User leandro - MathOverflow

Independent bond percolation on upper density zero subgraphs of the square lattice can have a non-trivial critical point ?

2013-04-02T06:42:24Z

Consider the two-dimensional lattice $G=(\mathbb{Z}^2,\mathbb{E}^2)$, where edge set $\mathbb{E}^2$ is given by the pairs of nearest neighbors in the $\ell^1$ norm in $\mathbb{Z}^2$. Let $V\subset\mathbb{Z}^2$ an infinite subset and suppose that $G[V]$, the induced subgraph, is connected. Let $\Lambda_n=([-n,n]\times[-n,n])\cap \mathbb{Z}^2$ be a sequence of squares on the two-dimensional lattice. Suppose additionally that $$ \limsup_{n\to\infty} \frac{|V\cap \Lambda_n|}{|\Lambda_n|}=0. $$ Question: under the above conditions is it true that the independent bond percolation, with parameter $p$, on $G[V]$ is trivial, in the sense that for any $p\in [0,1)$ we do not have almost surely an infinite cluster ?

I suspect that the answer is affirmative and this is considered in the literature, but until now I only found trivial examples of such graphs $G[V]$, basically constructed from the one-dimensional lattice, where there is no percolation.

Is this a $C^{\infty}$ function ?

2010-08-17T01:37:40Z

Let be $(a_n)\in\ell^2(\mathbb N)$ and consider the mapping $f:\ell^2(\mathbb N)\to\ell^2(\mathbb N)$ given by $$ f\Big((a_n)\Big)=(a_n^n). $$ Question: Is $f$ a Fréchet $C^{\infty}$ function in whole $\ell^2(\mathbb N)$ ?

If the answer for the previous question is no. Is there a non-linear $C^{\infty}$ function, defined in some Banach space that maps a closed bounded set onto a non bounded set ?

Lower bound for double sums with power law decay terms.

2012-05-18T06:46:51Z

This question is related to a work in progress about Ballistic-Diffusive phase transition for some random polymers with long range self-repulsion. The motivation to ask here if the inequality below holds is to avoid a length analysis in this paper. Let $0\leq i<j\leq N$ be integers and $\alpha \in (3,4]$ is there a constant $C>0$ independent of $i,j$ and $N$ such that $$ \frac{C}{|i-j|^{\alpha-2}}\leq \sum_{0\leq k\leq i< j\leq r \leq N} \frac{1}{|k-r|^{\alpha}} $$ holds ?

Remark: similar upper bound holds, to get it I simply compared this sum with a double integral. For the lower bound it seems that such comparison does not work.

Answer by Leandro for Tools for collaborative paper-writing

2012-02-27T04:06:57Z

There is a new project called Sharelatex the website is www.sharelatex.com

It is a web application where the collaborators can edit the text at the same time. We can get a free account to test the software.

Answer by Leandro for Measures that satisfy a 0/1 law

2012-02-27T03:47:53Z

I would like to point out that extremal Gibbs measures are interesting class of measures that can be defined over $2^{\omega}$ and satisfies a zero-one law, not in the whole $\sigma$-algebra generated by the cylinder sets, but in the tail $\sigma$-algebra.

In several cases this approach agree with the Anthony suggestion. In the Gibbs measure theory dependence is constructed in very geometric way. Another interesting feature is: no group action invariance is required. Some theorems of ergodic theory are proved in this context. There is also a Choquet's Theorem.

If you want to get the details the classical mathematical reference is Gibbs Measures and Phase Transitions by Hans-Otto Georgii. The zero-one law is proved in the page 115 of the first edition.

There are some online options which are not so general as Georgii, but covers the space you are interested in and proves the zero-one law

A. Bovier: Lecture notes Gibbs measures and phase transitions - part 1.
http://www-wt.iam.uni-bonn.de/~bovier/files/note1.pdf

A. Bovier: Lectures notes Gibbs measures and phase transitions - part 2.
http://www-wt.iam.uni-bonn.de/~bovier/files/note2.pdf

Roots of Taylor Polynomials of analytic function with finite radius of convergence

2011-10-25T03:49:33Z

In 1914 Jentzsch proved that if $$ g(z)=1+a_1z+\ldots+a_nz^n+\ldots $$ has the unit circle as circle of convergence then every point of this circle is a cluster-point of zeros of partial sums $$ s_n(z)=1+a_1z+\ldots+a_nz^n. $$ I was wodering if you could point me out an alternative English written reference for Untersuchungen zur Theorie der Folgen analytischer Functionen. Inaud.-diss., Berlin where the proof of this theorem is presented in details.

Answer by Leandro for How much universality is there for contact processes?

2011-10-15T07:23:12Z

Hi Gowers,

In this paper: Sidoravicius, V. and Kesten, H . A shape theorem for the spread of an infection. Annals of Mathematics, v. 167, p. 1-63, 2008,

the authors consider a different model, but they were interested in the shape problem like you. One time before publish thi paper Vladas told me that about a model, similar to what you describe and also that there is a believe about universality for those models, perhaps they discuss this on this reference.

Here is a link for the arxiv version of this paper.

Infinite dimensional vector spaces with compact unit ball

2011-10-13T06:33:44Z

Let $X$ be an infinite dimensional vector space over a field $\mathbb{K}$. Suppose that $(X,\|\cdot\|)$ is a complete normed vector space, in the sense that any Cauchy sequence is convergent. Suppose that the closed unit ball of $X$ is compact in the strong topology.

Question 1. Is $X$ necessarily isomorphic to some finite dimensional Banach space ?

Question 2. If the answer for the question 1 is no , can we always find in $\mathcal{L}(X,X)$ an unbounded linear operator ?

Different from this question Is there an infinite-dimensional Banach space with a compact unit ball? I would like to assume the axiom of choice.

Comment. The example I have in mind is $(\mathbb{R}^n,\|\cdot\|_{2})$ as $\mathbb{Q}$ vector space. Clearly $\text{dim}_{\mathbb{Q}}\ \mathbb{R}^n=\infty$ and the unit ball is compact and this space is complete with respect to the standard Euclidean norm $\|\cdot\|_2$, but in this example both questions 1 and 2 are trivial.

Comparing number of spanning subgraphs

2010-02-14T07:25:09Z

Hi all,

Let be $G_n=(V_n,E_n)$ a finite graph, where $V_n= \{0,1,\ldots, n\} \times\{0,1,\ldots,n\}$

and $E_n\subset V_n^{(2)}$ is the edge set of the nearest neighbors in the $\ell^1$ norm, that is, $\ E_n=\{\ \{z,w\}\subset V_n; \ \sum_{i=1}^2 |z_i-w_i| =1 \ \}.$

Fix a vertex $x=(x_1,x_2)\in G_n$ such that $x_2>x_1$ (up-diagonal). I would like to know if it is true the following inequality:

$\sharp[m,p]_{x}\leq \sharp[p,m]_x$, whenever $p < m$

where $[m,p]_{x}$ is the set of all spanning subgraphs of $G_n$ satisfying the following properties:

1- the spanning subgraph has $m$ horizontal edges and $p$ vertical edges;

2- the vertices $(0,0)$ and $x=(x_1,x_2)$ are in the same connected component,

and $\sharp A$ is the cardinality of $A$.

In other words, if I have avaliable more vertical edges than horizontal ones is it true that I can find more configurations connecting $0$ and $x$ if $x$ is up-diagonal than in case that the quanties of horizontal and vertical are inverted ?

Thanks in advance for any idea or reference.

Is this an injective function ?

2011-08-25T22:22:54Z

Hi all,

I got stuck with a problem that pop up in a paper about location of zeros for some analytic functions that I am working on.

The problem is the following: Fix two arbitrary positive integers $N$ and $n$. Let $\tilde{\ln}$ be the branch of the logarithm such that its imaginary part take values in $(0,2\pi)$. Is it true that for each $1\leq k\leq n$ the equation $$ z^N-\tilde{\ln}z-\frac{1}{N}-\frac{\ln_{\mathbb{R}}N}{N}=\frac{2\pi ik}{n}\qquad\qquad (1) $$ has exactly one solution in $$U_N(\varepsilon)=B(0,\sqrt[N]{N})- \bigcup_{j=1}^{N}B(w_j,\varepsilon)-\{0\},$$ where $w_j=\frac{1}{\sqrt[N]{N}}e^{\frac{2\pi ij}{N}}$, for all $\varepsilon<<1$ ?

Observation: In the case $N=1$, for any fixed $k$, if there are solutions for (1) it lives in the Szëgo curve, which is given by the equation $|ze^{1-z}|=1$. In this case to decide if the solution is unique, for a fixed $k$, in $U_1(\varepsilon):=B(0,1)-B(1,\epsilon)-\{ 0\}$ it is enough to show that $g:U_1(\varepsilon)\to g(U(\varepsilon))$, given by $$ g(z)=\frac{e^z}{z} $$ is a biholomorphism. I suspect that the inequality $\delta(\epsilon)<|g'(z)|$ for $z\in U_1(\varepsilon)$ it is enough to prove this fact. This condition remember me the Jacobian conjecture. I don't know about this conjecture for one complex variable functions. I mentioning this case, because of the uniqueness in (1) for general $N$, can be obtaining by proving that $g:U_N(\varepsilon)\to g(U_N(\varepsilon))$ given by $$ g(z)=\frac{e^{z^N}}{z} $$ is a biholomorphism. The choice of $U_N(\varepsilon)$ it certainly not the maximal domain where this function is bijective, I am considering just based on the observation that the zeros of $g'$ are not in the closure of $U_N(\varepsilon)$.

I appreciate any help.

Relation between Hausdorff dimension and Bowen's equation

2010-08-03T17:10:51Z

I am reading the paper Hausdorff dimension for Horseshoes, by McCluskey and Manning. In the following theorem
Theorem: Let $\Lambda$ be a basic set for a $C^1$ axiom A diffeomorphism $f:M^2\to M^2$ with $(1,1)$ splitting $$ T_{\Lambda}M=E^s\oplus E^u. $$ Define $\phi:W^u(\Lambda)\to\mathbb R$ by $$ \phi(x)=-\log(\|Df_x|_{E^u_x}\|) $$ then the Hausdorff dimension of $W^u(x)\cap\Lambda$ is given by the unique $\delta$ for which $$ P_{f|_{\Lambda}}(\delta\phi)=0 \qquad \qquad (1) $$ the authors compute the Hausdorff dimension of $\Lambda\cap W^u_x$ by using the Bowen's equation (1). I read the proof but I was not able to figure out the intuition behind the Bowen's equation in this theorem. Could you give me explanation (do not need to be rigorous) about that or point out a reference ?

Answer by Leandro for Ising model on groups

2011-06-16T09:36:29Z

Hi Marcin,

recently writing a paper about phase transition on the Ising model with positive non-uniform magnetic field in infinite graphs I discovered that joining some results in the literature we can relate amenability and Phase transition in the Ising model with positive magnetic field:

Theorem 1: If $G$ is a non-amenable infinite connected graph then there is a ferromagnetic Ising model on $G$, with constant positive magnetic field having phase transition.

There is a partial converse of this result for quasi-transitive amenable graphs. A infinite graph $G=(V,E)$ is quasi-transitive if there exist a finite number of vertices $x_1,\ldots,x_k$ such that for any $x\in V$, there is an automorphism of $G$ taking $x$ to some $x_i$.

Theorem 2: If $G$ is a amenable quasi-transitive infinite connected graph then all ferromagnetic Ising model on $G$, with constant positive magnetic field has no phase transition.

Theorem 2 is interesting converse because of the quasi-transitive hypothesis can not be removed since Bausev shown that we have phase transition in ferromagnetic Ising model with magnetic field being constant at all sites of the lattice $\mathbb{Z}^2 \times \mathbb{Z}_+$ .

Definitions.
An Ising model on a graph $G=(V,E)$ is defined as follows:

Let $\mathcal{L}$ be the set of finite parts of $V$ and suppose that $\Lambda_n\in\mathcal{L}$ is such that $\cup_{n\in\mathbb{N}}\ \Lambda_n=G$. The Hamiltonian of the Ising model in $\Lambda_n$ with a boundary condition $\omega\in \{-1,1\}^{V}$ is given by $$ H_{\Lambda_n}(\sigma|\omega)= -\sum_{\substack{i,j\in E:\\ i,j\in\Lambda_n}} J\ \sigma_i\sigma_j -\sum_{i\in \Lambda_n}h_i \ \sigma_i - \sum_{\substack{i,j\in E:\\ i\in\Lambda_n,j\in\Lambda_n^c}} J\ \sigma_i\omega_j, $$ where $\sigma=(\sigma_i)_{i\in V}\in\{-1,1\}^{V}$ , $J\in\mathbb{R}$ (the model is called ferromagnetic of $J>0$) and $h_i\in\mathbb{R}$ is said the magnetic field. Finally we say that this Ising model has phase transition if the closed convex hull of the set $$ \left\{w-\lim_{\Lambda_n\uparrow G}\ \mu_{\Lambda_n}^{\beta,\omega}:\omega\in\{-1,1\}^{V} \right\} $$ is singleton for all $\beta>0$. The measures $\mu_{\Lambda_n}^{\beta,\omega}$ are defined by $$ \mu_{\Lambda_n}^{\beta,\omega}(\sigma)= \left\{ \begin{array}{rl} \frac{\exp(-\beta H_{\Lambda_n}^{\omega}(\sigma))}{Z_{\Lambda_n}^{\omega}},&\text{if}\ \sigma_i=\omega_i\ \forall i\in\Lambda_n^c;\\ 0,& \text{otherwise}, \end{array} \right. $$

One nice reference about the results I stated above is Jonasson, J. and Steif, J. E.: Amenability and Phase Transition in the Ising Model. J. Theor. Probab. 12, 549-559 (1999).

How to prove this Poincare Inequality

2011-06-14T07:16:05Z

Hi,

I want to ask a question about a statement that I found on the paper: Principal Eigenvalues for Problems With indefinite Weight Function in $R^N$.

The statement is the following:

Suppose that $g:\mathbb{R}^2\to\mathbb{R}$ is a $C^{\infty}$ function which changes sing on $\mathbb{R}^2$ and there exist constants $K,R>0$ such that $g(x)\leq-K$ for $|x|>R$. Let $B$ a ball such that $$ \int_{B} g(x) dx <0 $$
and $g(x)<0$ if $x\in \mathbb{R}^2\setminus B$.

I would like to know why is true that there exist a positive constant $C_1>0$ such that $$ \int_{B} u^2 dx \leq C_1\int_{B}|\nabla u|^2 dx $$ for all $u\in H^1(B)$ with $\int_{B}g u^2 dx>0$ ?

The authors give the following reference for this result: K.J. Brown, S.S. Lin and A. Terkitas, Existence and noexistence of steady-state solutions for a selection-migration model in popular genetics. J. Math. Biol. 27 (1989), 91-104.

I can't have access to this paper that's why I am posting this question here.

I am trying to prove this inequality, but the results I know are based on arguments that requires compact support for $u$ or mean zero. I appreciate if some specialist can give me a hint on how to prove this inequality for the general case (neither compact support or mean zero) or provide an alternative reference where it is proved. Thanks.

A general Lipschtiz potential can be specified by a Gibbs specification ?

2011-06-03T19:23:49Z

I want to consider one-dimensional system on the lattice $\mathbb{L}=\mathbb{N}$. Let be $A:(\mathbb{S}^1)^{\mathbb{L}}\to\mathbb{R}$ a lipschtiz potential. Consider the Ruelle operator $$ \mathcal{L}_{A}(\psi)(x)=\int_{\mathbb{S}^1} e^{A(ax)} \psi(ax)\ da $$ where $x\in(\mathbb{S}^1)^{\mathbb{N}}$ and $ax=(a,x_1,x_2,\ldots)\in (\mathbb{S}^1)^{\mathbb{N}}$ and $da$ is the normalized Lebesgue measure on $\mathbb{S}^1$. Let $\mu$ the Gibbs measure constructed from $\mathcal{L}_A$ in the standard way.

Let $\Lambda_n=[0,\ldots,n]\cap\mathbb{Z}$ and $\mathscr{T}_{\Lambda_n}$ the external $\sigma$-algebra as defined in the Georgii's book.

Question. Is there any Gibbs specification $\gamma=(\gamma_{\Lambda_n})_{n\in\mathbb{N}}$ such that

$$ \mu(A|\mathscr{T}_{\Lambda_n})(x)=\gamma_{\Lambda_n}(A|x) \quad \mu-\text{a.s.} ? $$

The answer is simple if $A$ is a potential depending only on finite number of coordinates or other words, if $A$ is a short range potential.

Since I am considering $A$ is lipschtiz, it seems reasonable to fix some configuration $\omega_0\in\mathbb{S}^{\mathbb{N}}$ and consider a sequence of truncated potentials as follows

$$ \Phi_n=(\Phi^n_{\Gamma})_{\Gamma\subset \mathbb{N}}, $$ where $$ \Phi^n_{\Gamma}(x)= \left\{ \begin{array}{rl} -A(x_1,\ldots,x_{n},\omega_{n+1},\ldots),& \text{if}\ \Gamma=\{1,\ldots,n\}; \\ 0,&\text{otherwise}. \end{array} \right. $$ Now we consider the respective specifications given by these potentials and ask if the unique $\mu_n\in\mathcal{G}(\Phi_n)$, converges to the measure $\mu$ for any choice of $\omega_0$.

The main motivation to post this question is to know if there is a standard procedure to obtain the measure $\mu$ from the specification point of view. I also appreciate any comments about the aproximatoin scheme I described above. Any help or reference is welcome. Thanks.

Can be this operator extended to an unbounded self-adjoint operator ?

2011-05-04T22:19:00Z

Consider an enumeration $\{q_1,q_2,\ldots\}$ of $\mathbb{Q}\cap [1,\infty)$ and a orthogonal Schauder basis $\{e_1,e_2,\ldots\}$ of $\ell^2(\mathbb{N})$. Define $Ae_{2k-1}=e_{2k-1}$ and $Ae_{2k}=q_ke_{2k}$ for all $k\geq 1$.
Question 1: Is it possible to extend $A$ to a linear self-adjoint operator defined in some infinite dimensional subspace of $\ell^2(\mathbb{N})$ ?

If I am not wrong, this possible linear self-adjoint extension of $A$ can not be defined everywhere in $\ell^2(\mathbb{N})$ and I would like to know if the following set $$ \left\{ v\in \ell^2(\mathbb{N}); v=\lim_{n\to\infty} \sum_{i=1}^n\alpha_ie_i \ \text{and}\ \ \lim_{n\to\infty}\sum_{i=1}^n q_i\alpha_ie_i \in \ell^2(\mathbb{N}) \right\} $$ is a good candidate to be the domain of $A$ ?
Question 2: Is the point spectrum $\sigma_p(A)\supset \{q_1,\ldots,q_n,\ldots\}$ ?

Motivavation: I would like to know if there is an example of an unbounded self-adjoint operator such that the point spectrum is not composed only by isolated points in $\mathbb{R}$ and there is at least one eigenvalue with infinite dimensional eigenspace.

How many Hamiltonians Paths there are in almost regular graph ?

2010-06-18T18:10:06Z

Let be $G=(V,E)$, where $V=\{1,\ldots,n\}$ and $E=\{\{i,j\}\subset V;|i-j|\leq k\}$ and $k<n$ .
For which values of $k\geq 2$, can we count explicitly the number of Hamiltonian paths in $G$ ?

If $G$ is amenable, when $G\times G$ is amenable ?

2011-03-18T06:36:14Z

I am not specialist on Topological Group Theory, I apologize if this is a trivial question.

Question. If $G_1=G_2$ are amenable topological groups what additional hypothesis we have to consider on the group, in order to prove that $G_1\times G_2$ is amenable ?

Following Leinster, in this question Why are abelian groups amenable?,
"The direct product of two amenable groups is amenable. This isn't exactly trivial, but the measure on the product is at least constructed canonically from the two given measures."
So discreteness of the $G_1$ and $G_2$ are enough to prove that $G_1\times G_2$ is amenable and also we do not need to suppose that $G_1=G_2$.

Looking for a proof, in more general cases, I found the following statement:
"... direct product $G_1\times G_2$ of two countable amenable groups can not be amenable."
in the paper, On Subadditive Processes on Direct Product of Countable Amenable Groups by Seyit Temir - Publications De l'Institute Mathématique (2002), 119-122.

Since the author did not mention if the example needs two different groups and I have no access to the paper containing this information, I decided to post this question.

I would be grateful if you could point me out some references discussing about this problem.

Hamilton Paths in $K_{2n}$

2011-04-06T22:57:34Z

Hi,

I am teaching this semester graph theory for undergraduate students. Now, I am discussing with them about Hamilton Paths in finite graphs. Last time we meet, I presented the following theorem:

Theorem. For $n\geq 3$ the complete graph $K_n$ is decomposable into edge disjoint Hamilton cycles iff n is odd. For $n\geq 2$ the complete graph $K_n$ is decomposable into edge disjoint Hamiltonian paths iff $n$ is even.

During the class I noted that my argument to prove this theorem was not complete. I started proving that the second statement implies the first one, which is ok. But I had not a correct argument to show that there exist an edge disjoint decomposition of $K_n$ in $n/2$ Hamilton paths if $n$ is even.

Can we explicitly construct such decomposition or just present an existence argument ?

On generalisation of Aizenman-Higuchi Theorem

2010-11-01T05:23:51Z

Let $\mathbb Z^2$ denote the two-dimensional integer lattice with norm of $i=(i_1,i_2)$ given by $\|i\|=|i_1|+|i_2|$.

For each $x\in\mathbb Z^2$, we assign a uniform random variable, $\sigma_x$ taking values on the set $\{-1,1\}$ .

Fix $\omega\in\{-1,0,1\}^{\mathbb{Z}^2}$ and $\beta>0$. For each finite $\Lambda\subset\mathbb{Z}^d$, define a probability measure on the sigma algebra generated by the cylinder sets of $\{-1,1\}^{\mathbb{Z}^2}$ , such that for each $\sigma\in\{-1,1\}^{\mathbb{Z}^2}$ the probability of this configuration is given by $$ \mu_{\Lambda}^{\beta,\omega}(\sigma)= \left\{ \begin{array}{rl} \frac{\exp(-\beta H_{\Lambda}^{\omega}(\sigma))}{Z_{\Lambda}^{\omega}},&\text{if}\ \ \sigma_i=\omega_i\ \forall i\in\Lambda^c;\\ \\ \\ 0,& \text{otherwise}, \end{array} \right. $$ where $$ H_{\Lambda}^{\omega}(\sigma)=-\sum_{i,j\in\Lambda}J_{ij}\sigma_i\sigma_j-\sum_{i\in\Lambda, j\in\Lambda^c}J_{ij}\sigma_i\omega_j $$ with $J_{ij}\equiv J(\|i-j\|)\geq 0$ and $J_{ij}=0$ if $\|i-j\|\geq R$, for some positive $R$ and $Z_{\Lambda}^{\omega}$ is a normalizing constant so that $\mu_{\Lambda}^{\beta,\omega}$ is a probability measure.

Question 1: If $\Lambda_n\uparrow\mathbb{Z}^2$ and $\omega_i=0$ for all $i\in\mathbb{Z}^2$, sounds reasonable that any accumulation point of the sequence $\mu_{\Lambda_n}^{\beta,\omega}$, in the weak* topology, is translation invariant. Is this true for any finite $R$ ?

Question 2: Suppose $R$ finite and bigger than one, keeping the setting of Question 1 but $\omega_i=1$ (or $\omega_i=-1$) for all $i\in\mathbb{Z}^2$ is the weak* limit $$w-\lim_{n\to\infty} \mu_{\Lambda_n}^{\beta,\omega}$$
translational invariant ?

Question 3: For finite $R$ bigger than one is it true the Aizenman-Higuchi Theorem
$$w-\lim_{n\to\infty} \mu_{\Lambda_n}^{\beta,\omega}\in [\mu^{\beta,+},\mu^{\beta,-}] ?$$

Exponential bounds for the number of lattice animals with a given boundary.

2011-02-16T18:11:15Z

Hi all,

I am doing a work in collaboration with other mathematicians about phase transition in the Ising model and we need to know if exponential upper bounds exist for the number of lattice animals with boundary of size $n$.

To be precise, consider the square lattice $\mathbb{Z}^2$ as graph where the edges are pairs of points in the lattice having distance one from each other, where the distance is induced by the norm $\|(z_1,z_2)\|=|z_1|+|z_2|$.

We call a lattice animal the set of vertices of any connected subgraph of the square lattice. Given an animal $A$, we denote the boundary of $A$ by $\partial A$, that is, the set of vertices of distance one from $A$.

Fix a site $z\in \mathbb{Z}^2$ and let be
$$ f(n)=\sharp \{A\ \text{is lattice animal}; A\ni z\ \text{and}\ |\partial A|=n\} $$
Is it known if $f(n)=O(e^{k n})$ ?

I learned from google that this problem is also known in the combinatorics community as enumeration of polyominoes with a given site-perimeter.

All the papers I found about the upper bounds at some point have to impose some geometric hypotheses on the polyominoes such as convexity, starcase shape or bargraph shape.

I don't know yet if those hypotheses are being used in order to get sharp upper bounds or if they are the only ones available.

If the question about exponential upper bound is not yet solved, is there a specialist in this area who could tell me what they think about the upper bound for this problem.

Answer by Leandro for Vanishing Trace

2011-01-19T04:05:11Z

The Igor's answer also works if $a$ is compact self-adjoint and $\mathcal{H}$ is isomorphic to $\ell^2(\mathbb{N})$ .

Because of the spectral theorem and $\text{Tr}(a)=0$, for any $v\in \mathcal{H}$ we have :

$$ \langle v,av\rangle =\sum_{i=1}^{\infty}\lambda_i v^2_{r_i} -\sum_{i=1}^{\infty}\beta_i v^2_{s_i}\qquad (1) $$

where $(\lambda_i)_{i\in\mathbb{N}}\neq (0,0,\ldots)$ and $(\beta_i)_{i\in\mathbb{N}}\neq (0,0,\ldots)$ and for all $i$ we have $\lambda_i,\beta_i\geq 0$. Here $\lambda$'s and $\beta$'s form the spectrum of $a$.

Note that $\ell^2(\mathbb{N})=V_+\oplus V_{-}$, where the spaces $V_{+}=\bigoplus_{i=1}^{\infty}e_{r_i}$ and $V_{-}=\bigoplus_{i=1}^{\infty}e_{s_i}$

We can rewrite (1) as

$|||\pi_{1}(v)|||$- $||\pi_{2}(v)||=0,$

where the norms appearing above are defined on the subspaces $V_{+}$ and $V_{-}$ by two positive bilinear forms associated to the $\lambda$'s and $\beta$'s respectively and $\pi_1$ and $\pi_2$ are projections on the subspaces $V_{+}$ and $V_{-}$ .

Taking an orthonormal sets in both spaces in the unit ball, we can find the vectors you want. I guess this idea can be generalized using the functional calculus.

Answer by Leandro for Convergence of alternating harmonic sums

2011-01-13T06:16:58Z

This is not an answer, but it is too long for a comment.

Hi Wadim, nice problem. I was trying to obtain a partial answer for it based on the following

Proposition. Let be $\xi_1,\xi_2,\ldots$ a sequence of independent Bernoulli random variables with $\mathbb{P}(\xi_n=+1)=\mathbb{P}(\xi_n=-1)=\frac{1}{2}$, then the series $\sum \xi_n a_n$, with $|a_n|\leq c$, converges with probability 1, if and only if $\sum a_n^2<\infty$.

The idea it was consider $a_n=\frac{1}{n}$, and try to analyse the set $$ A:=\left\{\Big((-1)^{n+[n^{\alpha}]}\Big)_{n\in\mathbb{N}}; 0\leq \alpha<1 \right\} \subset \{-1,1\}^{\mathbb{N}}. $$
In case that $\mathbb{P}(A)\neq 0$, since this product measure has no atoms we could, at least, say that the set of $\alpha$'s for which the series is convergent is non-enumerable.

Answer by Leandro for Why are matrices ubiquitous but hypermatrices rare?

2010-12-03T03:03:30Z

Hi Joseph,

it seems that the basic difference between hypermatrices and matrices is the indexation.

Let me explain what I mean. Consider a finite graph $G=(V,E)$ (the graph structure is not important I mentioning this just to refer to some geometrical interpretation of the index) and a collection of numbers $$(A_{v,w})_{(v,w)\in V\times V}.$$

Despite this list have a "two dimensional" character (bi-indexed) it is in fact a hypermatrice.

The way to import to hypermatrices the results of standard theory of matrix is to define the operations as usual. The product, for instance, is $$ (AB)_{(v,w)}=\sum_{z\in V}A_{v,z}B_{z,w}. $$

The determinant and other important objects can also be defined in this way by replacing the group $\mathbb{S}_n$ by automorphisms of $V$ and so on.

So I guess that do not pop up any interesting feature that justify to give a great attention to this object. Because one can see them just as a replacement of the indexation process.

For the other hand in fields like statistical mechanics and percolation they are much more natural in high dimensional problems than the usual matrices. When we define $p_{uv}$ the probability to see the edge $\{u,v\}$ open or when we are leading with the coupling constants $J_{i,j}$ in the Ising model, all of them are hypermatrices and its linear algebra structure are frequently arise in proofs of important results in correlations inequalities.

Is there an example of Gibbs measure that is not a weak limit of finite volume Gibbs measure ?

2010-11-25T06:11:42Z

Consider the first neighbors Ising model in $\mathbb Z^2$, with the Hamiltonian in the finite volume $\Lambda\subset\mathbb{Z}^2$ given by $$ H_{\Lambda}(\sigma|\omega)=-J\sum_{i,j\in\Lambda\atop{\|i-j\|=1}}\sigma_i\sigma_j-J\sum_{i\in\Lambda, j\in\Lambda^c\atop{\|i-j\|=1}}\sigma_i\omega_j $$ where $\omega\in\{-1,1\}^{\mathbb{Z}^2}$ is a boundary condition.

By the Aizenman-Higuchi Theorem for any $\beta>0$, we have that closed convex hull of the weak limits of Gibbs measures in finite volume is the convex set $ [\mu^{\beta,+},\mu^{\beta,-}]. $

Question: Is there any $\beta>\beta_c$ and $\lambda\in(0,1)$ such that $$ \mu=\lambda\mu^{\beta,+}+(1-\lambda)\mu^{\beta,-} $$ and
$$ \mu\notin \left\{w-\lim_{\Lambda\uparrow\mathbb{Z}^2}\ \ \mu_{\Lambda}^{\beta,\omega}:\omega\in\{-1,1\}^{\mathbb{Z}^2} \right\} \ \ ? $$

What $Re(f(z))=c$ can be if $f$ is a holomorphic function ?

2010-11-26T04:24:21Z

Suppose that $f:U\subset\mathbb{C}\to\mathbb{C}$, where $U$ is a region in the complex plane, is a holomorphic function.

Of course, if $c\in\mathbb{R}$ is a regular value for $\text{Re}(f(z))$ then $\text{Re}(f(z))^{-1}(c)$ is an union of differentiable curves in the plane.

Question:

If $c$ is not a regular value and $\text{Re}(f(z))^{-1}(c)$ have at least one cluster point is this set also a piece-wise differentiable curve ?

For which values of $N$ is known the Lieb-Simon Inequality for $Z_N$ Models ?

2010-06-09T04:24:25Z

Background:

Let $\mathbb Z^d$ denote the $d$-dimensional integer lattice with norm $|x|=\sum_i|x_i|$. For each $x\in\mathbb Z^d$ we associate a spin variable, $\sigma_x$ taking values on the set $\{\exp(2\pi i j/N);1\leq j\leq N\}$ with uniform a priori distribution.

Consider the formal Hamiltonian given on the lattice $\mathbb Z^d$ by $$ H_{ \Lambda }({\sigma}) = -\sum_{\langle x,y\rangle} J_{xy} \vec\sigma_{x} \vec\sigma_{y}
$$ where $J_{xy}$ are nonnegative constants and $\vec\sigma_x \vec\sigma_y$ is inner product in $\mathbb R^2$. The sum is taken over all pair of first neighbors $\langle x,y\rangle$ means that $|x-y|=1$. The Partition function on a finite
$\Lambda\subset \mathbb Z^d$ is given by $$ Z_\Lambda = \int\exp\Big( \beta \sum_{\langle x,y\rangle \in \Lambda }J_{xy} \vec\sigma_x \vec\sigma_y\Big)\;d\sigma $$ where the integral is taken over all sites of $\Lambda$. The two point correlations are given by $$ \langle \vec\sigma_x \vec\sigma_y \rangle_{\Lambda}= Z_{\Lambda}^{-1}\int\vec\sigma_x \vec\sigma_y\exp\Big( \beta \sum_{\langle x,y\rangle \in \Lambda }J_{xy} \vec\sigma_x \vec\sigma_y\Big) d\sigma $$ Question:

For which values of $N$ is known that the Lieb-Simon Inequality is true or false ?

Lieb-Simon Inequality $$ \langle \vec\sigma_x \vec\sigma_y \rangle_{\Lambda} \leq \sum_{b \in \partial B} \bigl<\vec\sigma_x \vec\sigma_b \bigr>_B \bigl<\vec\sigma_b \vec\sigma_y \bigr>_\Lambda, $$ where $B\subset\Lambda\subset\mathbb Z^d$ are finite, $x,y\in\Lambda$, $\partial B=\{z\in B; d(z,B^c)=1\}$ and $\partial B$ separates $x$ and $y$ ($i.e.$ any path from $x$ to $y$ must intercept $\partial B$).

Condition for Uniqueness of Measures

2010-11-10T22:14:24Z

Let be $\Omega$ a compact metric space, $\mathcal{B}(\Omega)$ the $\sigma$-algebra of Borelian sets of $\Omega$ and $\mathcal{M}_1(\Omega)$ the set of all probabilities defined on $\mathcal{B}(\Omega)$.

Suppose that $\lambda,\mu\in\mathcal{M}_1(\Omega)$ are extremal points (in the sense of convex combinations) and there is a real number $c$ such that $$ \lambda(A)\leq c\mu(A)\qquad\text{and}\qquad \mu(A)\leq c \lambda(A)\qquad $$ for all $A\in\mathcal{B}(\Omega)$.

Is it true that $\mu=\lambda$ ?

I have proved the above equality in particular cases: 1) $\Omega$ is discrete; 2) $\Omega$ is some subset of $\mathbb{R}$ (the compacity it was not necessary here).

Poincare Recurrence Theorem on Infinite Measure Space

2010-10-30T00:14:17Z

Suppose that $(\Omega,\mathcal{A},\mu)$ is a $\sigma$-finite measure space of infinite measure and $T:\Omega\to\Omega$ a measure-preserving transformation with measurable inverse. Let be $\Omega_k\in \mathcal{A}$ an increasing sequence such that $\Omega_k\uparrow\Omega$ and $\mu(\Omega_k)<+\infty$ for all $k\in\mathbb{N}$.

Question 1: Given a set $A\in\mathcal{A}$, such that $\mu(A)>0$, is it true that the set $$E_k=\{\omega\in A; T^n(w)\notin A\ \forall n\in\mathbb{N}\ \text{and}\ T^{n_j}(w)\in \Omega_k \ \text{for some infinite sequence}\ (n_j(\omega)) \}$$ has zero measure ?

Question 2: If $T$ is not invertible is $\mu(E_k)=0$, in general ?

On The Convergence of Ergodic Measures

2010-05-04T15:24:19Z

I would like to know an example (not using the Gibbs measure Theory) of a sequence of measures $\mu_n:\mathcal B\to[0,1]$ , where $\mathcal B$ is the $\sigma$-algebra of the borelians of a compact space $X$ such that :

1) $\mu_n$ is ergodic, with respect to a fixed continuous function $T:X\to X$, for all $n\in\mathbb N$;

2) $\mu_n\to \mu$ in the weak-$*$ topology and $\mu$ is not ergodic.

Answer by Leandro for What's your favorite equation, formula, identity or inequality?

2010-08-21T04:28:55Z

Cauchy integral formula $$ f(z)=\frac{1}{2\pi i}\int_{\gamma}\frac{f(w)}{w-z} dw $$

Comment by Leandro

2013-04-02T15:36:36Z

Hi Vincent, thanks for the answer.

Comment by Leandro

2012-05-18T08:58:36Z

Thanks for the observation Ralph, you right such lower bound can not exists.

Comment by Leandro

2011-10-25T05:10:52Z

Hi Gerry thanks a lot for this reference.

Comment by Leandro

2011-10-21T07:07:25Z

@Sauravrt, see math.stackexchange.com/questions/74469/… I give you there a suggestion on how to prove this identity.

Comment by Leandro

2011-10-19T08:00:20Z

Hi Faisal, I think I will edit this answer and add the proof.

Comment by Leandro

2011-10-19T06:31:17Z

@John :) you right. I should be drunk !

Comment by Leandro

2011-10-19T04:57:19Z

Hi David, perhaps your question will be closed soon, because this site is used for research level questions in Mathematics, see FAQ. In any case it seems to me reasonable question for this site: math.stackexchange.com/questions

Comment by Leandro

2011-10-14T02:20:55Z

Thanks Matthew for the example. The last paragraph I have to think a bit more. Anyway your example answer the question since I did not made any assumptions on the norm.

Comment by Leandro

2011-10-13T11:27:43Z

@Xabier why my example does not fulfill the conditions ?

Comment by Leandro

2011-10-13T10:58:51Z

@Matthew as you noted the field $\mathbb{K}$ will be not so general, we have to suppose that there exist a modulus function defined on it.

Comment by Leandro

2011-10-13T07:05:44Z

@Ricky thanks for point this correction.

Comment by Leandro

2011-08-25T22:54:52Z

Hi pm, I am most interested in the case were they are arbitrary.

Comment by Leandro

2011-06-30T00:05:56Z

morph I misread your question, sorry about that.

Comment by Leandro

2011-06-18T05:32:13Z

Hi ljjpfx, thank you for the references, I don't knew them.

Comment by Leandro

2011-06-14T08:41:09Z

Hi Pietro, thanks a lot.