Best linear approximation methods for functions of Taikov classes in the Hardy spaces H q,ρ , q ≥ 1, 0 < ρ ≤ 1

For the Hardy spaces H _q,ρ , q ≥ 1, 0 < ρ ≤ 1, we develop best linear approximation methods for classes of analytic functions W ^r H _q Φ, r ∈ ?, in the unit disk (studied by L. V. Taikov) whose averaged second-order moduli of continuity of the angular boundary values of the rth derivatives are majorized by a given function ? satisfying certain constraints.     

Multiobjective programming with new invexities

(Φ, ρ)-invexity and (Φ, ρ) ^w -invexity generalize known invexity type properties and have been introduced with the intent of extending most of theoretical results in mathematical programming. Here, we push this approach further, to obtain authentic extensions of previously known optimality and duality results in multiobjective programming.     

Inversions of Infinitely Divisible Distributions and Conjugates of Stochastic Integral Mappings

The dual of an infinitely divisible distribution on ? ^d without Gaussian part defined in Sato (ALEA Lat. Am. J. Probab. Math. Statist. 3:67–110, 2007) is renamed to the inversion. Properties and characterization of the inversion are given. A stochastic integral mapping is a mapping μ=Φ _f ? ρ of ρ to μ in the class of infinitely divisible distributions on ? ^d , where μ is the distribution of an improper stochastic integral of a nonrandom function f with respect to a Lévy process on ? ^d with distribution ρ at time 1. The concept of the conjugate is introduced for a class of stochastic integral mappings and its close connection with the inversion is shown. The domains and ranges of the conjugates of three two-parameter families of stochastic integral mappings are described. Applications to the study of the limits of the ranges of iterations of stochastic integral mappings are made.     

On the coefficients of ternary cyclotomic polynomials

We study coefficients of ternary cyclotomic polynomials Φ_pqr(z)=∏_ρ(z−ρ), where p, q, and r are distinct odd primes and the product is taken over all primitive pqrth roots of unity ρ.     

Geometry and entropy of generalized rotation sets

For a continuous map f on a compact metric space we study the geometry and entropy of the generalized rotation set Rot(Φ). Here Φ = (?₁, ..., ? _m ) is a m-dimensional continuous potential and Rot(Φ) is the set of all µ-integrals of Φ and µ runs over all f-invariant probability measures. It is easy to see that the rotation set is a compact and convex subset of ? ^m . We study the question if every compact and convex set is attained as a rotation set of a particular set of potentials within a particular class of dynamical systems. We give a positive answer in the case of subshifts of finite type by constructing for every compact and convex set K in ? ^m a potential Φ = Φ(K) with Rot(Φ) = K. Next, we study the relation between Rot(Φ) and the set of all statistical limits Rot _Pt (Φ). We show that in general these sets differ but also provide criteria that guarantee Rot(Φ) = Rot _Pt (Φ). Finally, we study the entropy function w ? H(w),w ∈ Rot(Φ). We establish a variational principle for the entropy function and show that for certain non-uniformly hyperbolic systems H(w) is determined by the growth rate of those hyperbolic periodic orbits whose Φ-integrals are close to w. We also show that for systems with strong thermodynamic properties (sub-shifts of finite type, hyperbolic systems and expansive homeomorphisms with specification, etc.) the entropy function w ? H(w) is real-analytic in the interior of the rotation set.     

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.     

On smooth sets of integers

This work studies evenly distributed sets of integers—sets whose quantity within each interval is proportional to the size of the interval, up to a bounded additive deviation. Namely, for ρ,Δ∈R a set A of integers is (ρ,Δ)- smooth if for any interval I of integers; a set A is Δ-smooth if it is (ρ,Δ)-smooth for some real number ρ. The paper introduces the concept of Δ-smooth sets and studies their mathematical structure. It focuses on tools for constructing smooth sets having certain desirable properties and, in particular, on mathematical operations on these sets. Three additional papers by us are build on the work of this paper and present practical applications of smooth sets to common and well-studied scheduling problems.One of the above mathematical operations is composition of sets of natural numbers. For two infinite sets A,B⊆N, the composition of A and B is the subset D of A such that, for all i, the ith member of A is in D if and only if the ith member of N is in B. This operator enables the partition of a (ρ,Δ)-smooth set into two sets that are (ρ₁,Δ)-smooth and (ρ₂,Δ)-smooth, for any ρ₁,ρ₂ and Δ obeying some reasonable restrictions. Another powerful tool for constructing smooth sets is a one-to-one partial function f from the unit interval into the natural numbers having the property that any real interval X⊆[0,1) has a subinterval Y which is ‘very close’ to X s.t. f(Y) is (ρ,Δ)-smooth, where ρ is the length of Y and Δ is a small constant.     

Mean approximation of functions on the real axis by algebraic polynomials with Chebyshev-Hermite weight and widths of function classes

We obtain sharp Jackson-Stechkin type inequalities on the sets L _2,ρ ^r (?) in which the values of best polynomial approximations are estimated from above via both the moduli of continuity of mth order and K-functionals of rth derivatives. For function classes defined by these characteristics, the exact values of various widths are calculated in the space L _2,ρ (?). Also, for the classes \(W_{2,\rho }^r (\mathbb{K}_m ,\Psi )\) , where r = 2, 3, h3, the exact values of the best polynomial approximations of the intermediate derivatives f ^(ν), ν = 1,..., r ? 1, are obtained in L _2,ρ (?).     

Characterization of Wishart-Laplace distributions via Jordan algebra homomorphisms

For a real, Hermitian, or quaternion normal random matrix Y with mean zero, necessary and sufficient conditions for a quadratic form Q(Y) to have a Wishart-Laplace distribution (the distribution of the difference of two independent central Wishart W_p(m_i,Σ) random matrices) are given in terms of a certain Jordan algebra homomorphism ρ. Further, it is shown that {Q_k(Y)} is independent Laplace-Wishart if and only if in addition to the aforementioned conditions, the images ρ_k(Σ⁺) of the Moore-Penrose inverse Σ⁺ of Σ are mutually orthogonal: ρ_k(Σ⁺)ρ_?(Σ⁺)=0 for k≠?.     

Gamma series associated to elements satisfying regularized double shuffle relations

Let K be a field of characteristic 0, and R be a commutative K-algebra. Let Φ(x₀,x₁) be an element in R《x₀,x₁》 with regularized double shuffle relations. We define a gamma series ΓΦ(s)∈1+s²R?s? associated to Φ. We prove that the associated beta series is just the image of ΦY(x₀,x₁) in the commutative formal power series ring R?x₀,x₁?, where if Φ=1+Φ₀x₀+Φ₁x₁, then ΦY=1+Φ₁x₁. We also give some equivalent conditions for the reflection formula of the gamma series ΓΦ(s).     

Powers of a matrix and a formula for the moduli of its eigenvalues

The spectral radius of a complex square matrix A is given by ρ(A) = lim sup_{k → ∞} (TrA^k)^1/k. A more general result is proved which gives information about the moduli of all eigenvalues of A.     

Multiplication operators on vector measure Orlicz spaces

Let m be a countably additive vector measure with values in a real Banach space X, and let L¹(m) and L_w(m) be the spaces of functions which are, correspondingly, integrable and weakly integrable with respect to m. Given a Young's function Φ, we consider the vector measure Orlicz spaces L^Φ(m) and L^Φ_w(m) and establish that the Banach space of multiplication operators going from W = L^Φ(m) into Y = L¹ (m) is M = L^Ψ_w (m) with an equivalent norm; here Ψ is the conjugated Young's function for Φ. We also prove that when W = L^Φ_w(m), Y = L¹(m) we have M = L^Ψ_w (m), and when W = L^Φ_w(m), Y = L¹(m) we have M = L^Ψ (m).     

Point processes in general position

The aim of this note is to introduce for point processes in ? ^d the notions general position and reinforced general position, and to characterize these processes. As a consequence we show that Poisson processes P _ρ with an infinite intensity measures ρ are in general position iff ρ is diffuse in the sense that any affine subspace of dimension d ? 1 is a ρ-nullset. Moreover, P _ρ is in reinforced general position iff in addition any (d ? 1)-sphere is a ρ-nullset.     

Estimates of solutions of a general initial-boundary-value problem for the linear nonstationary system of Navier-Stokes equations in a half-space

In the half-spaceX ₃>0 one considers the initial-boundary-value problem for the Stokes system in which the boundary conditions are given by means of an arbitrary matricial differential operator of size 3×4. One proves that, under certain restrictions on this operator, the solution satisfies coercive estimates in the norms of W_ρ ^2?,?.     

Complexity Estimates for Representations of Schmüdgen Type

Let Φ={φ₁, …, φ_j} and let K be a closed basic set in Rⁿ given by the polynomial inequalities φ₁⩾0, …, φ_j⩾0. Let Σ{Φ} be the semiring generated by the φ_k and the squares in R[x₁, …, x_n]. For example, if Φ={φ₁} then Σ{Φ}=σ₁+σ₂φ₁, where σ₁, σ₂ are sums of squares of polynomials. Schmüdgen has shown that if K is compact then any polynomial strictly positive on K belongs to Σ{Φ}. This paper develops a result of Schmüdgen type for functions in one dimension merely nonnegative on K. For this, it is necessary to add additional hypotheses, such as the proximity of complex zeros, to compensate for the loss of strict positivity necessary for Schmüdgen's result.     

Nonsmooth minimax programming under locally Lipschitz (Φ, ρ)-invexity

Minimax programming problems involving locally Lipschitz (Φ, ρ)-invex functions are considered. The parametric and non-parametric necessary and sufficient optimality conditions for a class of nonsmooth minimax programming problems are obtained under nondifferentiable (Φ, ρ)-invexity assumption imposed on objective and constraint functions. When the sufficient conditions are utilized, parametric and non-parametric dual problems in the sense of Mond-Weir and Wolfe may be formulated and duality results are derived for the considered nonsmooth minimax programming problem. With the reference to the said functions we extend some results of optimality and duality for a larger class of nonsmooth minimax programming problems.     

Refined distributional approximations for the uncovered set in the Johnson–Mehl model

Let Φ^z be the uncovered set (i.e., the complement of the union of intervals) at time z in the one-dimensional Johnson–Mehl model. We derive a bound for the total variation distance between the distribution of the number of components of Φ^z∩(0,t] and a compound Poisson-geometric distribution, which is sharper and simpler than an earlier bound obtained by Erhardsson. We also derive a previously unavailable bound for the total variation distance between the distribution of the Lebesgue measure of Φ^z∩(0,t] and a compound Poisson-exponential distribution. Both bounds are O(z_β(t)/t) as t→∞, where z_β(t) is defined so that the expected number of components of Φ^z_β(t)∩(0,t] converges to β>0 as t→∞, and the parameters of the approximating distributions are explicitly calculated.     

Primitive stable representations of free Kleinian groups

In this paper, we give a complete criterion for a discrete faithful representation ρ: F _n →PSL(2, ?) to be primitive stable. This will answer Minsky’s conjectures about geometric conditions on ?³/ρ(F _n ) regarding the primitive stability of ρ.     

Inversion of Toeplitz operators,Levinson equations,and Gohberg-Krein factorization—A simple and unified approach for the rational case

The problem of the inversion of the Toeplitz operator T_Φ, associated with the operator-valued function Φ defined on the unit circle, is known to involve the associated Levinson system of equations and the Gohberg-Krein factorization of Φ. A simplified and self-contained approach, making clear the connections between these three problems, is presented in the case where Φ is matrix-valued and rational. The key idea consists in looking at the Levinson system of equations associated with Φ^?1(z^?1), rather than that associated with Φ(z). As a consequence, a new invertibility criterion for Toeplitz operators with rational matrix-valued symbols is derived.