spec of invariants for taking quotients by $\mathbb{G}\_m$. Instead, your question is "Why can't we get an even better quotient by going further, involving more characters somehow?... | 16 | https://mathoverflow.net/users/18060 | 253381 | 114,772 |
https://mathoverflow.net/questions/253155 | 5 | Let $K$ be a number field with ring of integers $O\_K$. Moreover consider an Arakelov divisor $\widehat{D}\in\overline{\operatorname{Div }(\operatorname {Spec }O\_K)}$, namely
$$D=\sum\_{\mathfrak p\;\text{prime}}n\_\mathfrak p\mathfrak p+\sum\_\sigma\lambda\_\sigma \sigma$$
where $\sigma$ is an archimedian place i.e. ... | https://mathoverflow.net/users/47136 | The notions of $H^0(\widehat{ D})$ and $h^0(\widehat{D})$ are not satisfactory | The Arakelov-theoretical analogue of Riemann's inequality is Minkowski's theorem. As Felipe Voloch's answer indicates, Tate's approach makes the Riemann-Roch theorem a consequence of the Poisson summation formula, and the self duality of the characteristic functions of (ultrametric) balls is crucial. Over archimedean f... | 6 | https://mathoverflow.net/users/10696 | 253386 | 114,774 |
https://mathoverflow.net/questions/253369 | 0 | I cannot prove the following equality on the Fourier transformation of measure:
let $\mu$ be a finite Borel measure on $R^d,$ then
$$\lim\limits\_{T\to \infty}\frac{1}{(2T)^d}\int\_{[-T,T]^d}|\widehat{\mu}(\xi)|^2d\xi=\sum\_{x\in R^d}\mu(\{x\})^2,$$
where $\widehat{\mu}(\xi)=\int e^{-2\pi i\xi\cdot x}d\mu(x)$, i.e, th... | https://mathoverflow.net/users/100416 | How to prove the equality on the Fourier transformation of measure? | It's a classic theorem of Wiener ([1933, p. 146](https://books.google.com/books?id=L1U4AAAAIAAJ&pg=PA146)).
Modern expositions see e.g. [Katznelson 1968](http://www.ams.org/mathscinet-getitem?mr=248482) ([Chap 6](http://math.stanford.edu/~katznel/23309/index23309.html), Thm 2.12), [Strichartz 1990](http://www.ams.org... | 3 | https://mathoverflow.net/users/19276 | 253387 | 114,775 |
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