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most recent 30 from 2018-07-21T14:21:01Z

Canonical map in the direct image of $\mathscr{D}_X$

Tue, 04/24/2018 - 10:36

Let $f : X \to Y$ be a proper holomorphic map between holomorphic manifolds. We work with $\mathscr{D}$-modules. Consider the transfer bi-modules $\mathscr{D}_{Y\leftarrow X}.$ Can one find a canonical arrow $$\Omega_Y \to \Omega_Y \otimes_{\mathscr{D}_Y} f_+(\mathscr{D}_X) := \Omega_Y \overset{L}\otimes_{\mathscr{D}_Y}Rf_*(\mathscr{D}_{Y \leftarrow X}) \,\,\, ?$$ In some sense, I would like to use the canonical section $1_{Y \leftarrow X}$ but the derived functor makes things more complicated. If it is impossible in the general case, one can assume that the manifolds are $\mathbb{C}$-vector spaces. Thanks for any help.

A measure of noncompactness by a convex function

Tue, 04/24/2018 - 09:23

Let $E, \left \| \right \|$ be a Banach space, and indicate a family of all nonempty bounded subset of $E$.

Definition: A mapping $\mu:\mathfrak{M}_E\rightarrow \mathbb R^+$ is said to be a measure of noncompactness in $E$ if it satisfies the following conditions:

1) The family $\operatorname{ker} \mu $ is nonempty and $\operatorname{ker} \mu \subset \mathfrak{N}_E$.

2) $X\subseteq Y\Rightarrow \mu (X)\leq \mu (Y)$.

3) $\mu (\bar{X})=\mu (X)$.

4) $\mu (\operatorname{Conv}(X))=\mu (X)$.

5) $\mu (\lambda X+(1-\lambda) Y)\leq \lambda \mu (X)+(1-\lambda) \mu(Y)$ for $\lambda\in [0,1]$.

6)If $\{X_n\}\subset \mathfrak{M}_E^c$, such that $X_{n+1}\subset X_n$ for $n=1,2,...$ and if $\displaystyle\lim_{n\rightarrow \infty} \mu(X_n)=0$ then $X_{\infty}=\bigcap_{n=1}^{\infty}X_n\neq \emptyset$.

Now, I have to prove this this:

Suppose $\mu_1,\mu_2$ are two measures of noncompactness in $E_1,E_2$ respectively. Moreover, assume that the function $F:\mathbb R^2\rightarrow \mathbb R^+$ is a convex function and $F(x_1,x_2)=0\Leftrightarrow (x_1,x_2)=0$. Then $\mu (X)=F(\mu_1(X_1),\mu_2(X_2))$ defines a measure of noncompactness in $E_1\times E_2$ where $X_i$ denote the natural projection of $X$ into $E_i$ for $i=1,2$.

It's okay with 1, 3, 4 and 6 of the definition. How can we prove 2 and 5?

EDIT: This result (without proof) comes from : Bana’s, J., Goebel, K., Measures of Noncompactness in Banach Spaces. Lecture Notes in Pure and Applied Mathematics, vol. 60, New York: Dekker 1980

polynomial goal programming?

Tue, 04/24/2018 - 09:22

I am learning about portfolio optimization considering skewness and kurtosis. I read several papers on this topic,

They all talked about this polynomial goal programming (PGP). However, from my previous study, I only know about LP,QP,MIP,SOCP, and SDP. Does this PGP belongs to any class of the method I mentioned (SOCP, and SDP) (I suppose not since they seem to mention that this is nonconvex)? I am wondering what is the state of the art situations now for PGP, are there any well developed solver/modelling language for PGP? (solvers like cvxopt,gurobi,cplex, modeling language like picos in python) Thanks.

Minimal refinements of open covers of $T_2$-spaces

Tue, 04/24/2018 - 08:33

Let $(X,\tau)$ be a topological space. We say ${\cal U}\subseteq \tau$ is an open cover if

  • $\bigcup {\cal U} = X$, and
  • $X\notin {\cal U}$.

${\cal U}$ is minimal if for all $U_0\in {\cal U}$ we have $\bigcup ({\cal U}\setminus \{U_0\}) \neq X$. Clearly, every $T_1$-space on more than $1$ point possesses a minimal cover: pick $x\neq y\in X$ and let ${\cal U} = \{X\setminus\{x\}, X\setminus\{y\}\}$.

Question. Given any open cover of a Hausdorff space $(X,\tau)$ with $|X|>1$, does it have a refinement that is a minimal cover?

Note. I didn't ask for subcovers in the question above: Let $X = \mathbb{R}$ with the Euclidean topology, then ${\cal U} = \big\{ \{x\in\mathbb{R}: x < n\} : n\in\mathbb{N}, n\geq 1 \big\}$ does not have a minimal subcover.

Reference for De Giorgi-Nash-Moser theory

Tue, 04/24/2018 - 08:16

I am interested in Holder regularity for equations of the form $$u_t - div A(x,t) \nabla u = 0$$ where $A(x,t)$ is bounded, measurable and elliptic.

This was proved in the seminal paper of John Nash and later on by J. Moser.

I am looking for references where Ennio De Giorgi's methods are implemented to obtain the same regularity estimates.

Topological cycles with Lagrangian support

Tue, 04/24/2018 - 08:15

For a compact Kähler manifold of dimension $2n$, is there a classification of the homological $n$-cycles which are supported in a compact Lagrangian submanifold?

The main example for this question is a projective manifold with the symplectic structure coming from Fubini-study metric. Using symplectic Picard-Lefschetz theory one can describe many Lagrangians which are vanishing cycles, and explicit computations of the monodromy group helps a lot.

Status of the three-body problem

Tue, 04/24/2018 - 08:01

I find many numerical results on the three-body problem, but what is rigorously proved? Especially I would be interested in the parameter domains for which we have rigorous lower bounds on the topological entropy or Lyapunov exponent of the system.

Status of two Banach space theory open problems posted by Pełczyński

Tue, 04/24/2018 - 07:48

In the book 'Open Problems in the Geometry and Analysis of Banach Spaces', I am interested in the following two problems.

Problem $1$: Let $X$ be a separable infinite-dimensional Banach space that is not isomorphic to a Hilbert space.

(i) (A. Pełczyński) Does there exist an infinite dimensional subspace of $X$ with Schauder basis that is not complemented in $X?$

(ii) (A. Pełczyński) Do there exist two infinite-dimensional subspaces of $X$ with Schauder basis that are not isomorphic?

I would like to know status of the two problems above. Are they solved or still open?

PhD dissertations that opened a new field of research [on hold]

Tue, 04/24/2018 - 05:55

I propose this as a companion wiki page to the one about PhD dissertations which contain a solution to an open problem in the style of big-list questions, thinking in terms of the well-known paradigm that splits mathematical research into problem solving and theory building. Theories are at times developed to solve famous open problems, but sometimes the concrete problems they solve are quickly dwarfed by the possibilities that a new theory opens.

Can you name modern mathematicians who already in their PhD theses (or earlier in their career) developed a substantial new theory or laid the foundations of a new field of research?

If the center of a finite group is trivial, are there two elements whose centralizers intersect trivially?

Tue, 04/24/2018 - 03:27

Let $G$ be a (discrete) group. Define $k^*(G)$ as the minimal cardinality of a set $S \subset G$ such that $C_G(S) = Z(G)$. Define $k(G) = k^*(G)$ if $G$ has trivial center (i.e. $|Z(G)| = 1$), and $k(G) = \bot$ otherwise. If $k(G) = \bot$, then the convention is that neither $k(G) \leq n$ nor $k(G) \geq n$ holds, for any cardinal $n$.

Question: Does there exist a finite group $G$ such that $k(G) \geq 3$, and more generally does every natural number $k(G) \geq 3$ occur for some finite group $G$?

I do not even know such examples for $G$ infinite, and would also be interested in such, though I do not have an immediate application for this. I am not an expert on group theory (especially finite group theory), so I do not know very effective search terms for this, and would also be interested in pointers to the literature.

What I have tried so far (though don't take my word on these):

  1. No abelian group or a p-group or a nilpotent group is an example, since they have nontrivial centers (in the finite case), thus $k(G) = \bot$.

  2. No finite simple group is an example, since they are all 2-generated (by CFSG), thus satisfy $k(G) \leq 2$ or $k(G) = \bot$.

  3. $k(G \times H) = \max(k(G), k(H))$ for any groups $G, H$ (by a simple proof).

  4. I did a quick search in GAP and seems that there are no finite groups of size up to $1151$ with this property (this is the first time I used GAP, so not sure how much proof value this has).

  5. $k(G) = 0$ for precisely the trivial group, and $k(G) = 1$ is impossible (since any $g$ commutes with itself).

  6. For infinite cardinal $\kappa$, $k(G) = \kappa$ where $G$ is the group of finite-support permutations on a set of cardinality $\kappa$, but of course $k(G)$ is finite (or $\bot$) for finite groups.

  7. Arbitrarily large $k^*(G)$ are provided by wreath products $\mathbb{Z}_2 \wr \mathbb{Z}_2^d$, where $\mathbb{Z}_2 = \mathbb{Z}/2\mathbb{Z}$, but I wrote a quick proof sketch that $k(G \wr H) = 2$ whenever $|H| \geq 2$ and $k(G) = 2$ so it seems one cannot use wreath products to get examples.

  8. For a single permutation $\pi$ on a set of size $n$, The centralizer of $\pi$ in $S_n$ never has $k(G) \geq 3$ (I prove this as Proposition 7.9 in this paper of mine, by a rather ad hoc case analysis).

  9. In terms of the commuting graph $\Gamma(G)$ with vertices $G \setminus Z(G)$ and edges $\{(g, h) \;|\; gh = hg\}$, the question of whether $k(G) \geq 3$ is possible is equivalent to whether there exists a finite group $G$ with trivial center such that $\mathrm{diam}(\Gamma(G)) = 2$, where $\mathrm{diam}$ is the diameter, i.e. maximal minimal distance between a pair of vertices. For a finite minimal nonsolvable group the diameter is always at least $3$ according to this paper which implies $k(G) = 2$ for a minimal nonsolvable group (a different definition is used in that paper, but it should be equivalent to mine for minimal nonsolvable groups, as they have trivial center). Most literature I know about this graph and its diameter are about finding upper bounds, but I do not know if that has a relation to $k(G)$. Larger values of $k(G)$ also correspond to statements about this graph, but not about its diameter.

For context, the question arose from the study of automorphism groups of one-dimensional subshifts. I am interested in quantitative versions (or lack thereof) of the so-called Ryan's theorem, which states that the center of the automorphism group of a mixing subshift of finite type consists of only the shift maps. I ask the above question about finite groups after Lemma 7.7 here and Lemma 7.7 is my application for it. The paper of Boyle, Lind and Rudolph is a standard reference for these groups.

What is the "dual" of the space of currents?

Tue, 04/24/2018 - 03:14

On a smooth maniflod $M$ of dimension $n$, a current of degree $n-p$ is a functional on the space of compactly supported differential $p$-forms which is continuos. We denote the space of currents of degree $n-p$ by $D^{'n-p}(M)$. If we consider the functionals on $D^{'n-p}(M)$ with an approperiate comapctness and continuity assumptions, then what are these functionals? Are they just differential $p$-forms, or could be more then that?

I am asking this question because I want to understand why there is no definition for the pullback of current $T$ in general (as far as I know). Let $f: M_1 \to M_2$ be a map between manifolds. Then the pullback $f^*T$ should be defined as (formally) $$\langle f^*T, u\rangle = \langle T, f_*u \rangle.$$ Here $u$ is a differential $p$-form, in particular, $f_*u$ is well-defined as a current under some compactness assumption. Thus the problem is to make sense of $\langle T, f_*u \rangle$, i.e. can a current be a functional on the space of current? (Of course, it is enough to have $T$ be a functional on the space $\{f_*u\}$, that is why pullback of current is well-defined for submersion maps).

PhD dissertations that solve an established open problem

Tue, 04/24/2018 - 01:19

I search for a big list of open problems which have been solved in a PhD thesis by the Author of the thesis (and her/his supervisor).

In my question I search for every possible open problem but I prefer (but not restricted) to receive answers about those open problems which had been unsolved for at least (about) 25 years and before the appearance of the ultimate solution, there had been significant attentions and efforts for solving it. I mean that the problem was not a forgotten problem.

I ask the moderators to consider this question as a wiki question.

$SU(2)\times SU(2)$ invariant $SU(3)$-structure on $\{t\} \times M^6$

Mon, 04/23/2018 - 13:36

I am reading Jason Lotay and Goncalo Oliveira's paper -$SU(2)^2$ invariant $G_2$-instantons, and have few questions from the same.

If we consider the space $M = S^3 \times S^3$. Then the cone metric can be written as

$g = dt^2 + g_t$, where $g_t = \sum_{i=1}^3(2A_i)^2\eta_i^+ \wedge \eta_i^+ + (2B_i)^2\eta_i^- \wedge \eta_i^-$ is the compatible metric given by the $SU(2)\times SU(2)$ invariant $SU(3)$ structure on $\{t\}\times M$.

Here $\eta_i^\pm$ are the $\eta_i^\pm$ is the standard coframe of 1-forms, while the functions $A_i(t), B_i(t)$ specify the deformation of the cone singularity. I have the following questions:

  1. How can we see that the metric $g_t$ (and in general the SU(3) structure as given) is $SU(2)\times SU(2)$ invariant?

  2. Why an extra $U(1)$ symmetry forces $A_2=A_3, B_2=B_3$?

  3. Why an extra $SU(2)$ symmetry forces $A_1=A_2=A_3, B_1=B_2=B_3$?


On solutions of the continuity equation

Mon, 04/23/2018 - 08:02

Can all square integrable solutions $(\rho(t,x),j(t,x))$ of the homogeneous continuity equation $$\dot\rho(t,x)+\nabla \cdot j(t,x)=0$$ in 1+3 dimensions be approximated by solutions with compact support (in both space and time)? What are the simplest nontrivial solutions with compact support?

A probabilistic angle inequality

Mon, 04/23/2018 - 07:24

Conjecture: There is a universal constant $c$ such that for any fixed nonzero real vector $q$ of any dimension $n$ and any random vector $p$ of the same dimension $n$ with independent components uniformly distributed in $[-1,1]$, we have $$(p^Tp)(q^Tq)\le cn(p^Tq)^2$$ with probability $\ge 1/2$. Simulation suggests that a constant $c\approx 8$ should work.

I'd be interested in techniques for proving this and related statements. In particular, can one determine the best constant $c$?

Reduction of structure group for stacks

Sat, 04/21/2018 - 12:38

Consider an action of a smooth linear algebraic group $G$ on a variety $X$ over an arbitrary field $k$, and the quotient stack $[X/G]$. Let $p$ be a $k$-point of $X$. If the action is transitive (i.e. $G(\bar{k})$ acts transitively on $X(\bar{k})$) then we have an isomorphism $[X/G]\cong BG_p$, where $G_p$ is the scheme-theoretic stabilizer of $p$, given as follows. In one direction, send a $G$-torsor $P$ with $G$-equivariant map $\pi: P\to X$ to $\pi^{-1}(p)$ and on the other direction send a $G_p$-torsor $Q$ to $Q\times^{G_p}G$.

This cannot be right, because if $q$ is another $k$-point of $X$ then this implies $G_p\cong G_q$, but I think we only have $(G_p)_{\bar{k}}\cong (G_q)_{\bar{k}}$ (it can happen that $p$ and $q$ are conjugate over $\bar{k}$, but not over $k$). If $[X/G]\cong BG_p$ is wrong, how do I correct it?

Who wins the Rubik's cube game?

Sat, 04/21/2018 - 11:53

This game has two players, Spoiler and Solver. We start with a solved 3x3x3 rubik's cube(to make the problem easier).

Spoiler and Solver make turns making 90 degree twists. The cube is forbidden from ever repeating a position (besides the start position). This guarantees the game is finite.

If at any point (besides the beginning), the rubik's cube is in a solved state, Solver wins. If the game ends before that (because a position is entered with no valid moves), Spoiler wins.

An example game would be F,F;F,F (using basic rubik's cube notation). Solver wins this game. If a game goes through each position that is one move away from the solved state, and afterwards goes to some unsolved state, Spoiler will win (since it is now impossible to get to the solved state).

So, which player has the winning strategy?

EDIT: It may be simpler to consider the same problem with the 15 puzzle first.

Resolution of Simplicial Commutative Rings

Sat, 04/21/2018 - 11:08

I have just started learning some derived algebraic geometry. I was told that (if $ \mathrm{char}(\mathbb{K})=0 $) using commutative differential graded algebras in negative degree (for short $ \mathrm{cdga}^{\leq 0} $) is easier than using simplicial commutative rings (for short $ sComm $). I suppose that it is easier because one have to put less effort in computing fibrant/cofibrant replacements (for instance when $ I\subseteq R $ is an ideal generated by a regular sequence then we can use Koszul complex for the replacement of $ {R}/{I} $). Are there any further advantages?

Here follows the main question I am struggling with: is there any canonical (in some sense) way of computing fibrant/cofibrant replacement of objects in $ sComm $?

Tate modules of commutative group schemes over finite field

Sat, 04/21/2018 - 09:38

Let $G$ be a finite type commutative group scheme over a finite field $k=\Bbb F_q$ with $\Gamma=Gal(\bar k/k)$ and $l$ be a prime such that $(l,q)=1$, we can also define the Tate module $T_lG =\varprojlim_nG[l^n](\bar k)$. Then we can also consider the natural map

$Hom_k(G_1,G_2) \otimes \Bbb Z_l \rightarrow Hom_\Gamma(T_lG_1,T_lG_2)$

for any two commutative group scheme over $k$, or more generally the map

$\text{Ext}^i_k(G_1,G_2) \otimes \Bbb Z_l \rightarrow \text{Ext}^i_\Gamma(T_lG_1,T_lG_2)$ for every natural number $i$ at least when the extension groups are well-defined (This question may be related).

My first question is: when is this map surjective, injective or an isomorphism? If $G_i$ are abelian varieties and $i=0$, it's well-known that this map is an isomorphism. If $G_2=\Bbb G_m$ and $G_1$ is an abelian variety, this is also true for $i=0$ as both sides are zero, and is true for $i=1$ and $\text{dim}G_1=1$ as $\text{Ext}^1(A,\Bbb G_m)$ is the dual of $A$ and one can compute both sides explicitly (see this master thesis). However, this map can fail to be injective for trivial reasons (For instance, right side could be zero for some finite abelian groups).

So my second question is: how to describe extension groups between commutative group schemes using linear algebra datas in a systematic way? For instance, how to describe extension groups of finite group schemes like $\alpha_p$, $\mu_p$?

The last question is about growth of $\#G[l^n]$ for commutative group schemes (in order to study the Tate module). Let $k$ be any field, $G$ be a commutative group scheme over a field $k$ (not necessarily finite type) and $l$ be a prime. Assume $G[p^n]$ is a finite group scheme for every $n$, does there always exist $C >0,h \geq 0$ such that $\#G[p^n]=Cp^{nh}$ for $n >> 0$? Here the order of a finite group scheme means the dimension of its global section ring. I know the result holds for etale group schemes and abelian varieties. For example, if $G$ is constant i.e an abstract abelian group and $\#G[p^n]$ is finite for every $n$, then we know $G[p^\infty]\cong (\Bbb Q_p/\Bbb Z_p)^h\oplus T$ for some $h \geq 0$ and finite $p$-group $T$ (this result is used when studying Tate-Shafarevich group of elliptic curves).

solving the equation f'(x)=f-1(x)^for x in R [duplicate]

Sat, 04/21/2018 - 09:36

This question already has an answer here:

what are the solutions f :R=>R ,that verifie the equation ,for x in R, f'(x)=f^-1(x) ,where f^-1 is the reciprocal function of f.