The Turán-type inequalities for complex polynomials in <Emphasis Type="Italic">L</Emphasis><Subscript>0</Subscript>-metrics

In this paper we obtain inequalities for measures of trigonometric polynomials of power (P _n (e ^iφ )) and general (T _n (t)) types with the help of measures and their mth derivatives.     

<Emphasis Type="Italic">σ</Emphasis>-finite and semimoderate variational measures

A characterization of σ-finite variational measures of Thomson’s type in ?^m is obtained. It is shown that the notions of σ-finite and semimoderate measures coincide for variational measures constructed from continuous functions of an interval with respect to a complete basis in ?^m.     

Simple polytopes without small separators,II: Thurston’s bound

We construct a diffeomorphism of T³ admitting any finite or countable number of physical measures with intermingled basins. The examples are partially hyperbolic with splitting TT³ = E ^cs ⊕ E ^u and can be made volume hyperbolic and topologically mixing.     

Measure partitions using hyperplanes with fixed directions

We study nested partitions of R^d obtained by successive cuts using hyperplanes with fixed directions. We establish the number of measures that can be split evenly simultaneously by taking a partition of this kind and then distributing the parts among k sets. This generalises classical necklace splitting results and their more recent high-dimensional versions. With similar methods we show that in the plane, for any t measures there is a path formed only by horizontal and vertical segments using at most t - 1 turns that splits them by half simultaneously, and optimal masspartitioning results for chessboard colourings of R^d using hyperplanes with fixed directions.     

Transfinite diameter notions in ℂ N and integrals of Vandermonde determinants

We provide a general framework and indicate relations between the notions of transfinite diameter, homogeneous transfinite diameter, and weighted transfinite diameter for sets in ? ^N . An ingredient is a formula of Rumely (A Robin formula for the Fekete–Leja transfinite diameter, Math. Ann. 337 (2007), 729–738) which relates the Robin function and the transfinite diameter of a compact set. We also prove limiting formulas for integrals of generalized Vandermonde determinants with varying weights for a general class of compact sets and measures in ? ^N . Our results extend to certain weights and measures defined on cones in ? ^N .     

Approximation of measures by Markov processes and homogeneous affine iterated function systems

It is shown that under certain conditions, attractive invariant measures for iterated function systems (indeed for Markov processes on locally compact spaces) depend continuously on parameters of the system. We discuss a special class of iterated function systems, the homogeneous affine ones, for which an inverse problem is easily solved in principle by Fourier transform methods. We show that a probability measureμ onR ⁿ can be approximated by invariant measures for finite iterated function systems of this class if \(\hat \mu (t)/\hat \mu (a^T t)\) is positive definite for some nonzero contractive linear mapa:R ⁿ →R ⁿ . Moments and cumulants are also discussed.     

Multipliers of spherical harmonics and energy of measures on the sphere

We consider the operator,f(Δ) for Δ the Laplacian, on spaces of measures on the sphere inR ^d , show how to determine a family of approximating kernels for this operator assuming that certain technical conditions are satisfied, and give estimates for theL²-norm off(Δ)μ in terms of the energy of the measure μ. We derive a formula, analogous to the classical formula relating the energy of a measure onR ^d with its Fourier transform, comparing the energy of a measure on the sphere with the size of its spherical harmonics. An application is given to pluriharmonic measures.     

The relations among the three kinds of conditional risk measures

Let(Ω,E,P)be a probability space,F a sub-σ-algebra of E,Lp(E)(1 p+∞)the classical function space and Lp F(E)the L0(F)-module generated by Lp(E),which can be made into a random normed module in a natural way.Up to the present time,there are three kinds of conditional risk measures,whose model spaces are L∞(E),Lp(E)(1 p+∞)and Lp F(E)(1 p+∞)respectively,and a conditional convex dual representation theorem has been established for each kind.The purpose of this paper is to study the relations among the three kinds of conditional risk measures together with their representation theorems.We first establish the relation between Lp(E)and Lp F(E),namely Lp F(E)=Hcc(Lp(E)),which shows that Lp F(E)is exactly the countable concatenation hull of Lp(E).Based on the precise relation,we then prove that every L0(F)-convex Lp(E)-conditional risk measure(1 p+∞)can be uniquely extended to an L0(F)-convex Lp F(E)-conditional risk measure and that the dual representation theorem of the former can also be regarded as a special case of that of the latter,which shows that the study of Lp-conditional risk measures can be incorporated into that of Lp F(E)-conditional risk measures.In particular,in the process we find that combining the countable concatenation hull of a set and the local property of conditional risk measures is a very useful analytic skill that may considerably simplify and improve the study of L0-convex conditional risk measures.∞     

Another connection between orthogonal polynomials and L-orthogonal polynomials

We consider a connection that exists between orthogonal polynomials associated with positive measures on the real line and orthogonal Laurent polynomials associated with strong measures of the class S³[0,β,b]. Examples are given to illustrate the main contribution in this paper.     

Composition limits and separating examples for some boolean function complexity measures

Block sensitivity (bs(f)), certificate complexity (C(f)) and fractional certificate complexity (C*(f)) are three fundamental combinatorial measures of complexity of a boolean function f. It has long been known that bs(f) ≤ C*(f) ≤ C(f) = O(bs(f)²). We provide an infinite family of examples for which C(f) grows quadratically in C*(f) (and also bs(f)) giving optimal separations between these measures. Previously the biggest separation known was \(C(f) = C*(f)^{\log _{4,5} 5}\). We also give a family of examples for which C*(f)= Ω (bs(f)^3/2).These examples are obtained by composing boolean functions in various ways. Here the composition fog of f with g is obtained by substituting for each variable of f a copy of g on disjoint sets of variables. To construct and analyse these examples we systematically investigate the behaviour under function composition of these measures and also the sensitivity measure s(f). The measures s(f), C(f) and C*(f) behave nicely under composition: they are submultiplicative (where measure m is submultiplicative if m(fog) ≤ m(f)m(g)) with equality holding under some fairly general conditions. The measure bs(f) is qualitatively different: it is not submultiplicative. This qualitative difference was not noticed in the previous literature and we correct some errors that appeared in previous papers. We define the composition limit of a measure m at function f, m ^lim(f) to be the limit as k grows of m(f ^(k))^1/k , where f ^(k) is the iterated composition of f with itself k-times. For any function f we show that bs ^lim(f) = (C*)^lim(f) and characterize s ^lim(f); (C*)^lim(f), and C ^lim(f) in terms of the largest eigenvalue of a certain set of 2×2 matrices associated with f.     

Patterson-Sullivan-type measures on Riccati foliations with quasi-Fuchsian holonomy

We consider Riccati foliations ?_ρ with hyperbolic leaves, over a finite hyperbolic Riemann Surface S, constructed by suspending a representation ρ: π ₁(S) → PSL(2,?) in a quasi-Fuchsian group. The foliated geodesic flow has a repeller-attractor dynamic with generic statistics µ⁺ and µ^? for positive and negative times, respectively. These measures have a common projection to a harmonic measure μ_ρ for the Riccati foliation. We describe μ _ρ ⁺ , μ _ρ ^- and μ_ρ in terms of the Patterson-Sullivan construction, and we show that the measures μ_ρ provide examples of the conformal harmonic measures introduced by M. Brunella.     

Large entropy measures for endomorphisms of ℂℙ k

Let f be a holomorphic endomorphism of ?? ^k . We construct by using coding techniques a class of ergodic measures as limits of non-uniform probability measures on preimages of points. We show that they have large metric entropy, close to log d ^k . We establish for them strong stochastic properties and prove the positivity of their Lyapunov exponents. Since they have large entropy, those measures are supported in the support of the maximal entropy measure of f. They in particular provide lower bounds for the Hausdorff dimension of the Julia set.     

Exotic analytic structures and Eisenman intrinsic measures

Using Eisenman intrinsic measures we prove a cancellation theorem. This theorem allows to find new examples of exotic analytic structures onC ⁿ under which we understand smooth complex affine algebraic varietiers which are diffeomorphic toR ²ⁿ but not biholomorphic toC ⁿ . We also develop a new method of constructing these structures which enables us to produce exotic analytic structures onC ³ with a given number of hypersurfaces isomorphic toC ² and a family of these structures with a given number of moduli.     

Caricature of hydrodynamics for lattice dynamics

We consider the lattice dynamics in the half-space, with zero boundary condition. The initial data are supposed to be random function. We introduce the family of initial measures {?? ₀ ^? , ? > 0} depending on a small scaling parameter ?. We assume that the measures ?? ₀ ^? are locally homogeneous for space translations of order much less than ? ^?1 and nonhomogeneous for translations of order ? ^?1. Moreover, the covariance of ?? ₀ ^? decreases with distance uniformly in ?. Given ?? ?? ? / 0, r ?? ? ₊ ^d , and ?? > 0, we consider the distributions of random solution in the time moments t = ??/? _?? and at lattice points close to [r/?] ?? ? ₊ ^d . Themain goal is to study the asymptotic behavior of these distributions as ? ?? 0 and to derive the limit hydrodynamic equations of the Euler or Navier-Stokes type.     

A BK inequality for randomly drawn subsets of fixed size

The BK inequality (van den Berg and Kesten in J Appl Probab 22:556?C569, 1985) says that, for product measures on {0, 1} ⁿ , the probability that two increasing events A and B ??occur disjointly?? is at most the product of the two individual probabilities. The conjecture in van den Berg and Kesten (1985) that this holds for all events was proved by Reimer (Combin Probab Comput 9:27?C32, 2000). Several other problems in this area remained open. For instance, although it is easy to see that non-product measures cannot satisfy the above inequality for all events, there are several such measures which, intuitively, should satisfy the inequality for all increasing events. One of the most natural candidates is the measure assigning equal probabilities to all configurations with exactly k 1??s (and probability 0 to all other configurations). The main contribution of this paper is a proof for these measures. We also point out how our result extends to weighted versions of these measures, and to products of such measures.     

Convolutions of random measures on compact groups

LetG be a compact group andM ₁(G) be the convolution semigroup of all Borel probability measures onG with the weak topology. We consider a stationary sequence {μ _n } _n=?∞ ^+∞ of random measures μ _n =μ _n (ω) inM ₁(G) and the convolutions $$v_{m,n} (\omega ) = \mu _m (\omega )* \cdots *\mu _{n - 1} (\omega ), m< n$$ and $$\alpha _n^{( + k)} (\omega ) = \frac{1}{k}\sum\limits_{i = 1}^k {v_{n,n + i} (\omega ),} \alpha _n^{( - k)} (\omega ) = \frac{1}{k}\sum\limits_{i = 1}^k {v_{n - i,n} (\omega )} $$ We describe the setsA _m ⁺ (ω) andA _n ⁺ (ω) of all limit points ofv _m,n(ω) asm→?∞ orn→+∞ and the setA _∞(ω) of its two-sided limit points for typical realizations of {μ _n (ω)} _n=?∞ ^+∞ . Using an appropriate random ergodic theorem we study the limit random measures ρ _n ^(±) (ω)=lim _k→∞ α _n ^(±k) (ω).     

Bounded projectors and duality in the spaces of functions holomorphic in the unit ball

The paper studies the Banach spaces h _∞(φ), h ₀(φ), and h ¹(η) of harmonic functions over the unit ball in R ⁿ . These spaces depend on a weight function φ and a weight measure η. For a given function φ from a sufficiently broad class of functions, we solve the duality problem. that is, we construct measures η such that h ¹(η)* ～ h _∞(φ) and h ₀(φ)* ～ h ¹(η).     

General Hausdorff functions,and the notion of one-sided measure and dimension

The main facts about Hausdorff and packing measures and dimensions of a Borel set E are revisited, using determining set functions \(\phi_\alpha\colon\mathcal{B}_E\to(0,\infty)\), where \(\mathcal{B}_E\) is the family of all balls centred on E and α is a real parameter. With mild assumptions on φ_α, we verify that the main density results hold, as well as the basic properties of the corresponding box dimension. Given a bounded open set V in ? ^D , these notions are used to introduce the interior and exterior measures and dimensions of any Borel subset of ?V. We stress that these dimensions depend on the choice of φ_α. Two determining functions are considered, φ_α(B)=Vol _D (B∩V)diam(B)^α-D and φ_α(B)=Vol _D (B∩V)^α/D , where Vol _D denotes the D-dimensional volume.