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most recent 30 from mathoverflow.net 2018-07-15T13:14:40Z

Different norms in decompositions of matrices

Mon, 04/09/2018 - 09:22

It is known that the nuclear norm (trace norm) $\|A\|_*$ of a complex matrix $A$ is less than 1 if and only if $A$ can be written as a convex combination $$A = \sum_i c_i x_i y_i^*$$ for non-negative coefficients $c_i$ such that $\sum_i c_i = 1$ and some vectors $x_i, y_i$ whose Euclidean norm satisfies $\|x_i\| = \|y_i\| = 1$.

Now, say that $\|A\|_*\leq 1$, and additionally we know that A can be written as a convex combination $$A = \sum_i c'_i x'_i y'^{*}_i$$ where now the vectors $x'_i, y'_i$ satisfy $\|x'_i\|_\infty \leq \mu$ and $\|y_i\|_\infty \leq \mu$ for some $\mu < 1$.

Does this necessarily mean that A can be written as a convex combination $$A = \sum_i c''_i x''_i y''^{*}_i$$ with the vectors satisfying both $\|x''_i\| = \|y''_i\| = 1$ and $\|x''_i\|_\infty \leq \mu$, $\|y''_i\|_\infty \leq \mu$ ? Or perhaps $\|x''_i\|_\infty \leq \mu'$, $\|y''_i\|_\infty \leq \mu'$ for a different choice of $\mu'$?

I am trying to understand what exactly we can say about $A$ if we know that it admits the first two convex decompositions as mentioned above. Could we say more in cases when, for example, A is positive semidefinite?

Any ideas and thoughts about this problem would be appreciated. I have been stuck on this for a long time and not able to come up with any good way to approach it at all.

My colleague and I have deduced some incredibly useful information based on a few widely-unknown formulae. Take a look! [on hold]

Mon, 04/09/2018 - 09:20

These deductions are found primarily with the laws of pentallelograms and aerodynamics.

Work

How to equate a text on left hand side to an inserted image on right hand side in latex? [on hold]

Mon, 04/09/2018 - 09:18

I want the text in left hand side to be equated to an inserted image on left hand side as shown below:

Here, i wish the text in left hand side to be written in latex and equate it horizontally to an inserted image on right hand side. Also, how to insert text between images horizontally?

Is the Milnor boundary map, a natural transformation?

Mon, 04/09/2018 - 09:00

Consider the Milnor $K_n$-functors for discrete valuiation fields. For any discrete valuation field $F$ we can associate an abelian group $K_n(F)$ and the construction is given thanks a universal property involving Steinberg map. We also have a map called $r$-th boundary map, which is given by: $$\partial_n: K_n(F)\to K_{n-1}(\overline F)$$

We can see $\partial_n$ as a transformation between functors, and my question is the following one:

Is $\partial_n$ a natural tranformation?

Suppose that we have an embedding of discrete valuation fields $F\to L$ which gives $\overline F\to \overline L$; then is the following diagram commutative?

Connection on a Principal bundle and transition functions, as in Hitchin's notes

Mon, 04/09/2018 - 09:00

This is along the lines of this question on gerbes.

Gerbes are not just topological objects: we can do differential geometry with them too. We shall next describe what a connection on a gerbe is. To begin with, let’s look at a connection on a line bundle which is given by transition functions $g_{\alpha\beta}:U_\alpha\cap U_\beta\rightarrow S^1\subseteq \mathbb{C}^*$.

A connection on consists of $1$ forms $A_\alpha$ defined on $U_\alpha$ such that on a twofold intersection $U_\alpha\cap U_\beta$ we have $iA_\alpha-iA_\beta=g_{\alpha\beta}^{-1}dg_{\alpha\beta}$

I know what is a connection $1$ form but not as in above version. I am trying to relate what I know with what is given here.

Let $\pi:P\rightarrow M$ be a principal $G$ bundle with $\mathfrak{g}$ being the lie algebra of $G$.

Transition functions $g_{\alpha\beta}:U_\alpha\cap U_\beta\rightarrow G$ are as in this question

Definition : A connection form on $P$ is a $\mathfrak{g}$ valued $1$ form $\omega$ on $P$ such that

  • $\omega(p)(A^*(p))=A$ for all $A\in \mathfrak{g}$ and $p\in P$
  • $(\delta_g^*\omega)(p)(v)=Ad_{g^{-1}}(\omega(p)(v))$ for all $p\in P,g\in G$ and $v\in T_pP$.

Given a connection $1$ form $\omega$ on $P$ I am trying to associate a collection of $1$ forms $\{A_\alpha\}$ with some compatability conditions (here $A_\alpha$ is a $1$ form on $U_\alpha$).

Given local trivialization $\psi_\alpha$, we have a section of $\pi$ namely $\sigma_\alpha:U_\alpha\rightarrow P$ defined as $\sigma_\alpha(x)=\psi_\alpha^{-1}(x,e)$. Given local trivialization $\psi_\beta$, we have a section of $\pi$ namely $\sigma_\beta:U_\beta\rightarrow P$ defined as $\sigma_\beta(x)=\psi_\beta^{-1}(x,e)$.

Suppose $x\in U_\alpha\cap U_\beta$ then, we have $\sigma_\alpha(x)\in \pi^{-1}(x)$ and $\sigma_\beta(x)\in \pi^{-1}(x)$. Thus, there exists $g\in G$ (depending on $x$) such that $\sigma_\alpha(x)=\sigma_\beta(x)g$. Given $x\in U_\alpha\cap U_\beta$ there is an obvious choice for an element of $G$ namely $g_{\alpha\beta}(x)$. I could not prove (I am missing something obvious) but have seen that the $g$ that satisfy the condition as mentioned above is actually $g_{\alpha\beta}(x)$ i.e., we have $\sigma_\alpha(x)=\sigma_\beta(x)g_{\alpha\beta}(x)$ for all $x\in U_{\alpha}\cap U_\beta$ i.e., $\sigma_\alpha=\sigma_\beta g_{\alpha\beta}$.

Given a $1$ form $\omega$ on $P$ and we can pull back $\omega$ to $U_\alpha$ under $\sigma_\alpha$ to get $1$ form $\omega_\alpha=\sigma_\alpha^*\omega$ on $U_\alpha$ similarly we can pull back to $U_\beta$ to get $1$ form $\omega_\beta=\sigma_\beta^*\omega$ on $U_\beta$.

As $\sigma_\alpha$ and $\sigma_\beta$ are related by $\sigma_\alpha=\sigma_\beta g_{\alpha\beta}$, one can expect that $\omega_\alpha$ and $\omega_\beta$ are related some how. Given $g_{\alpha\beta}:U_{\alpha\beta}\rightarrow G$ we can produce a $1$ form on $U_\alpha\beta$ as pull back of $\theta$ on $G$ i.e., the canonical $1$ form on $G$ which is a left invariant $1$ form determined by $\theta(e)(A)=A$ for all $A\in \mathfrak{g}$. Let us denote pull back of $\theta$ to $U_\alpha\cap U_\beta$ by $\theta_{\alpha\beta}$. Then, I am expecting some compatibility relation between $1$ forms $\omega_\alpha,\omega_\beta$ and $\theta_{\alpha\beta}$ that should come from $\sigma_\alpha=\sigma_\beta g_{\alpha\beta}$.

Question : What could be reasonable relation between $\omega_\alpha,\omega_\beta$ and $\theta_{\alpha\beta}$ and how do I see that relation giving $iA_\alpha-iA_\beta=g_{\alpha\beta}^{-1}dg_{\alpha\beta}$ as a sepcial case.

Any suggestion on how to see this is welcome.

Noncompact dual of $\mathrm{Spin}(2n)$ corresponding to $\mathfrak{so}^*(2n)$

Mon, 04/09/2018 - 08:50

Let $U=\mathrm{Spin}(2n)$, which is a simply connected compact simple Lie group, and let $\mathfrak{u}_0=\mathfrak{so}(2n)$, the Lie algebra of $U$. If $\mathfrak{g}_0$ is a noncompact dual of $\mathfrak{u}_0$ in the complexified Lie algebra $\mathfrak{g}=\mathfrak{so}(2n,\mathbb{C})$, namely $\mathfrak{g}_0=\mathfrak{u}_0^\theta+\sqrt{-1}\mathfrak{u}_0^{-\theta}$ for some involutive automorphism $\theta$ of $\mathfrak{u}_0$, where $\mathfrak{u}_0^{\pm\theta}=\{X\in\mathfrak{u}_0\mid\theta(X)=\pm X\}$. Then there exists a noncompact closed subgroup $G$ of $G_\mathbb{C}=\mathrm{Spin}(2n,\mathbb{C})$. For example, if $\mathfrak{g}_0=\mathfrak{so}(m,2n-m)$, then $G=\mathrm{Spin}(m,2n-m)$.

QUESTION

What is $G$ when $\mathfrak{g}_0=\mathfrak{so}^*(2n)$?

I am not sure whether the question fits the level of MathOverFlow. I would like to say sorry if the question is too fundamental to be posted here.

Singular Value Decomposition of product of a lower unitriangular and upper unitriangular matrices

Mon, 04/09/2018 - 08:27

Let L be an $n\times n$ matrix with $(L)_{ij}=\mathbb 1(i\le j)$. Let the singular value decomposition of $LL'$ be $UDU'$. Is there an analytical form for $U$ and $D$, only as function of $n$?

Number of collinear ways to fill a grid

Mon, 04/09/2018 - 08:18

A way to fill a finite grid (one box after the other) is called collinear if every newly filled box (the first excepted) is vertically or horizontally collinear with a previously filled box. See the following example:

Let $g(m,n)$ be the number of collinear ways to fill a $m$-by-$n$ grid. Note that $g(m,n) = g(n,m)$.

Question: What is an explicit formula for $g(m,n)$?

Conjecture (user44191): $g(m,n)=m!n!(mn)!/(m+n-1)!$.

Definition (user44191): Let $g(m,n,i)$ be the number of collinear ways to fill $i$ boxes in a $m$-by-$n$ grid such that every row and every column contain at least one filled box.

Remark: $g(m,n) = g(m,n,mn)$.

Proposition (user44191): Here is a recursive formula for $g(m,n,i)$:

  • $g(1,1,1) = 1$.
  • If $m=0$ or $n=0$ or $ i< \min(m,n)$, then $g(m,n,i) = 0$.
  • $g(m,n,i+1)=(mn-i) g(m,n,i) + mn g(m-1,n,i) + mn g(m,n-1,i).$

Proof: The two first points are obvious. We consider the number of collinear ways to fill $i+1$ boxes in a $m$-by-$n$ grid such that every row and every column contain at least one filled box.
There are three cases, corresponding to the three components of the recursive formula:

  1. The last filled box is not the only filled box in its row and not the only filled box in its column.
  2. The last filled box is the only filled box in its row.
  3. The last filled box is the only filled box in its column.

By the collinear assumption, 2. does not overlap 3. $\square$

One way to answer the question is to prove the conjecture using the above recursive formula.

We checked the conjecture for $1\le m \le n \le 5$, using the recursive formula (see below).

Remark: This question admits an extension to higher dimensional grids.
Remark: This question was inspired by that one.

Sage program

# %attach SAGE/grid.sage from sage.all import * import copy def grid(m,n,j): if [m,n,j]==[1,1,1]: return 1 elif j < min(m,n) or m==0 or n==0: return 0 else: i=j-1 return (m*n-i)*grid(m,n,i) + m*n*grid(m-1,n,i) + m*n*grid(m,n-1,i) def IsFormulaCorrect(m,n): return grid(m,n,m*n)==factorial(m)*factorial(n)*factorial(m*n)/factorial(m+n-1) def CheckFormula(M,N): for m in range(1,M+1): for n in range(M,N+1): if not IsFormulaCorrect(m,n): return False return True

Computation

sage: CheckFormula(5,5) True

Information on structure (CI-magma with (non surjective)homorphism) of chemical transformations

Mon, 04/09/2018 - 07:28

Thinking about the mathematical structure of chemical transformations, between all possible components (educts, products) it occurs to me, that this structure is a commutative-idempotent groupoid(=magma) (CI-groupoid). Call it $(\mathcal{X},\cdot)$.

Take reactants $a,b\in\mathcal{X}$ (one can imagine any kind of substance mixture) and react them. The set is closed under $\cdot$ by definition: $$ a\cdot b \in\mathcal{X},$$ its commutative $$ a\cdot b = b\cdot a ,$$ to say you react $a$ with $b$ is the same than to say you react $b$ with $a$ (though, "pouring one into the other" isn't, but this should be ignored here) and idempotent

$$ a \cdot a = a.$$

I cannot pinpoint much more at the moment, associativity does not hold and inversion does not exist in general. So that should be a CI-groupoid, or CI-magma. Dropping the dots from now on, second dilution of $a$ in $b$ is $$ (ab)b \ne a(bb) = ab.$$

$(\mathcal{X},\cdot)$ is now a kind of messy children chemistry where you end up mixing everything with everything and never separating anything.

Separation of components (carbon and oxygen gives CO$_2$ and CO) apparently is the identification of more structure:

$$ab=cd$$

Ultimately $\mathcal{X}$ is finitely generated, since ultimately there is only a finite number of molecules in the universe. Explicit generation from the finite number of elementary components (materials composed of only one element="elements"), however will afford inclusion of reaction conditions into the description.

Same reactants can yield different products under different reaction conditions. Reaction conditions can be described like applying an indexed map $\chi_i$ where in $i\in I$ the reaction conditions are encoded. The maps are apparently (non surjective) homomorphisms on $\mathcal{X}$ $$ \forall a,b\in \mathcal{X}:\; \chi_i(ab)=\chi_i(a)\chi_i(b) \in \mathcal{X}.$$

Now my question, is there any developed theory/body of literature about such an algebraic structure? I have searched the internet for quite a while but only found sparse mentions of the term "CI-groupoid". Is there another maybe more common term which escaped me? Is it expected that any interesting comes out of such a categorisation (since there is so to say not really much structure)?

(I am not aware of this description in chemical research, but there is quite a lot going on in "chemical information theory" a field which is usually "below the radar" for mosts Chemists, and I do not expect that anyone here might be aware of developments in this specific field.)

Does there exist concept of angles and sides in Non-archimedian field geometry?

Mon, 04/09/2018 - 06:45

Question about Non-Euclidean Geometry, particularly about Non-archimedian Geometry:

I am studying and trying to understand about Non-Euclidean Geometry for my future purpose.

An example of Non-archimedian field is the field of $ \ Q-adic \ \ numbers \ $ .

I have information about ultrametric space and the information that any triangle in an Non-archimedian field is Isoscale triangle. It is ok to me.

But right here I have a question.

Does there exist concept of angles and sides in Geometry over Non-archimedian field as in our ordinary Euclidean Geometry ?

Any help would be appreciated if you share something that can help me

Does the axiom of choice follow from the statement "Every simple undirected graph is either connected, or its complement is connected"?

Mon, 04/09/2018 - 06:23

Using the Well-Ordering Principle, which is equivalent to the Axiom of Choice, it can be proved that

(S): for every simple, undirected graph $G$, finite or infinite, either $G$ or its complement $\bar{G}$ is connected.

Does (S) imply (AC)?

When does the forgetful functor from algebras over a monad commute with homotopy geometric realizations?

Mon, 04/09/2018 - 06:12

Let $\mathcal{C}$ be a combinatorial model category and $\mathrm{T}$ a monad on $\mathcal{C}.$

Assume that the model structure on $\mathcal{C}$ lifts to a model structure on the category of $\mathrm{T}$-algebras $\mathrm{Alg}_{\mathrm{T} }(\mathcal{C}),$ where the weak equivalences and fibrations are those of $\mathcal{C}.$

Assume that $\mathrm{T}: \mathcal{C} \to \mathcal{C} $ preserves homotopy colimits indexed by $\Delta^{\mathrm{op}}.$

Does the forgetful functor $\mathrm{Alg}_{\mathrm{T} }(\mathcal{C}) \to \mathcal{C}$ preserve homotopy colimits indexed by $\Delta^{\mathrm{op}}?$

More generally one can ask the question replacing $\Delta^{\mathrm{op}}$ by an arbitrary category.

Remark: If $\mathrm{T}: \mathcal{C} \to \mathcal{C} $ preserves colimits indexed by some category $\mathrm{K}, $ the forgetful functor $\mathrm{Alg}_{\mathrm{T} }(\mathcal{C}) \to \mathcal{C}$ preserves colimits indexed by $\mathrm{K}.$

This also holds for $\infty$-categories with the appropriate notion of monad and algebras over a monad.

An inequality regarding projection

Mon, 04/09/2018 - 05:30

Let $a, b \in \mathbb{R}^k$ be two normalized vectors such that $a^T b \ll 1$. Define matrix $C$ such that $[a, b, C]$ is full column rank, and let matrix $D$ be positive definite. Define projection matrix $P_A:=A^T (AA^T)^{−1}A$. Can we say the following?

$$\frac{a^T D^{-1/2}\left(I - P_{D^{-1/2}C}\right)D^{-1/2}b}{ \left\|\left(I - P_{D^{-1/2}C}\right)D^{-1/2}b\right\|_2 \left\|\left(I - P_{D^{-1/2}C}\right)D^{-1/2}a\right\|_2 }\ll 1$$

Also posted on MSE.

Bound of the measure of the support of a set of divisors in a fixed linear system

Mon, 04/09/2018 - 03:28

Let $(X,L)$ be a compact polarized complex manifold of dimension $n$. Let $\varphi$ be a smooth positive metric on $L$. Define $\omega=dd^c\varphi$. We shall use $MA(\varphi)=\omega^n$ as the measure on $X$. Then there is a natural $L^2$-inner product on $H^0(X,L)$. Now fix $s\in H^0(X,L)$ of norm $1$. Consider the following set $$ A_\epsilon=\{s'\in H^0(X,L): \|s'\|=1, |(s,s')|<\epsilon\}, $$ where $\epsilon>0$. My question is: how can we get an upper bound of $$ MA(\varphi)\left(\bigcup_{s'\in A_{\epsilon}} Z_{s'}\right), $$ where $Z$ denotes the zero locus.

A question on completely continuous operators

Mon, 04/09/2018 - 03:12

An operator $T:X\rightarrow Y$ is said to be completely continuous if $T$ maps weakly convergent sequences to norm convergent sequences.

Let $Q: l_{1}\rightarrow l_{2}$ be any surjection and $J:l_{1}\rightarrow Y$ be an isomorphic embedding.

Question. Is there a completely continuous operator $S:Y\rightarrow l_{2}$ such that $Q=SJ$?

Thank you!

Hilbert functions of graded modules generated by mapped generators

Sun, 04/08/2018 - 18:20

I need to prove the following claim for my work. Intuitively, the claim should hold as it is analogous to the concept of an initial module, but a rigorous approach for the same I cannot find. Any help on a possible approach would be much appreciated. Thanks very much!

Given:

  1. The polynomial ring $A = \mathbb{R}[x, y]$. It can be graded according to total polynomial degree as $A = \oplus_{k\geq 0} A_k$, with the Hilbert function of the $i^{th}$ piece given by $HF_i(A)= i+1$.
  2. The $A$-module $M = \oplus_{i}[\alpha_i]A$ generated by formally independent elements $\alpha_i$ coming from some finite set of cardinality $m$.
  3. The submodule $M \supset N = \sum [\beta_i] A$, where each generator $[\beta_i] = [\alpha_j] - [\alpha_k]$ for some $j$ and $k$.
  4. A surjective map $f: \{[\alpha_i]\} \rightarrow \{[\gamma_j]\}$, where $\gamma_j$ are formally independent elements of another finite set of cardinality $p < m$. Assume that, $$ \forall [\beta_i] = [\alpha_j] - [\alpha_k], \quad f([\beta_i]) := f([\alpha_j]) - f([\alpha_k]) \neq 0. $$
  5. A map, $$ g : \sum_{i \in I} [\alpha_i]a_i \mapsto \alpha_{\max(I)} a_{\max(I)}\;, $$ where $A \ni a_{i} \neq 0$ for all $i \in I$, and $I \subseteq \{1, \dots, m\}$.
  6. The submodules $N_0$ and $N_1$, $$ N_0 = \sum f([\beta_i])A, \qquad N_1 = \sum f\circ g([\beta_i])A. $$

Claim: Assuming the natural grading on $N_0$ and $N_1$, the following inequality holds: $$ HF_i(N_0) \geq HF_i(N_1). $$

EDIT

A simplified/alternate version of the problem statement on has been posted on SE by posing the problem in terms of real vector spaces. If that helps on nailing down an approach, you can find it here: https://math.stackexchange.com/questions/2728748/vector-space-dimension-after-nonlinearly-mapping-spanning-vectors

A possible generalization of "Group Cohomolgy"

Sun, 04/08/2018 - 14:33

The group cohomology of a group $G$ is defined as the derived functor associated to the following left exact functor: $$FIX: \mathcal{M_G} \to \mathcal{Ab}$$ where $FIX$ is the functor from the category of $G$-modules to the category of Abelian groups sending each module $M$ to its subgroup consisting of all elements of $M$ which are fixed by $G$ action.

Now we fix a natural number $n\in \mathbb{N}$ and do an obvious generalization of the above construction:

Instead of the above left exact functor $FIX$ we consider the functor $P_n$ which send a $G$-midule $M$ to the following subgroup of $M$: $$P_n(M)=\{x\in M \mid g.x= nx, \forall g \in G\}$$.

In this way what would be the corresponding derived functor?What kind of cohomology theory would appear? Is this a trivial generalization of classical "Group Cohomology"?If not, what would be a topological analogy?

The later question is motivated by the fact that the group cohomology corresponds to singular cohomology of the corresponding Eilenberg-Mclane space)

Claim: $r_a(A_1,\cdots,A_n)\neq r_a(A_1^*,\cdots,A_n^*)$

Sun, 04/08/2018 - 08:49

Let $E$ be an infinite-dimensional complex Hilbert space.

The spectral radius of a commuting multivariable operator $A = (A_1,\cdots,A_n)\in\mathcal{L}(E)^n$ (i.e. $A_iA_j=A_jA_i$ for all $i,j$) is given by \begin{align*} r_a(A_1,\cdots,A_n) & =\displaystyle\lim_{m\to \infty}\left\|\displaystyle\sum_{|\alpha|=m}\frac{m!}{\alpha!}{A^*}^{\alpha}A^{\alpha}\right\|^{\frac{1}{2m}} \\ &=\sup\{\|\lambda\|_2,\;\;\lambda \in \sigma_{ap}(A)\}, \end{align*} where $$\sigma_{ap}(A)=\bigg\{\lambda\in \mathbb{C}^n: \;\exists\;(x_k)_k\subset E;\,\,\|x_k\|=1\;\;\hbox{such that}\;\;\\\lim_{k\longrightarrow \infty}\sum_{1\leq j\leq n}\|(A_j-\lambda_j)x_k\|=0\bigg\}.$$

If $n=1$, it is well known that $r(A)=r(A^*)$. I claim that if $A_iA_j=A_jA_i$ for all $i,j$, then in general $$r_a(A_1,\cdots,A_n)\neq r_a(A_1^*,\cdots,A_n^*).$$ I hope to find an example which show that the claim is true.

Cohomology of $ko,tmf,MSpin,MString$ with coefficients $\mathbb{Z}/p$ for odd primes $p$

Sat, 04/07/2018 - 22:16

It is well-known that $$H^*(ko,\mathbb{Z}/2)=\mathcal{A}\otimes_{\mathcal{A}(1)}\mathbb{Z}/2$$ $$H^*(tmf,\mathbb{Z}/2)=\mathcal{A}\otimes_{\mathcal{A}(2)}\mathbb{Z}/2$$ where $\mathcal{A}$ is the mod 2 Steenrod algebra.

$H^*(MSpin,\mathbb{Z}/2)$ and $H^*(MString,\mathbb{Z}/2)$ are closely related to the above because of the Atiyah-Bott-Shapiro orientation and Witten genus.

I find in Adams and Priddy's Uniqueness of BSO: $$H^*(ko,\mathbb{Z}/p)=\bigoplus_{s=0}^{\frac{p-3}{2}}\Sigma^{4s}\mathcal{A}_p/(\mathcal{A}_pQ_0+\mathcal{A}_pQ_1)=\bigoplus_{s=0}^{\frac{p-3}{2}}\Sigma^{4s}\mathcal{A}_p\otimes_{E(Q_0,Q_1)}\mathbb{Z}/p$$ where $\mathcal{A}_p$ is the mod $p$ Steenrod algebra for odd primes $p$ and $Q_0=\beta,Q_1=P^1\beta-\beta P^1$.

I want to know what is $H^*(MSpin,\mathbb{Z}/p)$, $H^*(tmf,\mathbb{Z}/p)$ and $H^*(MString,\mathbb{Z}/p)$ for odd primes $p$.

Any references and partial answers are appreciated.

symplectic topology of (perturbed) KAM tori

Fri, 04/06/2018 - 12:11

Consider a real analytic $H_0:\mathbb{R}^n\to \mathbb{R}$ whose Hessian is everywhere non-degenerate as well as a real analytic $F:\mathbb{T}^n\times \mathbb{R}^n\to \mathbb{R}$. KAM theory studies what happens to (Lagrangian) tori which are invariant under the Hamiltonian flow $\phi_{H_{\epsilon}}$, associated to $$ H_{\epsilon}(q,p)=H_0(p)+\epsilon F(q,p) \quad \forall \ (q,p)\in \mathbb{T}^n\times \mathbb{R}^n,$$ when $\epsilon$ varies. If the rotation vector of the restriction of the Hamiltonian flow $\phi_{H_{0}}$ to the torus $T(p_0):=\mathbb{T}^n\times \{p_0\}$ is Diophantine, then the KAM theorem (Kolmogorov-Arnold-Moser) guarantees, for all small enough $\epsilon>0$, the existence of an invariant Lagrangian torus $L\approx \mathbb{T}^n \subset \mathbb{T}^n\times \mathbb{R}^n$ which is "close" to $T(p_0)$ and invariant under $\phi_{H_{\epsilon}}$.

Is $L$ also guaranteed to be Hamiltonian isotopic to $T(p_0)$?

Any explanations/references will be much appreciated.

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