A priori bounds for some infinitely renormalizable quadratics: II. Decorations

Jeremy Kahn; Mikhail Lyubich

Displaying similar documents to “A priori bounds for some infinitely renormalizable quadratics: II. Decorations”

Cutting the loss of derivatives for solvability under condition $(Ψ)$

Nicolas Lerner (2006)

Bulletin de la Société Mathématique de France

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For a principal type pseudodifferential operator, we prove that condition $(ψ)$ implies local solvability with a loss of 3/2 derivatives. We use many elements of Dencker’s paper on the proof of the Nirenberg-Treves conjecture and we provide some improvements of the key energy estimates which allows us to cut the loss of derivatives from $ϵ + 3 / 2$ for any $ϵ > 0$ (Dencker’s most recent result) to 3/2 (the present paper). It is already known that condition $(ψ)$ doesimply local solvability with a loss of 1...

Some results on spaces with $ℵ_{1}$ -calibre

Wei-Feng Xuan, Wei-Xue Shi (2016)

Commentationes Mathematicae Universitatis Carolinae

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We prove that, assuming , if $X$ is a space with $ℵ_{1}$ -calibre and a zeroset diagonal, then $X$ is submetrizable. This gives a consistent positive answer to the question of Buzyakova in Observations on spaces with zeroset or regular $G_{δ}$ -diagonals, Comment. Math. Univ. Carolin. 46 (2005), no. 3, 469–473. We also make some observations on spaces with $ℵ_{1}$ -calibre.

On multiset colorings of generalized corona graphs

Yun Feng, Wensong Lin (2016)

Mathematica Bohemica

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A vertex $k$ -coloring of a graph $G$ is a if $M (u) \neq M (v)$ for every edge $u v \in E (G)$ , where $M (u)$ and $M (v)$ denote the multisets of colors of the neighbors of $u$ and $v$ , respectively. The minimum $k$ for which $G$ has a multiset $k$ -coloring is the $χ_{m} (G)$ of $G$ . For an integer $ℓ \geq 0$ , the $ℓ$ - of a graph $G$ , ${cor}^{ℓ} (G)$ , is the graph obtained from $G$ by adding, for each vertex $v$ in $G$ , $ℓ$ new neighbors which are end-vertices. In this paper, the multiset chromatic numbers are determined for $ℓ$ - of all complete graphs, the regular complete...

The parabolic Anderson model in a dynamic random environment: Basic properties of the quenched Lyapunov exponent

D. Erhard, F. den Hollander, G. Maillard (2014)

Annales de l'I.H.P. Probabilités et statistiques

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In this paper we study the parabolic Anderson equation $\partial u (x, t) / \partial t = κ 𝛥 u (x, t) + ξ (x, t) u (x, t)$ , $x \in ℤ^{d}$ , $t \geq 0$ , where the $u$ -field and the $ξ$ -field are $ℝ$ -valued, $κ \in [0, \infty)$ is the diffusion constant, and $𝛥$ is the discrete Laplacian. The $ξ$ -field plays the role of athat drives the equation. The initial condition $u (x, 0) = u_{0} (x)$ , $x \in ℤ^{d}$ , is taken to be non-negative and bounded. The solution of the parabolic Anderson equation describes the evolution of a field of particles performing independent simple random walks with binary branching: particles jump at rate $2 d κ$ ,...

On standard norm varieties

Nikita A. Karpenko, Alexander S. Merkurjev (2013)

Annales scientifiques de l'École Normale Supérieure

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Let $p$ be a prime integer and $F$ a field of characteristic $0$ . Let $X$ be theof a symbol in the Galois cohomology group $H^{n + 1} (F, μ_{p}^{\otimes n})$ (for some $n \geq 1$ ), constructed in the proof of the Bloch-Kato conjecture. The main result of the paper affirms that the function field $F (X)$ has the following property: for any equidimensional variety $Y$ , the change of field homomorphism $CH (Y) \to CH (Y_{F (X)})$ of Chow groups with coefficients in integers localized at $p$ is surjective in codimensions $< (dim X) / (p - 1)$ . One of the main ingredients of the proof is a computation...

Dimers and cluster integrable systems

Alexander B. Goncharov, Richard Kenyon (2013)

Annales scientifiques de l'École Normale Supérieure

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We show that the dimer model on a bipartite graph $Γ$ on a torus gives rise to a quantum integrable system of special type, which we call a. The phase space of the classical system contains, as an open dense subset, the moduli space $Ł_{Γ}$ of line bundles with connections on the graph $Γ$ . The sum of Hamiltonians is essentially the partition function of the dimer model. We say that two such graphs $Γ_{1}$ and $Γ_{2}$ areif the Newton polygons of the corresponding partition functions coincide up to translation....

Coincidence for substitutions of Pisot type

Marcy Barge, Beverly Diamond (2002)

Bulletin de la Société Mathématique de France

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Let $ϕ$ be a substitution of Pisot type on the alphabet $𝒜 = {1, 2, ..., d}$ ; $ϕ$ satisfies theif for every $i, j \in 𝒜$ , there are integers $k, n$ such that $ϕ^{n} (i)$ and $ϕ^{n} (j)$ have the same $k$ -th letter, and the prefixes of length $k - 1$ of $ϕ^{n} (i)$ and $ϕ^{n} (j)$ have the same image under the abelianization map. We prove that the strong coincidence condition is satisfied if $d = 2$ and provide a partial result for $d \geq 2$ .

Local-global divisibility of rational points in some commutative algebraic groups

Roberto Dvornicich, Umberto Zannier (2001)

Bulletin de la Société Mathématique de France

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Let $𝒜$ be a commutative algebraic group defined over a number field $k$ . We consider the following question:A complete answer for the case of the multiplicative group $𝔾_{m}$ is classical. We study other instances and in particular obtain an affirmative answer when $r$ is a prime and $𝒜$ is either an elliptic curve or a torus of small dimension with respect to $r$ . Without restriction on the dimension of a torus, we produce an example showing that the answer can be negative even when $r$ is a prime. ...

The equation $- Δ 𝑢 - λ \frac{𝑢}{{| 𝑥 |}^{2}} = {| \nabla 𝑢 |}^{𝑝} + 𝑐 𝑓 (𝑥)$ : The optimal power

Boumediene Abdellaoui, Ireneo Peral (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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We will consider the following problem $- Δ u - λ \frac{u}{{| x |}^{2}} = {| \nabla u |}^{p} + c f, u > 0 in Ω,$ where $Ω \subset ℝ^{N}$ is a domain such that $0 \in Ω$ , $N \geq 3$ , $c > 0$ and $λ > 0$ . The main objective of this note is to study the precise threshold $p_{+} = p_{+} (λ)$ for which there is novery weak supersolutionif $p \geq p_{+} (λ)$ . The optimality of $p_{+} (λ)$ is also proved by showing the solvability of the Dirichlet problem when $1 \leq p < p_{+} (λ)$ , for $c > 0$ small enough and $f \geq 0$ under some hypotheses that we will prescribe.

Piecewise linear approximation of smooth functions of two variables

Joseph H.G. Fu (2013)

Actes des rencontres du CIRM

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The normal cycle of a singular subset $X$ of a smooth manifold is a basic tool for understanding and computing the curvature of $X$ . If $X$ is replaced by a singular function on $ℝ^{n}$ then there is a natural companion notion called the of $f$ , which has been introduced by the author and by R. Jerrard. We discuss a few fundamental facts and open problems about functions $f$ that admit gradient cycles, with particular attention to the first nontrivial dimension $n = 2$ .

Limit theorems for one and two-dimensional random walks in random scenery

Fabienne Castell, Nadine Guillotin-Plantard, Françoise Pène (2013)

Annales de l'I.H.P. Probabilités et statistiques

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Random walks in random scenery are processes defined by $Z_{n} : = \sum_{k = 1}^{n} ξ_{X_{1} + \dots + X_{k}}$ , where $(X_{k}, k \geq 1)$ and $(ξ_{y}, y \in ℤ^{d})$ are two independent sequences of i.i.d. random variables with values in $ℤ^{d}$ and $ℝ$ respectively. We suppose that the distributions of $X_{1}$ and $ξ_{0}$ belong to the normal basin of attraction of stable distribution of index $α \in (0, 2]$ and $β \in (0, 2]$ . When $d = 1$ and $α \neq 1$ , a functional limit theorem has been established in ( (1979) 5–25) and a local limit theorem in (To appear). In this paper, we establish the convergence in distribution...

Arithmetic Properties of Generalized Rikuna Polynomials

Z. Chonoles, J. Cullinan, H. Hausman, A.M. Pacelli, S. Pegado, F. Wei (2014)

Publications mathématiques de Besançon

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Fix an integer $ℓ \geq 3$ . Rikuna introduced a polynomial $r (x, t)$ defined over a function field $K (t)$ whose Galois group is cyclic of order $ℓ$ , where $K$ satisfies some mild hypotheses. In this paper we define the family of ${r_{n} (x, t)}_{n \geq 1}$ of degree $ℓ^{n}$ . The $r_{n} (x, t)$ are constructed iteratively from the $r (x, t)$ . We compute the Galois groups of the $r_{n} (x, t)$ for odd $ℓ$ over an arbitrary base field and give applications to arithmetic dynamical systems.

Limit distributions for multitype branching processes of $m$ -ary search trees

Brigitte Chauvin, Quansheng Liu, Nicolas Pouyanne (2014)

Annales de l'I.H.P. Probabilités et statistiques

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Let $m \geq 3$ be an integer. The so-calledis a discrete time Markov chain which is very popular in theoretical computer science, modelling famous algorithms used in searching and sorting. This random process satisfies a well-known phase transition: when $m \leq 26$ , the asymptotic behavior of the process is Gaussian, but for $m \geq 27$ it is no longer Gaussian and a limit $W^{D T}$ of a complex-valued martingale arises. In this paper, we consider the multitype branching process which is the continuous time version of...

Equations in the Hadamard ring of rational functions

Andrea Ferretti, Umberto Zannier (2007)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

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Let $K$ be a number field. It is well known that the set of recurrencesequences with entries in $K$ is closed under component-wise operations, and so it can be equipped with a ring structure. We try to understand the structure of this ring, in particular to understand which algebraic equations have a solution in the ring. For the case of cyclic equations a conjecture due to Pisot states the following: assume ${a_{n}}$ is a recurrence sequence and suppose that all the $a_{n}$ have a $d^{th}$ root in the field...

Effective bounds for Faltings’s delta function

Jay Jorgenson, Jürg Kramer (2014)

Annales de la faculté des sciences de Toulouse Mathématiques

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In his seminal paper on arithmetic surfaces Faltings introduced a new invariant associated to compact Riemann surfaces $X$ , nowadays called Faltings’s delta function and here denoted by $δ_{Fal} (X)$ . For a given compact Riemann surface $X$ of genus $g_{X} = g$ , the invariant $δ_{Fal} (X)$ is roughly given as minus the logarithm of the distance with respect to the Weil-Petersson metric of the point in the moduli space $ℳ_{g}$ of genus $g$ curves determined by $X$ to its boundary $\partial ℳ_{g}$ . In this paper we begin by revisiting a formula derived...