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https://mathoverflow.net/questions/247977 | 1 | Can we find a Taylor Series expansion for $y(x)$ implicitly defined by:
$$\sum \_{i=1}^nA\_ie^{a\_ix+b\_iy} = 1 ?$$
In financial mathematics, the two-additive-factors Model G2++ is commonly used for interest rates forecasting. G2++ Model parameters are calibrated on market Caps and Swaptions prices. On the basis of ... | https://mathoverflow.net/users/22279 | Taylor Series expansion for an implicitely defined family of functions | The coefficient of the Taylor expansion of y at x=0 can be found recursively.
The convergence of the resulting series can be analyzed by mayorizing the coefficients and verifyig that the majorant series converges in a neighborhood of x=0. In your case, bounding the tail of the Taylor series may be important to decide h... | 1 | https://mathoverflow.net/users/47322 | 247983 | 112,787 |
https://mathoverflow.net/questions/247548 | 2 | In "Percolation in the hyperbolic plane" the authors study the properties of percolation in the hyperbolic plane. Smirnov and others proved convergence of isotropic percolation to SLE(6).
Do these results follow for the hyperbolic case too?
Findings:
1)L. Arosio, F. Bracci, "Infinitesimal generators and the Loe... | https://mathoverflow.net/users/78164 | Percolation on the hyperbolic plane and convergence to SLE(6) on hyperbolic plane | Smirnov's theorem assert the convergence to SLE *in the scaling limit*: one discretizes the domain with the triangular lattice of mesh size $\delta$, and lets $\delta$ go to zero. It is only proven for the triangular lattice; it's a major open problem to prove universality of this result (in fact, even to extend it to ... | 2 | https://mathoverflow.net/users/56624 | 247992 | 112,789 |
https://mathoverflow.net/questions/247327 | 1 | Suppose we have a poisson manifold $M$ whose Poisson cohomology is finite-dimensional in each degree? Does it mean that our manifold is symplectic? Are there other cohomological criteria of symplecticity? For example, suppose that all Casimir functions on a manifold are constant, does it imply that this manifold is sym... | https://mathoverflow.net/users/88385 | Finite-dimensional Poisson cohomologies | Consider on $\mathbb R^2$ the Poisson structure $\pi=(x^2+y^2)\partial\_x\wedge\partial\_y$. Then this Poisson manifold is not symplectic but with finite dimensional Poisson cohomology (It can be computed directly: the generator of nontrivial 1-dim Poisson cohomology is the modular vector field which is the rotation ve... | 7 | https://mathoverflow.net/users/6032 | 247995 | 112,790 |
https://mathoverflow.net/questions/247458 | 5 | Let $M$ be a compact Kahler Calabi-Yau variety which admit Ricci flat metric $\tilde\omega$, $Ric(\tilde \omega)=0$, then the Lelong number $\tilde \omega$ is zero?
In general if $\omega$ satisfies in the Kahler-Einstein equation $$Ric(\omega)=\lambda\omega$$ then is the Lelong number of $\omega$ vanishes?
| https://mathoverflow.net/users/nan | Lelong number of Ricci flat metric | ***Motto***: Canonical metric has vanishing Lelong number or (some times called Lelong-Demailly number)
Let me start with the definition of Lelong number.
Let
$W\subset \mathbb C^n$
be a domain, and $\Theta$ a positive current of degree $(q,q)$ on
$W$. For a point $p\in W$
one defines
$$\mathfrak v(\Theta,p,r)=\fra... | 8 | https://mathoverflow.net/users/nan | 247999 | 112,792 |
https://mathoverflow.net/questions/239068 | 1 | Consider the following Ito diffusion $X\_t$ satisfying
$$dX\_t=b(X\_t)dt+\sigma(X\_t)dB\_t,\quad X\_0=x\in \mathbb{R}^n,$$
with Lipschitz coefficients $b,\sigma$.
It can be shown that if $g$ is bounded and continuous, then $u(x)=E^x[g(X\_t)]$ is continuous. So any Ito diffusion is $C\_b$-[Feller continuous](https... | https://mathoverflow.net/users/91196 | Feller property for Ito diffusion with Lipschitz coefficients | Under the hypotheses:
1. $b$ and $\sigma$ are globally Lipschitz continuous;
2. $\sigma(x) \sigma(x)^T$ is positive definite for all $x \in \mathbb{R}^n$;
one can prove that the semigroup $P\_t$ associated to $X$ is, in fact, *strong* Feller, meaning that for any $t \ge 0$ the semigroup $P\_t$ maps $B\_b(\mathbb{R}... | 1 | https://mathoverflow.net/users/64449 | 248012 | 112,795 |
https://mathoverflow.net/questions/247873 | 2 | Do the first two columns of the character table of the permutation group $S\_n$ (the conjugacy classes () and (1 2) - that's a fairly standard order) always suffice to identify an irrep? (E.g. take $S\_6$: four irreps have dimension 5 but the second entry is 1,-1,3 or -3.) If yes, can this be generalized to other group... | https://mathoverflow.net/users/11504 | How to recognize different types of irreps | For $S\_{16}$ there appear (I asked GAP) to be two pairs of non-self-conjugate partitions where the value of the corresponding characters are zero on transpositions.
I haven't tried to figure out what the partitions are.
**Edit.** Suzuki ("The values of irreducible characters of the symmetric group", Arcata confer... | 6 | https://mathoverflow.net/users/22989 | 248024 | 112,798 |
https://mathoverflow.net/questions/247848 | 0 | To begin with, let us give the conceptual background needed to expose the problem. First of all, we shall consider the set $\mathbb{L}^{n} = \mathbb{R}^{n}\_{\geq0} = \{\overrightarrow{x}\in\mathbb{R}^{n}\_{\geq0}\mid\sum x\_{i} = 1\}$. More precisely, $\overrightarrow{x} = (x\_{1},x\_{2},\ldots,x\_{n})$ belongs to $\m... | https://mathoverflow.net/users/nan | Recurrence Equation and Matrix Convergence | Your example does not quite fit the problem description, as the matrix $\Lambda\_0$ you are adding has the absolute values of the eigenvalues of $A\_0$ on its diagonal. If this variation is the problem you are interested in, then the convergence is fairly trivial.
Your initial condition is a matrix $A\_0$ with nonne... | 1 | https://mathoverflow.net/users/85570 | 248031 | 112,800 |
https://mathoverflow.net/questions/248029 | 3 | A two dimensional TQFT can be described as a commutative Frobenius algebra by breaking up bordisms along submanifolds of codimension one into a set of generating bordisms. I was wondering if there is a similar way to break up bordisms of any dimension along codimension one submanifolds?
| https://mathoverflow.net/users/75887 | Generating bordims in TQFTs | András Juhász recently posted a [paper](http://arxiv.org/abs/1408.0668) to the arxiv with a result in this direction. Of course the answer is much much more complicated than the 2-dimensional case, even just in 3-dimensions.
Somewhat nicer answers can be found if you allow "extended TFTs" where you are allowed to con... | 7 | https://mathoverflow.net/users/22 | 248036 | 112,803 |
https://mathoverflow.net/questions/248043 | 25 | Where can I find relatively concise (i.e. not excessively wordy and waxing poetic about history and intuitions and such, doesn't spend an eternity carefully developing various parts of the theory of étale cohomology, etc.) English expositions of the proofs of the various Weil conjectures? The four Weil conjectures, acc... | https://mathoverflow.net/users/nan | Relatively concise English expositions of the proofs of the various Weil conjectures | What about Nick Katz' expose:
*Nicholas M. Katz*, MR 1831948 [**$L$-functions and monodromy: four lectures on Weil II**](http://dx.doi.org/10.1006/aima.2000.1979), *Adv. Math.* **160** (2001), no. 1, 81--132.
As well as [Kowalski's notes.](https://people.math.ethz.ch/~kowalski/deligne.pdf)
| 14 | https://mathoverflow.net/users/11142 | 248045 | 112,805 |
https://mathoverflow.net/questions/248037 | 0 | Clearly every finite group has a minimal generating subset.
1. Is there any formula for the number of minimal generating subsets of a finite group?
2. Is it known which groups have a unique minimal generating subset?
| https://mathoverflow.net/users/97333 | the number of minimal generating subsets of a group | 1.No, this is a hard question in general. It could maybe be done for special classes of groups, say nilpotent groups.
2.The only (finitely-generated) groups which have a unique minimal generating subset are the trivial group and the cyclic group of order 2.
Let $G$ be a group with a unique minimal generating subset... | 3 | https://mathoverflow.net/users/22377 | 248048 | 112,807 |
https://mathoverflow.net/questions/247461 | 6 | I'm learning about Lefschetz pencils and vanishing cycles and have looked at a few sources:
* <http://www.math.purdue.edu/~dvb/preprints/sheaves.pdf>
* <http://www3.nd.edu/~lnicolae/Morse2nd.pdf>
* Voisin's Complex Algebraic Geometry and Hodge Theory II
I want to be able to start computing explicit examples of vani... | https://mathoverflow.net/users/78824 | What tools can I use to compute the cohomology of the fibers of a Lefschetz Pencil? | Your example is a smooth hypersurface, which is much easier to understand than a general smooth projective variety. You ask for a computation of the cohomology of a Lefschetz pencil. This could mean either:
1. The cohomology groups of the total space of the pencil.
2. A description of the Gauss-Manin local system on ... | 2 | https://mathoverflow.net/users/30554 | 248051 | 112,809 |
https://mathoverflow.net/questions/248023 | 6 | For an uncountable compact metric space $X$ denote by $K(X)$ be the hyperspace of non-empty compact subsets of $X$, endowed with the Vietoris topology (which is generated by the Hausdorff metric).
In the countable power $K(X)^\omega$ consider the subspace $$cov(X)=\{(K\_n)\_{n\in\omega}\in K(X)^\omega:\bigcup\_{n\in... | https://mathoverflow.net/users/61536 | Is the space of countable closed covers of the Cantor set analytic? | For $X=2^\omega$ (and similarly for every uncountable $\sigma$-compact Polish space) this set is indeed $\Pi^1\_1$-complete:
Take a $G\_\delta$ set $B \subset 2^\omega \times 2^\omega$ so that $proj\_1(B)$ is $\Sigma^1\_1$-complete. Let $S=2^\omega \setminus proj\_1(B)$, then of course $S$ is $\Pi^1\_1$-complete and th... | 7 | https://mathoverflow.net/users/47760 | 248052 | 112,810 |
https://mathoverflow.net/questions/247778 | 3 | I would like compute the following
$$I\_{t,x,y} = \int\_{\mathbb S^2} e^{-i\left<t,\omega\right>} \, e^{-i\left< A(\omega)x,y\right>} \, d\sigma(\omega); $$
where $\mathbb S^2$ is the two-dimensional sphere, $t\in \mathbb R^3, \, x,y\in \mathbb R^4$ and $A(.)$ is the matrix given by $$ A(s\_1,s\_2,s\_3):= \left(
\begi... | https://mathoverflow.net/users/84558 | How to compute $\int_{\mathbb S^2} e^{-i\left<t,\omega\right>} \, e^{-i\left< A(\omega)x,y\right>} \, d\sigma(\omega)$ | One can get the answer with elementary means by direct calculation.
Define $A^\*(x,y) \in \mathbb{R}^3$ such that $\langle A(\omega) x, y\rangle = \langle A^\*(x,y), \omega \rangle$. Also, let $z = t + A^\*(x,y) \in \mathbb{R}^3$. Now, parametrize the $2$-sphere in the usual spherical coordinates, such that the polar... | 7 | https://mathoverflow.net/users/2622 | 248071 | 112,816 |
https://mathoverflow.net/questions/248074 | 5 | Let $X$ and $Y$ be smooth del Pezzo surfaces of the same degree $K\_X^2=K\_Y^2$.
>
> Are the sets $X(\mathbb{C})$ and $Y(\mathbb{C})$ homeomorphic, or at least homotopy equivalent?
>
>
>
| https://mathoverflow.net/users/97581 | Homeomorphism between del Pezzo surfaces | Yes, with precisely one exception.
If $K^2 \neq 8$, then the del Pezzo surface is the blow-up of the plane at $9-K^2$ points, so it is homeomorphic to the connected sum of $\mathbb{CP}^2$ with $9-K^2$ copies of $\overline{\mathbb{CP}^2}$.
If $K^2=8$, then we have either the quadric $\mathbb{P}^1 \times \mathbb{P}^1... | 14 | https://mathoverflow.net/users/7460 | 248076 | 112,817 |
https://mathoverflow.net/questions/248068 | 1 | Given a local field K and its finite galois extension L (we have non-archimedean valuation $v\_{K}$ and $v\_{L}$). We define $\Gamma=Gal(L/K)$, and for $i\in \mathbb{Z}$, we can define $\Gamma\_{i}$ (the $\Gamma\_{0}$ is the inertia group). I want to know if $\Gamma\_{i}$ is nontrivial, whether it strictly contains $\G... | https://mathoverflow.net/users/88343 | Is the filtration of inertia group a strict one? | No, the inclusion of $\Gamma\_{i+1}$ in $\Gamma\_i$ is often an equality. The simplest example is $K=\mathbf{Q}\_2$, $L=\mathbf{Q}\_2(\sqrt2)$. Here $\Gamma=\Gamma\_0=\Gamma\_1=\Gamma\_2$, and $\Gamma\_3$ is trivial.
It is possible to find a $K$ and a degree-$p$ cyclic extension $L$ of $K$ (where $p$ is the residual... | 3 | https://mathoverflow.net/users/2821 | 248089 | 112,822 |
https://mathoverflow.net/questions/248062 | 5 | Famously, the [fpqc topology](http://stacks.math.columbia.edu/tag/022A) is so big that no one *set* of fpqc covers of an affine scheme is cofinal in *all* fpqc covers. The Stacks Project [gives a construction](http://stacks.math.columbia.edu/tag/0BBK) using field extensions with arbitrarily large infinite transcendence... | https://mathoverflow.net/users/4177 | Is there a group scheme G such that there is a proper class of non-isomorphic fpqc G-bundles over some fixed base? | In more down-to-earth terms, you are asking whether there is always a *set* of $G$-torsors for the fpqc topology over a scheme $S$ such any every $G$-torsor for the fpqc topology over $S$ is isomorphic to one of those in the set. The answer is "yes" when $G$ is flat over $S$ (as is automatic over a field).
The quest... | 9 | https://mathoverflow.net/users/81332 | 248091 | 112,824 |
https://mathoverflow.net/questions/248079 | 7 | I am reading Class field theory - Bonn Lectures by Neukirch.
Given a $G$ module $A$ he defines Cohomology groups $H^i(G,A) : i\in \mathbb{Z}$ by considering some complete resolution of $G$ modules and applying hom functor.
He then calculated $H^{-1}(G,A),H^0(G,A),H^1(G,A),H^2(G,A)$
It says $H^2(G,A)$ is related ... | https://mathoverflow.net/users/nan | Is cohomology of groups all about $H^{i}: -2\leq i\leq 2$? | As Serre seems to be fond of saying, life begins at $H^3$.
The first appearance of a $3$-cocycle seems to be in Teichmüller's article *Über die sogenannte nichkommutative Galoissche Theorie...* in **Deutsche Mathematik 5**. He was trying to extend Galois Theory to extensions of a commutative field which are not thems... | 10 | https://mathoverflow.net/users/2821 | 248093 | 112,825 |
https://mathoverflow.net/questions/248092 | 1 | It seems to be widely known that every topological manifold can be embedded as a neighbourhood retract in euclidean space, I can not find a reference, though.
The reason, why I'm asking this, is that I would like to have a nice class of spaces containing topological manifolds and CW complexes.
And as it is still an o... | https://mathoverflow.net/users/91925 | Every topological manifold is a ENR? (Reference) | As it turns out, a close look at the [Wikipedia page](https://en.wikipedia.org/wiki/Retract#Absolute_neighborhood_retract_.28ANR.29) solves my Question.
There is an (indirect) reference to the paper:
Hanner, Olof (1951), "Some theorems on absolute neighborhood retracts", Arkiv för Matematik, 1: 389–408, [doi:10.1007/... | 1 | https://mathoverflow.net/users/91925 | 248100 | 112,826 |
https://mathoverflow.net/questions/248117 | 8 | A real $x\in2^\omega$ is random (or Solovay-random) over a model $M$ iff it's obtained via forcing by closed sets of reals, of positive measure, coded in $M$. Similarly, a pair $(x,y)$ of reals is random over a model $M$ iff it's obtained via forcing by closed sets in $2^\omega\times2^\omega$, of positive product measu... | https://mathoverflow.net/users/53091 | Solovay-random pairs of reals | Suppose $\kappa$ is the least ordinal such that for some $A \in 2^{\kappa} \cap M[x] \cap M[y]$, $A \notin M$. You already have that $\kappa = cf(\kappa) \geq \omega\_1$. Let $\tau$ be a Random name for $A \in M[x]$. WLOG, assume that the empty condition forces $\tau \in 2^{\kappa} \setminus M$ and all initial segments... | 2 | https://mathoverflow.net/users/97604 | 248123 | 112,832 |
https://mathoverflow.net/questions/248124 | 0 | I am having some trouble following how the expectation is being computed in the proof of Lemma 2.3 in [this](https://hal.archives-ouvertes.fr/hal-00506519/document) paper. For your convenience I am reproducing the relevant part here.
Let $B(t)$ denote an fBm process with $H=1$, then since $B(t) = t \mathcal{N}$, where ... | https://mathoverflow.net/users/18929 | Requesting Clarification on how Expectation for fbm with H = 1 is being computed in a paper | The function $xt-\frac{1}{2}t^2$ looks like a parabola opening down, with vertex at $x$. If $x \le 0$ then the function is decreasing on $[0, \sqrt{2}aT]$ so its maximum occurs at $t=0$. If $x \ge \sqrt{2}aT$ then its maximum occurs at $t=\sqrt{2}aT$. Otherwise it occurs at $x$.
| 1 | https://mathoverflow.net/users/4832 | 248126 | 112,833 |
https://mathoverflow.net/questions/248125 | 4 | A colleague of mine was stating there are no known undecidable statements that have explicit connection with prime numbers. What does this mean? I understand that it is unknown whether Goldbach conjecture, twin prime conjecture etc are not known to be undecidable.
Is there non-trivial examples of statements in number... | https://mathoverflow.net/users/10035 | Provably undecidable problems within prime numbers context | I believe there are not (or rather, if there are they are highly contrived), but that this is mostly a symptom of our lack of techniques for showing undecidability.
How can I prove that $PA$ does not prove $\varphi$? Well, this requires that I build a model of $PA+\neg\varphi$. Unfortunately, we don't really know how... | 9 | https://mathoverflow.net/users/8133 | 248128 | 112,835 |
https://mathoverflow.net/questions/248114 | 7 | A rational representation $(G,V)$ of a complex reductive linear algebraic group is called **multiplicity-free** if the decomposition of $\mathbb C[V]$ into irreducible $G$-modules contains each irreducible $G$-module with multiplicity at most one. Part of the importance of such representations lies in the fact that the... | https://mathoverflow.net/users/15155 | Are multiplicity-free representations weight multiplicity free? | It is a theorem of Brion and, independently, of Vinberg that varieties with an open $B$-orbit (a.k.a. spherical varieties) have in fact only finitely many orbits. A shorter argument is due to Matsuki (see his ICM talk) and independently (using the same idea) by me (On the set of orbits $\ldots$). Thus multiplicity free... | 10 | https://mathoverflow.net/users/89948 | 248130 | 112,837 |
https://mathoverflow.net/questions/239365 | 3 | Some months ago I studied Beilinson's paper about generators for the derived category of $\mathbb{P}^n$, "Coherent Sheaves on $\mathbb{P}^n$ and problems of linear algebra".
As next step, I moved to the paper by Kapranov regarding derived categories of Grassmanians and quadrics, "On the derived categories of coheren... | https://mathoverflow.net/users/91935 | K theory and derived categories | The connection is via the additivity theorem in (connective) algebraic $K$-theory or, better, the fact that algebraic $K$-theory is the universal additive invariant as proven in [BGT](https://arxiv.org/abs/1001.2282). Let me try to give an answer that blackboxes $K$-theory in terms of its universal property.
So Beili... | 3 | https://mathoverflow.net/users/24706 | 248131 | 112,838 |
https://mathoverflow.net/questions/248136 | 2 | This question is inspired by [this post](https://math.stackexchange.com/questions/1899199/a-basis-with-rational-entries-for-the-niemeier-lattices). A good answer to this question might help answer the linked post as well.
Question: Given a $24\times24$ matrix corresponding to an even, self-dual lattice in $24$ dimens... | https://mathoverflow.net/users/64186 | Finding which Niemeier lattice a given basis corresponds to | Most Niemeier matrices are characterized by their number of "roots"
(vectors of norm 2). In this case we're lucky: we find
(e.g. using the **qfminim** function in **gp**) that there are
$528$ roots, and thus that each of the simple root lattices
contained in the lattice has Coxeter number $528/24 = 22$;
and this deter... | 3 | https://mathoverflow.net/users/14830 | 248137 | 112,840 |
https://mathoverflow.net/questions/248007 | 1 | **Motivation**: Currents with mild singularities seems to have vanishing Lelong number, so the following question seems to be natural.
Definition(**Minimal singularities**):Let $T$ be a closed positive $(1,1)$ current on projective variety $X$. $T$ is said to be of
minimal singularities, if for every closed positive... | https://mathoverflow.net/users/nan | A question about minimal singularities | No, in any $(1,1)$ cohomology class which is big but not nef, any closed positive $(1,1)$ current with minimal singularities has nontrivial Lelong number somewhere, by Boucksom (Propositions 3.1 and 3.8 [here](https://arxiv.org/abs/math/0204336)).
For an explicit example, take $X$ a projective algebraic surface, $L$ ... | 2 | https://mathoverflow.net/users/13168 | 248155 | 112,842 |
https://mathoverflow.net/questions/248101 | 6 | The finiteness of the Tate-Shafarevich group is known to be equivalent to BSD for elliptic curves over function fields over $\mathbb{F}\_{q},$ this result is due to Kato and Trihan if I am not mistaken.
Question: Is any result in that direction known to hold for number fields?
| https://mathoverflow.net/users/70751 | Relationship between Tate-Shafarevich group and the BSD conjecture | Firstly, the functional field result your state is due to Tate in his Bourbaki talk. In fact he proves that the finiteness of the $p$-primary part of Sha is enough for $p$ different from the characteristic.
For elliptic curves over a number field, the finiteness of Sha (over larger fields) gives us the parity on the ... | 11 | https://mathoverflow.net/users/5015 | 248160 | 112,844 |
https://mathoverflow.net/questions/248166 | 2 | I'm reading the paper [On the relation between the Itō and Stratonovich integrals in Hilbert spaces](http://www.sbc.org.pl/Content/33707/2004_04.pdf) and there is something I don't understand.
In the notation of the paper, let
* $H,H\_1$ be separable $\mathbb R$-Hilbert spaces
* $Q\in\mathfrak L(H)$ be nonnegative ... | https://mathoverflow.net/users/91890 | Correction term in the relation between the Itō and Stratonovich integrals in Hilbert spaces | To paraphrase what you said, the paper you cite introduces an Ito SDE in (2.3) and a second Ito SDE in (2.4), which has an Ito-Stratonovich correction term. The paper then states that assumptions (A1)-(A4) imply existence and uniqueness of a mild solution for *both* (2.3) and (2.4). They claim that the latter statement... | 1 | https://mathoverflow.net/users/64449 | 248179 | 112,849 |
https://mathoverflow.net/questions/248111 | 2 | It is known that a reduced commutative unitary Baer ring $A$ is normal iff for every prime ideal $\mathfrak p$ of $T(A)$ (here $T(A)$ is the total quotient of $A$) one has $A/(\mathfrak p\cap A)$ is integrally closed in $T(A)/\mathfrak p$.
I would be interested in an example for which one implication does not hold i... | https://mathoverflow.net/users/1245 | Normal reduced rings | If $A=\{(a,b)∈\mathbb Z^2∣a+b≡0 \mod2\}$, then $A$ is a unital subring of $\mathbb Z×\mathbb Z$ with ring of quotients $\mathbb Q×\mathbb Q$. There are only two prime ideals in the latter, and their intersections with $A$ yield $\mathbb Z⊂\mathbb Q$ as the factor. However, $A$ is not integrally closed in $\mathbb Q× \m... | 2 | https://mathoverflow.net/users/42278 | 248180 | 112,850 |
https://mathoverflow.net/questions/248140 | 0 | I asked this [question](https://math.stackexchange.com/questions/1901209/meromorphic-extension-of-a-dirichlet-series) in SEM but I got no answer, so I'm trying my luck here.
Let the Dirichlet series $\phi(s)=\sum\_{n\ge 1}\frac{a(n)}{n^s}$ be absolutely convergent for $\Re(s)>1$ and extend to a meromorphic function o... | https://mathoverflow.net/users/95654 | Meromorphic continuation of a Dirichlet series | * Assuming the Riemann hypothesis is false, let $a\_n = \ln n$, and you have a counter-example.
* Assuming the Riemann hypothesis is true, let $a\_p = \ln p + p^{-1/4}\ln p, a\_{p+1} = \ln(p+1) - p^{-1/4}\ln p, a\_n = \ln n$ otherwise, and you get a counter-example.
| 1 | https://mathoverflow.net/users/84768 | 248182 | 112,852 |
https://mathoverflow.net/questions/246937 | 1 | I have a question about the equation (1.24) in [the paper](http://arxiv.org/pdf/math/9910188.pdf) about classical r-matrices.
It is said that when we put $\overline{r} = Pr$ in the equation (1.24):
$$
\overline{r}\_{23}\overline{r}\_{12}P\_{23} + \overline{r}\_{23}P\_{12}\overline{r}\_{23} + P\_{23}\overline{r}\_{12}... | https://mathoverflow.net/users/11877 | How to obtain the classical Yang-Baxter equation from a related equation | The index notation refers to the tensor factors these maps act on, i.e. $(Pr)\_{23}=P\_{23}r\_{23}$ etc. To prove the claim, you collect all P factors on the left, where they give the total inversion permutation M on both sides of the equation. Multiplying by M from the left then gives the CYBE.
| 2 | https://mathoverflow.net/users/23753 | 248184 | 112,854 |
https://mathoverflow.net/questions/248165 | 8 | i think i saw something like the statement of the question above but i am not sure.
Given two graph G and H if they have the same characteristic polynomial and it does not have any repeated roots,then G and H are isomorphic ?
if any one could give me the reference to the proof or even if its a true statement.
| https://mathoverflow.net/users/95470 | if two graphs have the same distinct eigenvalues, can we conclude that they are isomorphic? | Here are the edge sets for two trees on 12 vertices.
Graph 1:
$\left[\left(0, 1\right), \left(1, 2\right), \left(2, 3\right),
\left(2, 10\right), \left(3, 4\right), \left(3, 11\right), \left(4, 5\right), \left(5, 6\right), \left(6, 7\right), \left(7, 8\right), \left(8, 9\right)\right]$
Graph 2: $\left[\left(0, 1\... | 15 | https://mathoverflow.net/users/1266 | 248193 | 112,858 |
https://mathoverflow.net/questions/248216 | 2 | Considering certain random walks I came up with the following question: Given a finite set $A$ containing positive and negative integers, how many representations of zero as the sum of $n$ integers in $A$ exist. In the case that $A$ has two elements the question is easy to answer but for set with larger cardinality the... | https://mathoverflow.net/users/23542 | Representations of zero as the sum of integers | This is basically covered in section VII.8.1 of Flajolet and Sedgewick's "Analytic Combinatorics". You're looking at the generating function for bridges in their terminology and the form of the generating function is given by equation (97) on page 511 and is obtained using the kernel method.
| 6 | https://mathoverflow.net/users/1907 | 248218 | 112,864 |
https://mathoverflow.net/questions/248199 | 11 | The series $\sum\_{n\geq 1} \frac{\sin n}{n}$ is easily seen to be conditionally convergent, e.g. by Abel summation. But how about $\sum\_{n\geq 1} \frac{\sin(n^2)}{n}$? (for which Abel summation fails)
I came across the problem of showing that $\sum\_{n\geq 1} \frac{|\sin (n^2)|}{n}$ is divergent in [convergence of ... | https://mathoverflow.net/users/95413 | Conditional convergence of $\sum_{n\geq 1} \frac{\sin(p(n))}{n}$? | This is something I learned from fedja (artofproblemsolving.com):
We will show that if $x$ is not Liouvillian then the sum
$$
\sum\_{n \geq 1} \frac{\sin (2 \pi x n^{2})}{n}
$$
converges.
An argument goes like this:
rewrite
$$
\sum\_{k =1}^{n} \frac{\sin (2 \pi x k^{2})}{k} = \frac{1}{n}\sum\_{k=1}^{n}\sin (2 \pi ... | 11 | https://mathoverflow.net/users/50901 | 248219 | 112,865 |
https://mathoverflow.net/questions/248215 | 5 | Let $2Ring$ denote the 2-category of cocomplete categories with monoidal structures that preserve colimits in each argument. The morphisms are cocontinuous and strong monoidal functors (which are categorified ring homomorphisms). This notion is described in detail in Martin Brandenburg's thesis *Tensor categorical foun... | https://mathoverflow.net/users/84563 | (Co)completness of the 2-category of ``2-Rings" | I would argue that the "correct" version of (co)limits is the "pseudo" or "bi" or "weak" or "strong" or "homotopy" version. (As far as I can tell, all of these words mean the same thing.) The bicategories of cocomplete or of locally presentable categories are closed under limits: indeed, you can show by hand that the l... | 6 | https://mathoverflow.net/users/78 | 248224 | 112,868 |
https://mathoverflow.net/questions/248207 | 3 | A subset $A \subset \mathbb{R}$ is relative dense if there exists a real number $L>0$ such that for every $t \in \mathbb{R}$ the set $A \cap [t,t+L]$ is not empty. Such number $L$ is called an inclusion interval.
A real number $\omega\_\varepsilon \in \mathbb{R}$ is an $\varepsilon$-almost period for a function $f$ i... | https://mathoverflow.net/users/85336 | Almost periodic functions and the property A | If I deciphered the definitions right, you want to show that if $\omega>0$ is an $\varepsilon$ period of $f$, then either every $\omega'\in[-\omega,\omega]$ is, or every interval of length $C\omega$ contains a $C\varepsilon$ period of $f$. Unfortunately, this is false even for $e^{2\pi it}+e^{2\pi i\lambda t}$ if $\lam... | 5 | https://mathoverflow.net/users/1131 | 248228 | 112,870 |
https://mathoverflow.net/questions/248206 | 2 | For each positive integer n, let E(n) be n-dimensional Euclidean space-with its standard metric-and let M(n) be the separable metric space whose "points" are all the non-empty compact subsets of E(n) and whose "distance" is the Hausdorff distance between each pair of these subsets. Let F(n) be the mapping of M(n) onto ... | https://mathoverflow.net/users/4423 | A question about the centroids of compact subsets of Euclidean spaces | Even if you restrict to convex sets, I don't think there's a definition of centroid that makes this work for $n > 1$.
For instance, in $\mathbb{R}^2$ consider rectangles of the form $[0, 1] \times [0, \epsilon]$ versus triangles with vertices $(0, 0), (1, 0), (0, \epsilon)$.
As $\epsilon \to 0$, both families of sh... | 3 | https://mathoverflow.net/users/1227 | 248229 | 112,871 |
https://mathoverflow.net/questions/247876 | 8 | I am trying to show that a topological group which is also a (not necessarily smooth) manifold is automatically orientable. I know of a proof involving transition functions for *smooth* manifolds, in which case the object in question is a Lie group.
I am using Hatcher's definition of orientability: An $n$-manifold $M... | https://mathoverflow.net/users/97468 | A topological group which is also a (not necessarily smooth) manifold is orientable | In fact beside your question there is a beautiful theorem in homotopy theory. This theorem due to T. Bauer, N. Kitchloo, D. Notbohm and E. K. Pedersen guarantees that any loop space $X=\Omega B$ where $B$ is a $CW$-complex and such that $H\_\*(X)=\oplus H\_i(X;\mathbb{Z})$ is a finitely generated abelian group is homot... | 11 | https://mathoverflow.net/users/27816 | 248242 | 112,878 |
https://mathoverflow.net/questions/248246 | -1 | Consider the simple symmetric random walk on $\mathbb{Z}$. That is, let $X\_1, X\_2, \dots$ be i.i.d. random variables with
$$
P(X\_i=1)=P(X\_i=-1)=1/2,
$$
and define $S\_n=X\_1+\dots+X\_n$ with $S\_0=0$. As is well known, the sum $S\_n$ is (null) recurrent and satisfies the law of the iterated logarithm
$$
P\left(\li... | https://mathoverflow.net/users/45947 | Surely recurrent random walks and the law of the iterated logarithm | Consider a path satisfying $S\_{k^3} = (-1)^k k^2$ for all sufficiently large $k$. Since $$(k+1)^3 -k^3 \approx 3k^2 \gg 4k \approx |S\_{(k+1)^3} - S\_{k^3}|$$
this is clearly possible. Moreover, we can ensure that $|S\_n| \le |S\_{k^3}|$ for all $1 \le n \le k^3$ simply by replacing all excursions outside that value b... | 3 | https://mathoverflow.net/users/4832 | 248268 | 112,886 |
https://mathoverflow.net/questions/248244 | 3 | The question we are considering concerns a sum over all permutation $\sigma \in S\_n$ (symmetric group) of a certain rational function:
$$\sum\_{\sigma \in S\_n} \frac{\sigma\_{j1} \sigma\_{j2}... \sigma\_{jk}}{\prod\_{i=1}^{n-1} (\sigma\_i - \sigma\_{i+1})} \stackrel{?}{=} 0,$$ with $k < n-1$ and $jm \in \{1,2,...,n... | https://mathoverflow.net/users/97670 | sum over permutations equals zero? | Perhaps you wish to look at the paper
>
> Tewodros Amdeberhan, *Explicit computations with the Divided Symmetrization operator*, Proc. Amer. Math. Soc., **144** no 7 (2016) pp 2799-2810, doi:[10.1090/proc/12931](https://doi.org/10.1090/proc/12931)
> arXiv:[1406.0447](https://arxiv.org/abs/1406.0447)
>
>
>
In ... | 2 | https://mathoverflow.net/users/66131 | 248270 | 112,887 |
https://mathoverflow.net/questions/246925 | 2 | I have a question about compactness of semigroups and boundary conditions.
Let $\Omega$ be an **unbounded** domain of $\mathbb{R}^d$ with smooth boundary and $m(\Omega)=\infty$. Then we can define two bilinear forms (Dirichlet forms) on $L^{2}(\Omega)$.
\begin{align\*}
&\mathcal{E}^{D}(f,g)=\int\_{\Omega}(\nabla f, \... | https://mathoverflow.net/users/68463 | Compactness of semigroups, boundary conditions | You will find examples in this paper:
<http://math.caltech.edu/SimonPapers/212.pdf>
| 2 | https://mathoverflow.net/users/12120 | 248279 | 112,891 |
https://mathoverflow.net/questions/248280 | 10 | Are there structures $\mathcal{A}$ and $\mathcal{B}$ such that $\mathcal{A}$ and $\mathcal{B}$ are elementary equivalent, not isomorphic but bi-embeddable?
| https://mathoverflow.net/users/94393 | If $\mathcal{A} \equiv \mathcal{B}$ and $\mathcal{A} \not \cong \mathcal{B}$, is it possible that $\mathcal{A}$ and $\mathcal{B}$ are bi-embeddable? | Take the free groups $F\_n$ and $F\_m$ where $m\neq n$ and both are bigger than 1.
This paper:
<http://www.math.mcgill.ca/olga/p3new.PDF>
proves that they both have the same elementary theory. The groups are not isomorphic, but each one can be embedded in the other.
| 8 | https://mathoverflow.net/users/41644 | 248281 | 112,892 |
https://mathoverflow.net/questions/248272 | 3 | Let $S^{n-1}$ be the unit sphere in $\mathbb{R}^n$. For $m>n$, we can define $\Lambda$ to be the set
$$\{(\lambda\_1, ..., \lambda\_m):\sum\_{i=1}^m \lambda\_i=1, \lambda\_i\ge0, and \mbox{ there exist}\, v\_i \in S^{n-1}, i=1, ..., m \, \mbox{such that}\, \sum\_{i=1}^m \lambda\_i v\_i=0 \}.$$
The motivation of the ... | https://mathoverflow.net/users/51546 | How to find extreme points of a set related to Minkowski's Theorem? | I'll assume $m, n \ge 2$.
I claim that
$$\Lambda = \left\{\lambda \in \mathbb R^m: \; \sum\_{i=1}^m \lambda\_i = 1,\; 0 \le \lambda\_i \le \frac{1}{2} \ \text{for all}\ i\right \}$$
The necessity of the condition $\lambda\_i \le 1/2$ comes from the fact that for $v\_j \in S^{n-1}$, $$\left\|\sum\_{j=1}^m \lambda\... | 2 | https://mathoverflow.net/users/13650 | 248297 | 112,903 |
https://mathoverflow.net/questions/248262 | 16 | $\require{AMScd}$One nice thing about $\infty$-categories is that spaces are themselves $\infty$-categories. What's the analogue for spectra? Presumably this would be the stabilization of the $\infty$-category of $(\infty,1)$-categories, fitting into a cube:
$$
\begin{CD}
ILS\_\* @>>> \text{Spaces}\_\* @>>> \text{$(\... | https://mathoverflow.net/users/18702 | What's the stabilization of the $\infty$-category of $\infty$-categories? | In a project in progress with Matan Prasma and Joost Nuiten concerning the abstract cotangent complex formalism we compute the stabilization of the $\infty$-category $\infty\mathrm{Cat}\_{/C}$ of $\infty$-categories over a fixed $\infty$-category $C$, and show that it is equivalent to the $\infty$-category of functors ... | 21 | https://mathoverflow.net/users/51164 | 248298 | 112,904 |
https://mathoverflow.net/questions/248039 | 5 | A theorem of Corson and Lindenstrauss in:
Corson, H. H. and Lindenstrauss, J. “On weakly compact subsets of Banach spaces”. In: Proceedings of the American Mathematical Society 17.2 (1966), pp. 407–412.
<http://www.ams.org/journals/proc/1966-017-02/S0002-9939-1966-0199669-9/S0002-9939-1966-0199669-9.pdf>
states ... | https://mathoverflow.net/users/94358 | Corson-Lindenstrauss : Weakly compact sets as intersection of finite unions of cells | Let $x\_n=-\frac{\sum\_{k=1}^n e\_k}{n}$. Then $\|e\_i-x\_n\|^2=\frac{n+3}{n}$ if $i \leq n$ and $\|e\_i-x\_n\|^2=\frac{n+1}{n}$ if $i>n$. So your set is the intersection of the sets $B(x\_n,\sqrt{\frac{n+1}{n}}) \cup \bigcup\_{i=1}^n B(e\_i,\frac{1}{n})$.
Nice question, btw.
| 4 | https://mathoverflow.net/users/54609 | 248340 | 112,918 |
https://mathoverflow.net/questions/248351 | 2 | If there is nontrivial isotrivial elliptic fibration, what will be type of singular fiber(since we have the kodaira's table of singular fiber). Is it a contradiction since in kodaira's table, smooth fiber around singular fiber seems non isomorphic. I am a bit confused?
| https://mathoverflow.net/users/78863 | isotrivial elliptic fibration and kodaira's table of singular fiber | I think one ususally uses isotrivial in this case to mean that the smooth fibers are all isomorphic. Then it is certainly the case that there can be singular fibers.
Which singular fibers can appear depends on the $j$ invariant. The $j$ invariants at which they can appear are the same as in a non-isotrivial fibration... | 6 | https://mathoverflow.net/users/18060 | 248354 | 112,920 |
https://mathoverflow.net/questions/248355 | 6 | Fix a first-order language $\mathcal{L} = \{R\}$ whose signature consists of equality and another binary relation symbol $R$. For an axiom schema $A$ and a sentence $\tau$ in this language, we write $A \vdash\_n \tau$ if there exists a proof of length $n$ of $\tau$ from $A$. Define the following invariant for a given a... | https://mathoverflow.net/users/nan | Does there exist inconsistent axiom schemata which require arbitrary long proofs of their inconsistency? | This depends on the precise proof system, and notion of proof length, that you use. For example, the proof system that allows you to deduce any valid conclusion in one step is certainly sound and complete, and provides a counterexample.
But for any reasonable proof system, the answer will be **yes**: specifically, a... | 10 | https://mathoverflow.net/users/8133 | 248360 | 112,923 |
https://mathoverflow.net/questions/248361 | 4 | The $n$Lab writes (prop. 2.2 in <https://ncatlab.org/nlab/show/category+of+monoids>) :
Let $C$ be a monoidal category with countable coproducts that are preserved by the tensor product. Then the forgetful functor $U\_C$ has a left adjoint $F\_C:C\rightarrow Mon(C)$.
Suppose now $C$ is a symmetric monoidal category... | https://mathoverflow.net/users/84563 | Left adjoint for categories of commutative monoids? | The left adjoint is the functor Sym, defined as $Sym(X) = I \coprod X \coprod (X \otimes X)/\Sigma\_2 \coprod (X\otimes X\otimes X)/\Sigma\_3 \coprod \dots$, where $I$ is the unit. I only ever work in closed, cocomplete settings, where this always exists, but in general I think it should exist under the same conditions... | 6 | https://mathoverflow.net/users/11540 | 248362 | 112,924 |
https://mathoverflow.net/questions/248306 | 6 | Any reference that we can find the following $$\Bigr[-\log(1-t)\Bigr]^x = t^x + x t^x \sum\_{k=0}^\infty \psi\_k(x+k)\,t^{k+1}; \quad \mbox{for all} \, x\in \mathbb R, \, |t|<1$$
where $\psi\_k(.)$ are the Stirling Polynomials ([Stirling Polynomials](http://mathworld.wolfram.com/StirlingPolynomial.html)).
Thank in ad... | https://mathoverflow.net/users/84558 | Any reference for the series expansion of $\Bigr[-\log(1-t)\Bigr]^x$? | I edit my post to answer Carlo Beenakker's remark and also because I would like to add a reference, possibly more accurate than the two below. Theorem 7.1 p.13 of
A. Adelberg, A finite difference approach to degenerate Bernoulli and Stirling polynomials, Discrete Math. 140 (1995), 1-21,
states that, for $s$ a compl... | 10 | https://mathoverflow.net/users/89429 | 248369 | 112,928 |
https://mathoverflow.net/questions/248345 | 5 | Let $M$ be a non-trivial monoid and $\mathbb ZM$ its monoid ring. All modules are left modules in what follows. Suppose that $M$ contains a zero element (or absorbing element) $z$. That is $mz=z=zm$ for all $m\in M$. For example the zero element of a ring is absorbing for multiplication. It is well known that the trivi... | https://mathoverflow.net/users/15934 | Can the trivial module be stably free for a monoid ring? | I've had a little more time to think about your questions, and I believe the following is true: **Contracted monoid rings have IBN.** This might appear implicitly in Cohn's paper, but I couldn't find it on a causual reading.
We will need the following lemma, which appears (with proof) as Exercise 1.5 in "Exercises in... | 4 | https://mathoverflow.net/users/3199 | 248371 | 112,930 |
https://mathoverflow.net/questions/248343 | 3 | I know there is an almost exactly same question [here](https://mathoverflow.net/questions/144963/how-to-solve-a-matrix-equation-with-both-inverses-and-a-hadamard-product) but I have further specifications. So my problem is as follows:
$$
\Omega^{-1}=\dfrac{1}{n}\left(\Omega\odot \mathbf{W}+\mathbf{X}'\mathbf{X}+\lambda... | https://mathoverflow.net/users/85893 | Matrix equation with Hadamard product and its own inverse involved | Removing all unnecessary parameters, we come to the equation $\Omega^{-1}=2 W\odot \Omega + B$ where $B$ is positive definite. We need to find a solution in the cone $M\_+$ of positive definite matrices. The solutions are exactly the stationary points of
$F(\Omega)=\log\det\Omega- \operatorname{Tr} [(W\odot\Omega)\Ome... | 6 | https://mathoverflow.net/users/1131 | 248373 | 112,931 |
https://mathoverflow.net/questions/248291 | 1 | We say that $\mathbb{P}$ is a complete suborder of $\mathbb{Q}$, if it is a suborder, and maximal antichains in $\mathbb{P}$ remain maximal antichains in $\mathbb{Q}$
As the title says, is every c.c.c. non-atomic partial order of size $\omega\_1$ a union of countable complete suborders?
If the answer is no, what a... | https://mathoverflow.net/users/70946 | Is every c.c.c. non-atomic partial order of size $\omega_1$ a union of countable complete suborders? | In response to Joel's comment, here is an argument showing that it is consistent that every ccc poset of size $\omega\_1$ is the union of its nonatomic countable complete subposets.
Suppose $\text{MA}\_{\omega\_1}$ holds. It follows that there are no Suslin algebras, so any ccc poset must add a real. Furthermore, by ... | 6 | https://mathoverflow.net/users/1058 | 248374 | 112,932 |
https://mathoverflow.net/questions/248341 | 2 | Let $Pres$ denote the 2-category of locally presentable categories (categories that are accessible/cocomplete), cocontinuous functors, and natural transformations. Let $Cat$ denote the 2-category of (small) categories, functors, and natural transformations. There is an inclusion functor $i:Pres\hookrightarrow Cat$. Doe... | https://mathoverflow.net/users/84563 | Adjunction between locally presentable and ordinary categories? | Size conditions are not such a big deal when it comes to formulating the question, at least. We just let Cat be the non-locally-small 2-category of locally-small categories (perhaps we require that the object sets be no bigger than the universe, it turns out not to matter). The inclusion $\mathsf{Pres}^L \to \mathsf{Ca... | 2 | https://mathoverflow.net/users/2362 | 248380 | 112,934 |
https://mathoverflow.net/questions/248379 | 3 | Let $X$ be a Banach space and $Y$ be a separable closed subspace of $X^{\*}$. Is there a separable closed subspace $Z$ of $X$ such that $Y$ is isomorphic to a subspace of $Z^{\*}$? Thank you!
| https://mathoverflow.net/users/41619 | Separable subspaces in dual spaces | Yes. Take a countable dense subset $A$ in $Y$. For each element $a\in A$ take a countable sequence $x\_n$ in a unit ball of $X$ such that $a(x\_n)$ tends to $\|a\|$. Let $Z$ be a closed span of all these sequences.
| 3 | https://mathoverflow.net/users/4312 | 248383 | 112,935 |
https://mathoverflow.net/questions/248346 | 0 | This was a statement I came across in a paper on what are called Transnormal curves by Bernd Wegner.
**Two immersions $f\_1$ and $f\_2$ of a smooth manifold $M$ into $\mathbb{R}^n$ such that at each point $p \in M$, both immersions are connected by a parallel section of their respective normal bundles** iff **the no... | https://mathoverflow.net/users/39232 | Parallel sections of normals bundle | I found the result you quoted, also quoted in
Bernd Wegner: Parallel and transnormal curves on surfaces. New developments in differential geometry (Debrecen, 1994), 423–434, Math. Appl., 350, Kluwer Acad. Publ., Dordrecht, 1996.
There a reference for the proof is given:
B. WEGNER: Some remarks on parallel immers... | 2 | https://mathoverflow.net/users/78645 | 248384 | 112,936 |
https://mathoverflow.net/questions/248259 | 7 | For any space $X$ there is a fibration
$$
\Omega X\to LX\stackrel{ev}{\to} X
$$
where $LX=Map(S^1,X)$ is the free loop space, $\Omega X = Map\_\*(S^1,X)$ is the based loop space, and $ev:LX\to X$ is the evaluation map sending $\omega$ to $\omega(1)$. There is a canonical section $\sigma\_0:X\to LX$ sending $x\in X$ to ... | https://mathoverflow.net/users/8103 | Classification of sections of free loop fibration over the two-sphere | The space of sections of $\Lambda S^2\to S^2$ is a subspace of the space of all maps
$$S^2\to \Lambda S^2=Map(S^1,S^2).$$
When viewed as a subspace of the space of all maps
$$S^1\to Map(S^2,S^2),$$
it becomes $\Omega\_1 Map(S^2,S^2)$, the based loopspace of the space of self-maps of $S^2$, with the identity map as ba... | 6 | https://mathoverflow.net/users/6666 | 248390 | 112,938 |
https://mathoverflow.net/questions/247985 | 2 | I am looking for a proof of weighted version of [the inequality](https://mathoverflow.net/questions/245092/a-conjecture-generalization-of-karamata-inequality) as following.
Let $I$ be an interval of the real line and let $f$ denote a real-valued, convex function defined on $I$.
If $x\_1, . . . , x\_n$ and $y\_1, .... | https://mathoverflow.net/users/76698 | Weighted version of the Karamata Inequality | This is false by some stupid reasons. Say, if $\lambda\_1=\lambda\_n=0$, then what we have to prove does not depend on $x\_1,x\_n,y\_1,y\_n$, but changing these four variables with fixed other variables we may easily satisfy all the conditions.
| 2 | https://mathoverflow.net/users/4312 | 248402 | 112,942 |
https://mathoverflow.net/questions/248387 | 10 | I have encountered the following linear algebra/number theory question in my work (low-dimensional topology), so I thought I should ask the experts.
Let $A \in SL(n,\mathbb{Z})$ be a matrix , $n \geq 3$. One can easily show that there is some $j \in \mathbb{N}$ such that $Tr(A^j) \geq 3$ (proof given at the end). My ... | https://mathoverflow.net/users/56571 | Traces of powers of integral marices | Yes: for every $d\ge 3$ and all $A\in\mathrm{GL}\_d(\mathbf{Z})$, there exists $n\in\{1,\dots,12^d\}$ such that the trace of $A^n$ is $\ge 3$. (This is probably far from sharp, and I don't know if we can do better than exponential.)
First part: I claim that if $t=(t\_1,\dots,t\_d)$ is a $d$-tuple in the unit circle, ... | 4 | https://mathoverflow.net/users/14094 | 248408 | 112,943 |
https://mathoverflow.net/questions/248375 | 1 | Suppose:
$$
dX\_t = a(t,X\_t)dt + b(t,X\_t)dW^H\_t
$$
is an SDE with values in a separable Hilbert Space $H$, and $W^H\_t$ is an $H$-valued cylindrical Wiener process. Then can we write the dynamics for $X\_t$ in terms of a basis $\{e\_i\}$ of $H$?
That is, if $\{e\_i\}$ is a basis for $H$ then is the previous SDE eq... | https://mathoverflow.net/users/36886 | Hilbert-Space Values SDE in terms of Basis | In principle, the *variational form* of the SPDE problem allows you to do this. What you do is expand the solution in terms of basis vectors, and choose the test functions in the variational formulation to be basis vectors. This is how projection-based methods for SPDEs are constructed.
For more detail see, e.g., De... | 1 | https://mathoverflow.net/users/64449 | 248412 | 112,945 |
https://mathoverflow.net/questions/248401 | 2 | I am looking for a reference concering the definition of epi-1-morphisms and mono-1-morphisms in an arbitrary bicategory. These concepts should be defined somewhere, but I am not able to find a reference. It is not hard to come up with a definition of mono/epi-1-morphisms, but somehow it's not written down.
Thanks fo... | https://mathoverflow.net/users/97765 | Reference for the definition of epi-1-morphisms in bicategories | A definition similar to the one you give in your comment can be found in paragraph 2.8 of the paper
>
> Carboni, Aurelio; Johnson, Scott; Street, Ross; Verity, Dominic. Modulated bicategories. J. Pure Appl. Algebra 94 (1994), no. 3, 229--282.
>
>
>
A functor $F : A \longrightarrow B$ is said to be **pseudomoni... | 1 | https://mathoverflow.net/users/57405 | 248419 | 112,948 |
https://mathoverflow.net/questions/248418 | 4 | A well known fact in probability is that a uniform random variable on $[0,1]$ can be used to simulate any other probability distribution on $\mathbb{R}$.
A standard way of doing this is to define, given $\mu$ a probability on $\mathbb{R}$, the random variable $F(u,\mu) = \min\lbrace x \in \mathbb{R}: \mu((-\infty,x]... | https://mathoverflow.net/users/7631 | Simultaneous simulation of all probability measures on a compact metric space | This is a result of Blackwell and Dubins, ["An extension of Skorohod's almost sure representation theorem"](http://www.ams.org/journals/proc/1983-089-04/S0002-9939-1983-0718998-0/). In fact, your function F can be constructed to be almost surely continuous in the measure argument, and X can be any Polish space.
| 7 | https://mathoverflow.net/users/44169 | 248421 | 112,949 |
https://mathoverflow.net/questions/248424 | 3 |
>
> What are some relations between the existence of Motivic t-structures and Motivic galois groups?
>
>
>
I heard that indeed the existence of the Motivic t-structure implies the isomorphism between Ayoub's Motivic galois group and Nori's. Is this correct?
| https://mathoverflow.net/users/83957 | Relations between Motivic Galois groups and Motivic t-structure? | The argument sketched in Example 3.20 of [J. P. Pridham, *Tannaka duality for enhanced triangulated categories*, [arXiv:1309.0637](http://arxiv.org/abs/1309.0637)] demonstrates the comparison assuming the existence of the motivic t-structure.
On the other hand, an unconditional proof is given in [Utsav Choudhury, Mar... | 8 | https://mathoverflow.net/users/2503 | 248425 | 112,950 |
https://mathoverflow.net/questions/248422 | 1 | Let p(n) be the number of unrestricted partitions of n. p(0) is taken to be 1.
Let set 1 and set 2 be two empty sets.
Here's an algorithm. Put p(n) into set 1. On each successive step, k=1,2,3,..., n, put p(n-k) into that set which has the smaller sum of elements, or into set 1 if the two sets have equal sums.
Aft... | https://mathoverflow.net/users/40145 | Will this greedy algorithm always work? | Let a nondecreasing sequence (we'll say of positive integers) $a\_n$ starting from $a\_1=1$
be subdoubling if for all $n>2$ it satisfies the relation
$a\_n <= 2\*a\_{n-1}$. Let $d$ be an integer with $-a\_n \leq d \leq a\_n$ and put $d$ in set 2, then run your algorithm. I claim at the
finish of the algorithm the diff... | 3 | https://mathoverflow.net/users/3402 | 248427 | 112,951 |
https://mathoverflow.net/questions/191963 | 8 | I am interested in the complexity of the following problem:
>
>
> >
> > Given an $m\times n$ binary matrix $M$, can we permute its rows/columns to obtain a triangular matrix?
> >
> >
> >
>
>
>
I am also interested in the case where "triangular" is replaced by **unitriangular**.
It says [here](http://mat... | https://mathoverflow.net/users/5152 | Transforming a binary matrix into triangular form using permutation matrices | Luckily, the problem is no more in the "NP-limbo". The paper [Obtaining a triangular matrix by independent row-column permutations](https://hal.archives-ouvertes.fr/hal-01189621/document) Fertin, Rusu, and Vialette shows that the problem is NP-complete for binary square matrices.
**EDIT**: Also, the same problem was ... | 5 | https://mathoverflow.net/users/8784 | 248432 | 112,953 |
https://mathoverflow.net/questions/248415 | 5 | Let $R$ be a Noetherian ring and $P$ a prime ideal. Then the $n$-th symbolic power of $P$ is
$$P^{(n)} = P^n R\_P \cap R = \{ f \mid sf \in P^n \text{ for some } s \in R - P\}$$
(cf. [wiki](https://en.wikipedia.org/wiki/Symbolic_power_of_a_prime_ideal)). We have $P^{(n)}$ is just the $P$-primary component of $P^n$.
... | https://mathoverflow.net/users/17901 | a question on symbolic power | Question 1 is answered affirmatively in Eisenbud's "Commutative Algebra with a View Toward Algebraic Geometry" (p. 106, attributed to Zariski, Nagata). The text refers to "A Nullstellensatz with Nilpotents" [1979] by Eisenbud and Hochster for generalizations.
| 4 | https://mathoverflow.net/users/59248 | 248434 | 112,954 |
https://mathoverflow.net/questions/248433 | 3 | This is basically a restatement of [this question](https://mathoverflow.net/questions/248002/when-are-descriptions-of-formal-unramifiedness-smoothness-via-lifting-properties).
Two arrows $f,g$ are orthogonal, i.e satisfy $f\perp g$, iff the square below is a pullback
$$\require{AMScd} \begin{CD}
\mathsf C(B,X) @>{... | https://mathoverflow.net/users/69037 | Internal characterizations of lifting properties? | The second square being a pullback is a strictly stronger condition than $f\perp g$. Mapping out of the unit object shows that it implies $f\perp g$; mapping out of other objects says that the strong condition is equivalent to $(W\otimes f) \perp g$ for all objects $W$ (or $W\times f$ in the cartesian-closed case; I'm ... | 6 | https://mathoverflow.net/users/49 | 248441 | 112,956 |
https://mathoverflow.net/questions/248367 | 3 | I had asked this question in math.stackexchange (link: <https://math.stackexchange.com/questions/1902276/bounds-on-the-moore-penrose-inverse-of-a-product> ) but I did not get any response so I am trying my luck here.
Let $A^{\dagger}$ denote the Moore-Penrose inverse of a real matrix and let $\|A\|$ denote the usual ... | https://mathoverflow.net/users/6627 | Norm of Moore-Penrose inverse of a product | Here’s a small counterexample involving two rank-deficient matrices $A$ and $B$: $$A = \left( {\begin{array}{\*{20}c}
0 & { - 1} & 0 \\
0 & 0 & 0 \\
0 & { - 1} & 0 \\
\end{array}} \right), B = \left( {\begin{array}{\*{20}c}
4 & 0 & 2 \\
2 & 0 & 1 \\
4 & 0 & 2 \\
\end{array}} \right)$$
The corresponding Moore–Pe... | 2 | https://mathoverflow.net/users/19739 | 248445 | 112,958 |
https://mathoverflow.net/questions/248444 | 7 | Let $M$ be a manifold. Let's say $M$ is smooth, connected, oriented. We can also assume that $M$ is closed if that makes things easier.
Let $\mathit{Diff}(M)$ denote the group of diffeomorphisms of $M$ and $\mathit{Diff}\_0(M)$ denote its identity component, consisting of the isotopically trivial diffeomorphisms of $... | https://mathoverflow.net/users/25590 | Homotopically trivial vs isotopically trivial diffeomorphisms | The quotient group $Diff\_1(M)/Diff\_0(M)$ is a discrete group since $ Diff\_0(M)$ is a path component of $Diff(M)$, hence also a connected component since $Diff(M)$ is locally path-cconnected, and $Diff\_1(M)$ is a union of components of $Diff(M)$, making the quotient discrete. [It is easy to see that $Diff\_0(M)$ is ... | 14 | https://mathoverflow.net/users/23571 | 248447 | 112,959 |
https://mathoverflow.net/questions/248458 | 2 | Let $G$ be a locally compact group. Is there any correspondence between closed two-sided ideals of $L^1(G)$ and closed normal subgroups of $G$? (Especially, is there any correspondence between finite codimension closed two-sided ideals of $L^1(G)$ and closed normal subgroups of $G$?)
| https://mathoverflow.net/users/27066 | Ideals of $L^1(G)$ and normal subgroups of $G$ | The following is about the best you can hope for, I think.
For discrete groups a homomorphism between groups gives rise to a homomorphism between algebras. So although we do not have a "correspondence", in this setting a normal subgroup $N\lhd G$ gives rise to an ideal $J\_N:= \ker( \ell^1(G) \to \ell^1(G/N))$. A sim... | 4 | https://mathoverflow.net/users/763 | 248468 | 112,966 |
https://mathoverflow.net/questions/248464 | 3 | Let $\Gamma$ be a countable discrete group and $\beta \Gamma$ be its Stone–Čech compactification.
My question is that
**Does the $\sigma$-algebra generated by clopen sets in $\beta \Gamma$ equal to the Borel $\sigma$-algebra (generated by open sets) on $\beta \Gamma$?**
Note that the Stone–Čech compactification ... | https://mathoverflow.net/users/9401 | Sigma algebras on the Stone–Čech compactification of a countable discrete group | No.
First of all, take note that for compact zero-dimensional spaces $X$, the $\sigma$-algebra generated by all clopen sets is precisely the $\sigma$-algebra of Baire sets (recall that in a completely regular space, the $\sigma$-algebra of Baire sets is the $\sigma$-algebra generated by collection of all zero sets). ... | 8 | https://mathoverflow.net/users/22277 | 248470 | 112,968 |
https://mathoverflow.net/questions/248430 | 0 | let $\quad-1=x\_0 < x\_1 <\ ...\ < x\_n<1\quad$ be a set of abscissas
and $\quad(y\_0, y\_1,\ ...\,y\_n)\quad$ a sequence of the corresponding ordinates.
>
> **Question:**
>
>
> what can be said about the existence and calculation of a pole-free rational function $R(x)$,
>
> with the following properties?... | https://mathoverflow.net/users/31310 | $C^\infty$ Periodic Pole-free Rational Interpolation | No such function exists, except possibly when $R(x)$ is a constant. Your condition on the derivatives implies periodicity (the periodic extensions of $R(x)$ from $[-1,1]$ to the whole real line is analytic and agrees with $R(x)$ on $[-1,1]$ and must be equal to $R(x)$ by uniqueness of analytic continuation). No non-con... | 1 | https://mathoverflow.net/users/2622 | 248476 | 112,972 |
https://mathoverflow.net/questions/248083 | 1 | Mutations of valued quivers are defined in [cluster algebras II](https://arxiv.org/pdf/math/0208229v2.pdf), Proposition 8.1 on page 28. I have a question about the number $c'$. For example, let $a = 2, b=1, c=1$ and consider the quiver $Q$:
$1 \overset{a}{\to} 2 \overset{b}{\to} 3$ and there is an arrow $1 \overset{c}... | https://mathoverflow.net/users/11877 | Mutation of valued quivers | I don't think your diagram is coming from a skew-symmetrizable matrix. Let $B$ be a skew-symmetrizable matrix and assume $Q = \Gamma(B)$ so your quiver is the diagram of $B$ as defined in Definition 7.3 of the linked paper. Then we must have
$$B = \begin{bmatrix}0 & x & y \\ \frac{-1}{x} &0 & z \\ \frac{-1}{y} & \frac{... | 2 | https://mathoverflow.net/users/51668 | 248492 | 112,976 |
https://mathoverflow.net/questions/248511 | 0 | Let $P\in R^{n\times n}$ be an orthogonal matrix. I want to ask whether or not there exists some vector $x\in R^n$ containing no zero entries such that $Px$ also contains no zero entries.
| https://mathoverflow.net/users/75264 | A question on orthogonal matrix | Instead of Zariski topology consider usual topology. Every open ball will contain a vector with all entries non-zero. (I leave this to you to verify).
As $P$ is invertible it is a homeomorphism. Image of open ball will contain open ball. QED
| 6 | https://mathoverflow.net/users/22878 | 248512 | 112,982 |
https://mathoverflow.net/questions/248302 | 23 | It is well known that $BU$ is an infinite loop space, and as such it has an action of an $E\_\infty$ operad. An explicit construction of such an action is given, for example, in an answer to [this MO question](https://mathoverflow.net/questions/94295/infinite-loop-space-structure-of-bu) . My question is whether it is ... | https://mathoverflow.net/users/6668 | Is it possible to construct an action of an $E_\infty$ operad on $BU$ that respects filtration by $BU(n)$? | Unfortunately there is no such filtration.
At first, this looks very similar (but not as strong as) asking for a map of $E\_\infty$ spaces $\coprod BU(n) \to BU$ which would become a splitting map $ku \to bu$ of spectra. We know that doesn't happen, because there's a nontrivial $k$-invariant.
We can look more caref... | 17 | https://mathoverflow.net/users/360 | 248518 | 112,984 |
https://mathoverflow.net/questions/248508 | 4 | I have seen outlined in [this comment os mathoverflow](https://mathoverflow.net/a/145251/97814) how to solve quadratic matrix equations of the form
$$
XCX + AX = I
$$
where $X \in \mathbb{R}^{n\times n}$, $C = C^T > 0 \in \mathbb{R}^{n\times n}$, and $I$ is identity of corresponding size.
I need to look this up more th... | https://mathoverflow.net/users/97814 | Source of equation - theorems about solving quadratic matrix equations | Bini, Iannazzo, Meini, [*Numerical Solution of algebraic Riccati equations*](http://bookstore.siam.org/FA09/), SIAM books, seems a good starting point to me. It is a monograph that deals both with the symmetric and the non-symmetric case and assumes no previous knowledge in control theory.
| 3 | https://mathoverflow.net/users/1898 | 248524 | 112,985 |
https://mathoverflow.net/questions/248521 | 12 | Assume that $H$ is a Lebesgue measurable additive subgroup of $\mathbb{R}$. Is $H$ necessarily a Borel subset of $\mathbb{R}$?
| https://mathoverflow.net/users/36688 | Is a Lebesgue measurable subgroup of $\mathbb{R}$ a Borel measurable set? | No. Let $\langle A \rangle$ be a proper Borel subgroup of ${\mathbb R}$ generated by an algebraically independent Borel set $A$ with the cardinality of the continuum (such a set was constructed in
*J. v. Neumann*, MR 1512442 [**Ein System algebraisch unabhängiger zahlen**](http://dx.doi.org/10.1007/BF01459089), *Math... | 15 | https://mathoverflow.net/users/766 | 248526 | 112,986 |
https://mathoverflow.net/questions/238061 | 7 | Let $k$ be a field of characteristic 0. Let $\mathtt{DGA}\_{k}^{+}$ denote the category of non-negative graded DG algebras and $\mathtt{CDGA}\_{k}^{+}$ denote the category of non-negative graded commutative DG algebras. It is well known that there are model structures on them that the weak equivalences are the quasi-is... | https://mathoverflow.net/users/91245 | Cofibrations in the model structures for non-negative graded (commutative) DG algebras | The confusion between your desired statement and the statement from the nlab is arising because of the confusion between chain complexes and cochain complexes.
The answer to your first questions is "yes." I'm not sure about the second question, but I would guess the answer is "no."
---
Because your fibrations ... | 3 | https://mathoverflow.net/users/3075 | 248527 | 112,987 |
https://mathoverflow.net/questions/248497 | 20 | I heard several times the inverse function theorem fails in algebraic geometry. Now I realize I'm pretty confused by this. This question has two parts. The first part asks for the correct formulation of synthetic inverse function theorems. The second asks for clarifications and intuitions about [Penon's paper](http://a... | https://mathoverflow.net/users/69037 | What is meant by the inverse function theorem in algebraic geometry? | Penon's thesis "[De l'infinitésimal au local](http://www.numdam.org/item?id=DIA_1985__S13__1_0)" is often cited in this context. A rough overview of some things he does there (which I got from skimming through):
* He defines the *infinitesimal nbhd of $x\_0\in X$* to be all $x\in X$ such that $\neg \neg (x=x\_0)$ on ... | 9 | https://mathoverflow.net/users/745 | 248532 | 112,989 |
https://mathoverflow.net/questions/248558 | 5 | Let $H$ be a Hopf Algebra (over a field $K$), with comultiplication $\Delta$, counit $\varepsilon$, and antipode $S$.
A $K$-subspace V is said to be:
* A right ideal if $VH \subseteq V$
* A right coideal if $\Delta (V) \subseteq V \otimes H$ (some call this a left coideal...)
Suppose that there is a nonzero right... | https://mathoverflow.net/users/97853 | If a Hopf Algebra has a nontrivial, finite-dimensional right ideal, then it is finite dimensional | This is a part of [Lemma 5.3.1 in Hopf Algebra: an introduction.](https://www.dropbox.com/s/duxytpoxr61ukgu/hopfideal.pdf?dl=0)
```
Dascalescu, Sorin, Constantin Nastasescu, and Serban Raianu. Hopf algebra: An introduction. CRC Press, 2000.
```
| 2 | https://mathoverflow.net/users/11142 | 248567 | 113,002 |
https://mathoverflow.net/questions/248529 | 2 | In financial mathematics, the inverse series of: $$b(x) = -\frac{\log(1-e^{-x})}{x}$$ is needed in order to perform fast calculation on swaptions for G2++ calibration model. (see [this post](https://mathoverflow.net/questions/247977/taylor-series-expansion-for-an-implicitely-defined-family-of-functions) for further ref... | https://mathoverflow.net/users/22279 | Compute inverse series for implicit equation $b=-\log(1-e^{-x})/x$ | Writing $e^{-x} = t$, your equation is $t + t^b = 1$. I'll assume $0 < b < 1$ (for the case $b > 1$, write $s = t^b$ and the equation becomes $s + s^{1/b} = 1$).
Now the slightly more general equation $t + \epsilon t^b = 1$ has a nice series solution
$$ t = 1 - \sum\_{n=1}^\infty \left( \prod\_{j=0}^{n-2} (j-nb)\r... | 6 | https://mathoverflow.net/users/13650 | 248572 | 113,004 |
https://mathoverflow.net/questions/248578 | 8 | *The context for this question is the theory ZFC + a measurable cardinal, although answers not in this context would also be interesting to me.*
In a project I'm working on, the following class of reals has emerged, and I'd like to understand it better:
Say that a real $r$ is **hyperarithmetic modulo ordinals** (an... | https://mathoverflow.net/users/8133 | Which reals are "hyperarithmetic modulo ordinals"? | In fact $\Delta^{1, ord}\_1 = \mathbb{L} \cap \mathbb{R}$.
For suppose $(\phi(x, \overline{y})$, $\psi(x, \overline{y})$, $\overline{\alpha})$ is as you describe, defining $r \subset \omega$. We show $r \in \mathbb{L}$. For convenience we suppose $\overline{\alpha} = \alpha$ is a single ordinal; this is no loss, by c... | 9 | https://mathoverflow.net/users/26705 | 248584 | 113,006 |
https://mathoverflow.net/questions/248599 | 1 | We denote $X\_{T}$ the vector space of all $T$-periodic function with zero mean in $L^2$ ( we know that $X\_{T}$ is spawn by $(e^{2i\pi nt/T})$). Let be $$X=X\_{2\pi}+X\_{3\pi}.$$
I think that $X\_{2\pi}+X\_{3\pi}$ is closed in $L^2(0,4\pi)$ but i can't prove it.
| https://mathoverflow.net/users/78847 | Almost periodic function and closed spaces | I think that it is just a hyperplane $H$ defined by the condition $\int\_0^{\pi}+2\int\_{\pi}^{3\pi}+\int\_{3\pi}^{4\pi}=0$. Obviously all functions in $X\_{2\pi}$ and $X\_{3\pi}$ lie in this hyperplane $H$. Choose any function $f\in H$. Choose functions $a(t),b(t)$ on $[0,\pi)$ so that $f(t+3\pi)-b(t)=f(t)-a(t)$ on $[... | 1 | https://mathoverflow.net/users/4312 | 248606 | 113,016 |
https://mathoverflow.net/questions/236769 | 5 | I wanted to ask you, if you have any good references (book or pdf) to learn about the KK theroy of Kasparov. I think the presentation of Blackadar is too close from the commutative theory.
I was searching for a C\* proof of the Bott periodicity and arrived [in this thread](https://mathoverflow.net/questions/131763/t... | https://mathoverflow.net/users/86526 | reference for KK theory | If you want to learn more about KK-theory there is a book by Kjeld Knudsen Jensen and Klaus Thomsen with the title "Elements of KK-theory". You can have a look there.
I would also recommend those surveys:
<http://www.personal.psu.edu/ndh2/math/Papers_files/Higson%20-%201990%20-%20A%20primer%20on%20KK-theory.pdf>
<h... | 3 | https://mathoverflow.net/users/37905 | 248608 | 113,017 |
https://mathoverflow.net/questions/248607 | 6 | Does anyone maybe have a reference to the proof of the following result by Tate?
Let $\Gamma$ be the absolute Galois group of the rationals. Then the second cohomology group (for trivial $\Gamma$-action) H$^2(\Gamma, \mathbb{Q}/\mathbb{Z})$ is trivial.
Unfortunately I couldn't find it online or in the library. Any ... | https://mathoverflow.net/users/97869 | Reference request to proof that H$^2(\Gamma, \mathbb{Q}/\mathbb{Z}) = 0$ | By the Galois cohomology long exact sequence, this is isomorphic to $\operatorname{H}^3(\Gamma,\mathbb{Z})$, and the vanishing of this is Chapter I, Corollary 4.17 in Milne's *Arithmetic Duality Theorems*.
| 6 | https://mathoverflow.net/users/17907 | 248610 | 113,018 |
https://mathoverflow.net/questions/248547 | 3 | Let us consider, a closed Riemannian surface $(\Sigma,h)$ and a compact Riemannian manifold $(N,g)$ with dimension greater than $3$. If we are given a sequence of harmonic maps $u\_n:(\Sigma,h) \rightarrow (N,g)$ with bounded energy, i.e.
$$E(u\_n)=\int\_\Sigma \vert du\_n\vert^2 \, dv < C,$$
it is well know that we ha... | https://mathoverflow.net/users/9253 | Bubbling example for harmonic maps | Yes. The genus of $\Sigma$ is not really relevant. Here's an example: Let $f$ and $g$ be two meromorphic functions on $\Sigma$, where $g$ is nonconstant, and consider the sequence of maps $u\_n: \Sigma\to N^4 = \mathbb{CP}^1\times\mathbb{CP}^1$ given by
$$
u\_n(p) = \bigl([f(p)],[n\,g(p)]\bigr).
$$
(Here, $N$ is given ... | 8 | https://mathoverflow.net/users/13972 | 248613 | 113,019 |
https://mathoverflow.net/questions/235722 | 0 | Let $\pi:X\to Y$ be a surjective holomorphic map with connected fibers
and let fibers are singular Calabi-Yau varieties (i.e. numerical dimension is zero) then is it possible to construct canonical metric like Weil-Petersson metric on moduli space of such fibers which centeral fiber $X\_0$ has not mild singularity ([ca... | https://mathoverflow.net/users/nan | Canonical metric on moduli space of singular Calabi-Yau varieties | This question has been solved in the paper of [Y.Odaka,](http://annals.math.princeton.edu/2013/177-2/p06) saying that a polarized Calabi-Yau have at worst canonical singularities since they are K-stable. So if the central fibre $X\_0$ be Calabi-Yau variety with canonica singularities then all general fibers $X\_t$ are ... | 2 | https://mathoverflow.net/users/nan | 248615 | 113,021 |
https://mathoverflow.net/questions/196048 | 19 | I'm trying to understand the historical context behind the word *pencil* in matrix pencils, or pencil of curves so on.
I am aware that even Gantmacher 1959 has this terminology however I don't know where it originates from. I am also curious what he uses in the original Russian version in place for that word (though... | https://mathoverflow.net/users/17020 | Why is a matrix pencil called a pencil? | The *Oxford English Dictionary* has an example from 1665 of "pencil" in the sense of "A group of rays or a beam of radiation converging to or diverging from a point." And one from 1840 in the geometric sense of "A set of lines meeting in a point"
| 14 | https://mathoverflow.net/users/454 | 248618 | 113,022 |
https://mathoverflow.net/questions/248561 | 1 | This question has been bogging me down lately. I'm not sure how to come up with an approach to tackle the proof exactly. I'm without a proof, butI think the result I'm searching for is true. Similarly, I do believe I'm not the only person to consider this question and I expect it is already answered somewhere in the ve... | https://mathoverflow.net/users/nan | Is the $n$'th super root analytic in a half plane? | This is not a complete answer, but will give you some pointers in any case.
In our paper *[Bifurcations in the space of exponential maps](http://dx.doi.org/10.1007/s00222-008-0147-5)* (Invent. math. (2009) 175, doi:10.1007/s00222-008-0147-5; see also [arXiv:math/0311480](http://arxiv.org/abs/math/0311480)) with Schl... | 2 | https://mathoverflow.net/users/3651 | 248621 | 113,024 |
https://mathoverflow.net/questions/224075 | 10 | Let $X$ be a scheme over a finite field $\mathbb{F}\_q$ and let $F : X \to X$ be the absolute Frobenius morphism. If $\mathcal{L}$ is an invertible $\mathcal{O}\_X$-module, then there is a natural isomorphism $F^\*(\mathcal{L}) \cong \mathcal{L}^{\otimes q}$.
>
> Is there also a general formula for $F^\*(\mathcal{M... | https://mathoverflow.net/users/2841 | Pullback along Frobenius morphism | Consider the $p$-th tensor power $\mathcal{M}^{\otimes p}$. The group $\mathbb{Z}/p\mathbb{Z}$ acts by cyclic permutations on it. Denote its generator by $\sigma$. There is a map from coinvariants to invariants $$(\mathcal{M}^{\otimes p})\_{\sigma}\xrightarrow{1+\sigma+\dots+ \sigma^{p-1}}(\mathcal{M}^{\otimes p})^{\si... | 8 | https://mathoverflow.net/users/39304 | 248626 | 113,027 |
https://mathoverflow.net/questions/248574 | 5 | Let $f(x)$, $g(x)$ be two univariate, coprime, integer polynomials and let $I=\big(f(x),g(x)\big)$ the ideal of $\mathbb{Z}[x]$ generated by $f, g$. Let $I \cap \mathbb{Z}$, that is, the elements of $\mathbb{Z}$ which can be expressed as linear combinations of $f(x), g(x)$ with coefficients in $\mathbb{Z}[x]$. $I \cap ... | https://mathoverflow.net/users/85967 | Ideal generated by two univariate, coprime, integer polynomials | For question 2, compute a Gröbner basis over $\mathbb{Z}$ for the ideal generated by $f(x)$ and $g(x)$, which gives the required generator. You can do this easily in SageMathCloud (available free to use). The algorithm behind this computation is exactly what you would be doing by hand to find the generator. This also a... | 5 | https://mathoverflow.net/users/8112 | 248628 | 113,029 |
https://mathoverflow.net/questions/248629 | 2 | Let $f$ be a primitive form of an even weight $k\geq 2$ for the full modular group $SL\_2(Z)$. Following from , Proposition 2.3 from
[Z. Rudnick and P. Sarnak, Zeros of principal $L$-functions and random matrix theory, Duke Math. J. 81(1996), 269-322] and a standard
Riemann-Stieltjes partial integration, we plainly ha... | https://mathoverflow.net/users/76102 | Relation between these two sums over prime numbers | Yes, and we don't even need that the RHS tends to infinity. If we perform the partial integration carefully (namely if we examine how the $O(1)$ term behaves at infinity), we find that
$$ \sum\_{p\leq x} \frac{|\lambda\_f(p)|^2}{p}= \log{\log{x}}+c\_f+O\left(\frac{1}{\log x}\right), $$
where $c\_f$ is a constant. Simil... | 3 | https://mathoverflow.net/users/11919 | 248639 | 113,032 |
https://mathoverflow.net/questions/248452 | 4 | So the original question I wanted to ask was this one:
I'm currently a bit puzzled about the normalization for the Gelfand transform $U$:
So if we have a periodic Schrödinger operator $H$, then we can decompose it into a direct integral over the (what the physicists call) Brioullin zone $B$ (that is roughly the idea)... | https://mathoverflow.net/users/97785 | Question about normalization factors in the direct integral of operators | Well I think the point is that a "direct integral of operators" just means that you have a bounded measurable family of operators on the fibers of a bundle of Hilbert spaces, and you apply them all at once to get a single operator on the space of $L^2$ sections of the bundle. Of course the space of $L^2$ sections of th... | 1 | https://mathoverflow.net/users/23141 | 248642 | 113,033 |
https://mathoverflow.net/questions/247959 | 2 | A well known result in Symbolic Dynamics asserts that every **two-sided** subshift on a finite alphabet necessarily consists of all doubly infinite words not containing any finite word from a given set of "forbidden" words (Proposition 1.3.4, D. Lind and B. Marcus, An introduction to symbolic dynamics and coding, Cambr... | https://mathoverflow.net/users/97532 | Reference for one-sided subshifts | My friend Marcelo Sobottka came up with the reference:
>
> W. Ott, M. Tomforde, P. N. Willis; One-sided shift spaces over
> infinite alphabets, NYJM Monographs (5) 2014.
>
>
>
which answers my question in:
>
> Theorem 3.16. A subset $X \subseteq \Sigma\_\mathcal A$ is a shift space if and only if $X = X\_... | 2 | https://mathoverflow.net/users/97532 | 248643 | 113,034 |
https://mathoverflow.net/questions/248632 | 4 | A graph is a **minimal graph of diameter $2$** if it has diameter $2$ and the deletion of any edge increases its diameter.
Let $G$ be a minimal graph of diameter $2$, then every edge of $G$ belongs to a path $u$-$v$ in $G$ which is the unique path of length at most $2$ with endpoints $u$ and $v$.
My question is: If... | https://mathoverflow.net/users/58096 | A question about minimal graph of diameter $2$ | If I understand the question correctly, the answer is *no*.
Let $H$ be the 5-vertex graph consisting of a triangle with "tails" on two vertices; more explicitly let $V(H)=\{1,2,3,4,5\}$ and $E(H)=\{\{1,2\},\{2,3\},\{1,3\},\{1,4\},\{2,5\}\}$.
Then $H$ has the stated property: each edge is part of a path of length $\... | 2 | https://mathoverflow.net/users/12419 | 248649 | 113,035 |
https://mathoverflow.net/questions/248571 | 3 | I want to evaluate the following sum:
$\sum\_{\lambda}e\_k(\lambda)$ where the sum is over all partitions $\lambda$ that fit in an $m$ by $n$ grid and where $e\_k(\lambda):=e\_k(\lambda\_1,\lambda\_2,\dots,\lambda\_m)$ for $\lambda$ possibly padded with zeros if $\lambda$ does not have $m$ parts. Also $k\leq m$.
| https://mathoverflow.net/users/82020 | elementary symmetric function identity | I do not expect a nice closed formula for arbitrary $k$, in general. However, we could use the following recursive relation to extract some evaluations. This recurrence, we're alluding to, is already immediate from that of the elementary symmetric functions. Denote the given sum in the problem by
$$F\_k(m,n)=\sum\_{\la... | 4 | https://mathoverflow.net/users/66131 | 248652 | 113,036 |
https://mathoverflow.net/questions/248616 | 7 | Let $\text{SO}\_{p,q}$ be the special orthogonal group of type $(p,q)$. In [D. Prasad's lecture notes on algebraic groups](http://www.math.tifr.res.in/%7Edprasad/alg-grp.pdf) the statement
$$\text{"The group $\text{SO}(p, q)$ is quasi-split if and only if $|p − q| ≤ 2$."}$$
is given without proof or further explanation... | https://mathoverflow.net/users/nan | Condition for quasi-splitting of special orthogonal group | Since the OP asked for a reference, I would like to provide some. First of all, a group is quasi-split iff its Satake diagram does not contain black dots. The list of Statake diagrams can be found in many places but most of them are difficult to understand out of context. Of the more useful kind are the tables in
Oni... | 2 | https://mathoverflow.net/users/89948 | 248656 | 113,039 |
https://mathoverflow.net/questions/248485 | 9 | There's something that's been confusing me about symplectic field theory, and I'm not sure if I can formulate it exactly, but I'll try asking an imprecise question.
Symplectic field theory, as described in the seminal paper (<https://arxiv.org/abs/math/0010059>) of Eliashberg-Givental-Hofer, appears superficially ver... | https://mathoverflow.net/users/88749 | Relation of SFT and Gromov-Witten theory | What follows is a guess. I don't know enough about Symplectic Field Theory to be sure.
In perturbative string theory, one describes physics in the target spacetime $X$ by summing over maps from Riemann surfaces into $X$. These sums have an interpretation in terms of a 2d CFT living on the Riemann surfaces, but for p... | 5 | https://mathoverflow.net/users/35508 | 248664 | 113,041 |
https://mathoverflow.net/questions/248662 | 6 | The model structure on sSet for quasi-categories (the Joyal model structure) is not enriched over sSet with the Quillen model structure, so ordinary internal Hom of simplicial sets is not a correct homotopy function complex of quasi-categories. We can give a correct definition using either cosimplicial or simplicial re... | https://mathoverflow.net/users/62782 | Homotopy function complex for quasi-categories | **Yes**, you can compute the mapping spaces in ∞-categories by taking the biggest Kan subcomplex of the internal hom.
The trick is not to use the Joyal model structure, but instead the model structure on marked simplicial sets defined in Higher Topos Theory, proposition 3.1.3.7 (in the case $S=\Delta^0$). By proposit... | 8 | https://mathoverflow.net/users/43054 | 248675 | 113,044 |
https://mathoverflow.net/questions/247331 | 7 | This question is similar to another that I asked, but should be, I think, very much easier.
Start with the generating function for unrestricted partitions and replace some of the plus signs with minus signs to get:
\begin{align}
&(1\pm x\pm x^2\pm x^3\pm\cdots)(1\pm x^2\pm x^4\pm x^6\pm\cdots)(1\pm x^3\pm x^6\pm x^... | https://mathoverflow.net/users/40145 | Another question related to the generating function for unrestricted partitions | It is possible. Select positive signs for all factors but the first. The product of all these factors is then the generating function for partitions without 1's. Since this sequence is sub-doubling, G Pasemans' answer to question
[Will this greedy algorithm always work?](https://mathoverflow.net/questions/248422/will-t... | 1 | https://mathoverflow.net/users/97889 | 248679 | 113,046 |
https://mathoverflow.net/questions/248661 | 4 | Let $G\subset O(n)$ be a finite orthogonal group.
>
> Is there a regular CW-complex structure on $S^{n-1}$ on which $G$ acts cellularly which is in any sense "natural"?
>
>
>
What I'm looking for is inspired by the following examples and ideally would generalize them:
(a) If $G$ is a reflection group, then ... | https://mathoverflow.net/users/12419 | Given finite $G\subset O(n)$, is there a "standard" cell structure on $S^{n-1}$ with $G$ acting cellularly? | For a first pass at an answer:
Fix a point of the sphere with trivial stabilizer in $G$ and take the Voronoi diagram of its orbit under $G$. The top-dimensional cells are the Voronoi cells, which are polyhedra, and the $d$-dimensional cells are the $d$-faces of these polyhedra.
This construction has property (1) - ... | 0 | https://mathoverflow.net/users/12419 | 248680 | 113,047 |
https://mathoverflow.net/questions/248651 | 8 | Let $$ \cdots \to \Gamma\_n \to \Gamma\_{n-1} \to \cdots \to \Gamma\_0$$ be an inverse system countable groups and let's assume (for this post) that all homomorphisms in such an inverse system are surjective. We say that $\Gamma$ is *an* inverse limit if there exists compatible surjective homomorphisms $\varphi\_n \col... | https://mathoverflow.net/users/8176 | Amenable inverse limits of torsionfree amenable groups | Suppose that $\Gamma\_n$ is the free nilpotent group on $k$ generators of nilpotency class $n$. Any finite $k$-generated $p$-group ($p$ a prime) is nilpotent, hence the free $k$-generated pro-$p$ group is surjected by the (universal) inverse limit of $\Gamma\_n$. Hence any inverse limit of $\Gamma\_n$ maps to a dense s... | 8 | https://mathoverflow.net/users/1345 | 248681 | 113,048 |
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