Non-wandering points and the depth for graph maps

Let G be a graph and f:G→G be continuous.Denote by R(f) andΩ(f) the set of recurrent points and the set of non-wandering points of f respectively.LetΩ_0(f) = G andΩ_n(f)=Ω(f|_(Ω_(n-1)(f))) for all n∈N.The minimal m∈NU {∞} such thatΩ_m(f)=Ω_(m 1)(f) is called the depth of f.In this paper,we show thatΩ_2 (f)=(?) and the depth of f is at most 2.Furthermore,we obtain some properties of non-wandering points of f.     

On matroid intersections

This paper exploits and extends results of Edmonds, Cunningham, Cruse and McDiarmid on matroid intersections. Letr ₁ andr ₂ be rank functions of two matroids defined on the same setE. For everyS ⊂E, letr ₁₂(S) be the largest cardinality of a subset ofS independent in both matroids, 0≦k≦r ₁₂(E)−1. It is shown that, ifc is nonnegative and integral, there is ay: 2 ^E →Z ⁺ which maximizes and , subject toy≧0, ∀j∈E, .     

Un teorema di Radon-Nikodym per funzioni d’insieme subadditive

Riassunto In questo articolo si danno delle condizioni necessarie e sufficienti affinchè per una fissata coppia di funzioni d’insieme ν, μ crescenti esista una funzionef tale che ν=∫fdμ. Si ottiene cosi una proposizione comprendente il teorema di R-N. classico e dei teoremi di R-N., presentati da altri autori, riguardanti le funzioni d’insieme finitamente additive e le funzioni d’insieme subadditive e continue per successioni crescenti.

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Résumé Soient ν, μ:A→[0,+∞) deux fonctions d’ensemble croissantes sur une σ-algèbre d’ensemblesA⊂T(X), telles que pour chaqueA∈A avec ν(A)=μ(A)=0 on a l’égalité μ(A)=μ(A∪S) ∀S∈A (c’est le cas des fonctions sousadditives!). Dans cet article on démontre qu’il existe une fonctionf A-measurable telle que ν=∫fdμ si et seulement si pour chaquer∈(0, + ∞) il y a un ensembleA _r∈A qui vérifie les trois conditions suivantes: (1) ,B∈A avecB⊂A; (2) A (3) limν(A _r)=0. On déduit ainsi une proposition qui a été donnée parI. Forana: ?Si ν, μ sont simplement additives, il existe une fonctionf telle que ν=∫fdμ si et seulement si ν≪μ et la fonction d’ensemble additive a une decomposition de Hahn pour chaquer∈(0, + ∞), c’est-á-dire il y aA _r∈A tel que ?.

     

On linear isometries of Banach lattices in <Emphasis Type="Italic">C</Emphasis><Subscript>0</Subscript>(Ω)-spaces

Consider the space C0(Ω) endowed with a Banach lattice-norm ‖ · ‖ that is not assumed to be the usual spectral norm ‖ · ‖_∞ of the supremum over Ω. A recent extension of the classical Banach-Stone theorem establishes that each surjective linear isometry U of the Banach lattice (C ₀(Ω), ‖ · ‖) induces a partition Π of Ω into a family of finite subsets S ⊂ Ω along with a bijection T: Π → Π which preserves cardinality, and a family [u(S): S ∈ Π] of surjective linear maps u(S): C(T(S)) → C(S) of the finite-dimensional C*-algebras C(S) such that

On one extremal problem for numerical series

Let Γ be the set of all permutations of the natural series and let α = {α _j} _j∈ℕ, ν = {ν_j} _j∈ℕ, and η = {η_j} _j∈ℕ be nonnegative number sequences for which

A note on regularity for a class of quasilinear elliptic equations

The following regularity of weak solutions of a class of elliptic equations of the form are investigated.     

Weighted estimates for multilinear commutators of Marcinkiewicz integral

Let μ be the n-dimensional Marcinkiewicz integral and μb the multilinear commutator of μ. In this paper, the following weighted inequalities are proved for ω ∈ A∞ and 0 〈 p 〈 ∞,
｜｜μ（f）｜｜LP（ω）≤C｜Mf｜LP（ω） and ｜｜μb（f）｜｜LP（ω）≤C｜｜ML（log L）^1/r f｜｜LP（ω）.
The weighted weak L（log L）^1/r -type estimate is also established when p=1 and ω∈A1.     

Description of the lipschitz and zygmund classes in terms of the modulus of continuity

Let C be the space of 2π-periodic continuous real-valued functions, let

Nonexistence of invariant graphs in all supercritical energy levels of mechanical Lagrangians in T 2

Let (T², g) be a smooth Riemannian structure in the torus T². We show that given ε > 0 and any C^∞ function U : T² → ℝ there exists a C¹ function U_ε with Lipschitz derivatives that is ε-C⁰ close to U for which there are no continuous invariant graphs in any supercritical energy level of the mechanical Lagrangian L_ε : TT² → ℝ given by . We also show that given n ∈ ℕ, the set of C^∞ potentials U : T² → ℝ for which there are no continuous invariant graphs in any supercritical energy level E ≤ n of is C⁰ dense in the set of C^∞ functions. Partially supported by CNPq, FAPERJ-Cientistas do nosso estado.     

Directional approximation of functions of two variables in the spaces ℒ p (R2) and ℒ p (R + 2 )

Let r ∈ N, α, t ∈ R, x ∈ R ², f: R ² → C, and denote $ \Delta _{t,\alpha }^r (f,x) = \sum\limits_{k = 0}^r {( - 1)^{r - k} c_r^k f(x_1 + kt\cos \alpha ,x_2 + kt\sin \alpha ).} $ In this paper, we investigate the relation between the behavior of the quantity $ \left\| {\int\limits_E {\Delta _{t,\alpha }^r (f, \cdot )\Psi _n (t)dt} } \right\|_{p,G} , $ as n → ∞ (here, E ? R, G ∈ {R ², R ₊ ² }, and ψ _n ∈ L ₁(E) is a positive kernel) and structural properties of function f. These structural properties are characterized by its “directional” moduli of continuity: $ \omega _{r,\alpha } (f,h)_{p,G} = \mathop {\sup }\limits_{0 \leqslant t \leqslant h} \left\| {\Delta _{t,\alpha }^r (f)} \right\|_{p,G} . $ Here is one of the results obtained. Theorem 1. Let E and A be intervals in R ₊ such that A ? E, f ∈ L _p (G), α ∈ [0, 2π] when G =R ² and α ∈ [0, π/2] when G = R ₊ ² Denote Δ _{n, k} = ∫ _A t ^k ψ _n (t)dt. If there exists an r ∈ N such that, for any m ∈ N, we have Δ _{m, r} > 0, Δ _{m, r + 1} < ∞, and $ \mathop {\lim }\limits_{n \to \infty } \frac{{\Delta _{n,r + 1} }} {{\Delta _{n,r} }} = 0,\mathop {\lim }\limits_{n \to \infty } \Delta _{n,r}^{ - 1} \int\limits_{E\backslash A} {\Psi _n = 0} , $ then the relations $ \mathop {\lim }\limits_{n \to \infty } \Delta _{n,r}^{ - 1} \left\| {\int\limits_E {\Delta _{t,\alpha }^r (f, \cdot )\Psi _n dt} } \right\|_{p,G} \leqslant K, \mathop {\sup }\limits_{t \in (0,\infty )} t^r \omega _{r,\alpha } (f,t)_{p,G} \leqslant K $ are equivalent. Particular methods of approximation are considered. We establish Corollary 1. Let p, G, α, and f be the same as in Theorem 1, and $ \sigma _{n,\alpha } (f,x) = \frac{2} {{\pi n}}\int\limits_{R_ + } {\Delta _{t,\alpha }^1 (f,x)} \left( {\frac{{\sin \frac{{nt}} {2}}} {t}} \right)^2 dt. $ Then the relations $ \mathop {\underline {\lim } }\limits_{n \to \infty } \frac{{\pi n}} {{\ln n}}\left\| {\sigma _{n,\alpha } (f)} \right\|_{p,G} \leqslant K Let r ∈ N, α, t ∈ R, x ∈ R ², f: R ² → C, and denote In this paper, we investigate the relation between the behavior of the quantity as n → ∞ (here, E ⊂ R, G ∈ {R ², R ₊²}, and ψ _n ∈ L ₁(E) is a positive kernel) and structural properties of function f. These structural properties are characterized by its “directional” moduli of continuity: Here is one of the results obtained. Theorem 1. Let E and A be intervals in R ₊ such that A ⊂ E, f ∈ L _p (G), α ∈ [0, 2π] when G =R ² and α ∈ [0, π/2] when G = R ₊² Denote Δ _{n, k} = ∫ _A t ^k ψ _n (t)dt. If there exists an r ∈ N such that, for any m ∈ N, we have Δ _{m, r} > 0, Δ _{m, r + 1} < ∞, and then the relations are equivalent. Particular methods of approximation are considered. We establish Corollary 1. Let p, G, α, and f be the same as in Theorem 1, and Then the relations and are equivalent. Original Russian Text ? N.Yu. Dodonov, V.V. Zhuk, 2008, published in Vestnik Sankt-Peterburgskogo Universiteta. Seriya 1. Matematika, Mekhanika, Astronomiya, 2008, No. 2, pp. 23–33.     

On the local Artin conductor f<Subscript>Artin</Subscript> (Χ) of a character Χ of Gal(E/K) — II: Main results for the metabelian case

This paper which is a continuation of [2], is essentially expository in nature, although some new results are presented. LetK be a local field with finite residue class fieldK _k. We first define (cf. Definition 2.4) the conductorf(E/K) of an arbitrary finite Galois extensionE/K in the sense of non-abelian local class field theory as wheren _G is the break in the upper ramification filtration ofG = Gal(E/K) defined by . Next, we study the basic properties of the idealf(E/K) inO _k in caseE/K is a metabelian extension utilizing Koch-de Shalit metabelian local class field theory (cf. [8]). After reviewing the Artin charactera _G : G → ℂ ofG := Gal(E/K) and Artin representationsA _g G → G →GL(V) corresponding toa _G : G → ℂ, we prove that (Proposition 3.2 and Corollary 3.5) where Χ_gr : G → ℂ is the character associated to an irreducible representation ρ: G → GL(V) ofG (over ℂ). The first main result (Theorem 1.2) of the paper states that, if in particular,ρ : G → GL(V) is an irreducible representation ofG(over ℂ) with metabelian image, then where Gal(E^ker(ρ)/E^ker(ρ)•) is any maximal abelian normal subgroup of Gal(E^ker(ρ)/K) containing Gal(E^ker(ρ) /K)′, and the break n_G/ker(ρ) in the upper ramification filtration of G/ker(ρ) can be computed and located by metabelian local class field theory. The proof utilizes Basmaji’s theory on the structure of irreducible faithful representations of finite metabelian groups (cf. [1]) and on metabelian local class field theory (cf. [8]). We then discuss the application of Theorem 1.2 on a problem posed by Weil on the construction of a ‘natural’A _G ofG over ℂ (Problem 1.3). More precisely, we prove in Theorem 1.4 that ifE/K is a metabelian extension with Galois group G, then Kazim İlhan ikeda whereN runs over all normal subgroups of G, and for such anN, V _n denotes the collection of all ∼-equivalence classes [ω]∼, where ‘∼’ denotes the equivalence relation on the set of all representations ω : (G/N)^• → ℂ^Χ satisfying the conditions Inert(ω) = {δ ∈ G/N : ℂ_δ} = ω =(G/N) and where δ runs over R((G/N)^•/(G/N)), a fixed given complete system of representatives of (G/N)^•/(G/N), by declaring that ω₁ ∼ ω₂ if and only if ω₁ = ω _2,δ for some δ ∈ R((G/N)^•/(G/N)). Finally, we conclude our paper with certain remarks on Problem 1.1 and Problem 1.3.     

Automorphic L-Functions in the Weight Aspect

Let S_k(Γ) be the space of holomorphic Γ-cusp forms f(z) of even weight k ≥ 12 for Γ = SL(2, ℤ), and let S_k(Γ)⁺ be the set of all Hecke eigenforms from this space with the first Fourier coefficient a_f(1) = 1. For f ∈ S_k(Γ)⁺, consider the Hecke L-function L(s, f). Let

The individual ergodic theorem forp-sequences

Let (Ω,f,P) be a probability space and letT be a measure-preserving weak mixing transformation. We define a large class of sequences of integers calledp-sequences, such that iff∈L ₁ there exists a set Ω′⊂Ω of probability one and for ω∈Ω′ we have for everyp-sequence {k_n}.     

A joint norm control Nehari type theorem forN-tuples of Hardy spaces

If N ∈ ℕ, 0 < p ≤ 1, and(X_k) _k=1 ^N are r.i.p-spaces, it is shown that there is C(= C(p, N)) > 0, such that for every ƒ ∈ ∩ _k=1 ^N X_k, there exists with , for every 1 ≤ k ≤ N. Also, if ⊓ is a convex polygon in ℝ², it is proved that the N-tuple (H(X₁),…, H(X_n)) is K_⊓-closed with respect to (X₁,…, X_N) in the sense of Pisier. Everything follows from Theorem 2.1, which is a general analytic partition of unity type result.     

Some remarks on the identity between a variational integral and its relaxed functional

Summary Letf: (x, z)∈R ⁿ×Rⁿ→f(x, z)∈[0, +∞] be measurable inx and convex inz. It is proved, by an example, that even iff verifies a condition as|z| ^p≤f(x, z)≤Λ(a(x)+|z|^q) with 1<p<q,a∈L _loc ^s (R ⁿ),s>1, the functional that isL ¹(Ω)-lower semicontinuous onW ^1,1(Ω), does not agree onW ^1,1(Ω) with its relaxed functional in the topologyL ¹(Ω) given by inf

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Riassunto Siaf: (x, z)∈R ⁿ×Rⁿ→f(x, z)∈[0, +∞] misurabile inx e convessa inz. Si mostra con un esempio che anche sef verifica una condizione del tipo|z| ^p≤f(x, z)≤Λ(a(x)+|z|^q) con 1<p<q,a∈L _loc ^s (R ⁿ),s>1, il funzionale , che èL ¹(Ω)-semicontinuo inferiormente suW ^1,1(Ω), non coincide suW ^1,1(Ω) con il suo funzionale rilassato nella topologiaL ¹(Ω) definito da inf

     

The Dual Spaces of Sets of Difference Sequences of Order <Emphasis Type="Italic">m</Emphasis> and Matrix Transformations

Let p = （pk）k=0^∞ be a bounded sequence of positive reals, m C N and u be s sequence of nonzero terms. If x = （xk）k=0^∞ is any sequence of complex numbers we write Δ（m）x for the sequence of the m th order differences of x and Δu^（m）X = {x=（x）k=0^∞ uΔ（m）x ∈ X} for any set X of sequences. We determine the α-, β- and γ-duals of the sets Δμ^（m）X for X=co（p）,c（p）,l∞（p） and characterize some matrix transformations between these spaces Δ^（m）X.     

Heterogeneous ubiquitous systems in ℝ d and Hausdorff dimension

Let {x_n}_n∈ℕ be a sequence in [0, 1]^d , {λ_n}n∈ℕ a sequence of positive real numbers converging to 0, and δ > 1. The classical ubiquity results are concerned with the computation of the Hausdorff dimension of limsup-sets of the form

Higher integrability for minimizers of variational integrals with nonstandard growth

We consider a (possibly) vector-valued function u: Ω→R ^N, Ω⊂R ⁿ, minimizing the integral , whereD _iu=∂u/∂x _i, or some more general functional retaining the same behaviour; we prove higher integrability forDu:D ₁u,…,D_n−1u∈L^q, under suitable assumptions ona _i(x).

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Sunto Consideriamo una funzione u: Ω→R ^N, Ω⊂R ⁿ che minimizzi l'integrale , doveD _iu=∂u/∂x_i, o un funzionale con un comportamento simile; sotto opportune ipotesi sua _i(x), dimostriamo la seguente maggiore integrabilità perDu:D ₁u,…,D_n−1uεL^q.