A finite element method scheme for boundary value problems with noncoordinated degeneration of input data

A finite element method scheme is constructed for boundary value problems with noncoordinated degeneration of input data and singularity of a solution. We look at a rate with which an approximate solution by the proposed finite element method converges toward an exact R _ν -generalized solution in the weight set W _2,ν*+β ^2+1/1 (Ω, δ), and establish estimates for the finite element approximation.     

W-Sobolev spaces

Fix strictly increasing right continuous functions with left limits and periodic increments, Wi:R→R, i=1,…,d, and let for x∈Rd. We construct the W-Sobolev spaces, which consist of functions f having weak generalized gradients ∇Wf=(W_1∂f,…,Wd_∂f). Several properties, that are analogous to classical results on Sobolev spaces, are obtained. Existence and uniqueness results for W-generalized elliptic equations, and uniqueness results for W-generalized parabolic equations are also established. Finally, an application of this theory to stochastic homogenization is presented.     

Explicit solution of the inverse kinematic problem in the non-Herglotz case

The inverse kinematic problem is solved in the half space R ₊ ^ν+1 ={(x,z)|z?0,x∈R^ν, ν?1 under the assumption that the index of refraction can be represented in the form $$n^2 (x,z) = K^2 (z) + \sum\limits_{j = 1}^\nu {\Phi _j^2 (x_j ),} n_z< 0.$$ . The solution obtained is a generalization of the Herglotz-Wiechert formula. A formula is presented for the solution of the inverse kinematic problem in the general case of separation of variables in the eikonal equation.     

Optimality conditions for a class of composite multiobjective nonsmooth optimization problems

In this paper, a class of composite multiobjective nonsmooth optimization problems with cone constraints is considered. Necessary optimality conditions for weak minimum are established in terms of Semi-infinite Gordan type theorem. η-generalized null space condition, which is a proper generalization of generalized null space condition, is proposed. Sufficient optimality conditions are obtained for weak minimum, Pareto minimum, Benson’s proper minimum under K-generalized invexity and η-generalized null space condition. Some examples are given to illustrate our main results.     

Ramsey graphs and block designs

Symmetric (ν, κ, λ)-block designs admitting polarity maps are shown to be closely related to certain Ramsey numbers for bipartite graphs. In particular, if there exists a (ν, κ, λ)-difference set in an abelian group of order ν, then the Ramsey number R(K_2,λ+1, K_1,ν?k+1) is either 1 + ν or 2 + ν.     

Maurey-Rosenthal factorization of positive operators and convexity

We show that a Banach lattice X is r-convex, 1<r<∞, if and only if all positive operators T on X with values in some r-concave Köthe function spaces F(ν) (over measure spaces (Ω,ν)) factorize strongly through L_r(ν) (i.e., T=M_gR, where R is an operator from X to L_r(ν) and M_g a multiplication operator on L_r(ν) with values in F). This characterization of r-convexity motivates a Maurey-Rosenthal type factorization theory for positive operators acting between vector valued Köthe function spaces.     

Dirichlet problem with degeneration of the input data on the boundary of the domain

We define an R_?-generalized solution of the first boundary value problem for a second-order elliptic equation with degeneration of the input data on the entire boundary of the two-dimensional domain and prove the existence and uniqueness of the solution in the weighted set Open image in new window.     

Wavelet characterization of growth spaces of harmonic functions

We consider the space h _ν ^∞ of harmonic functions in R ₊ ⁿ⁺¹ with finite norm ‖u‖ _ν = sup |u(x, t)|/v(t), where the weight ν satisfies the doubling condition. Boundary values of functions in h _ν ^∞ are characterized in terms of their smooth multiresolution approximations. The characterization yields the isomorphism of Banach spaces h _ν ^∞ ～ l ^∞. The results are also applied to obtain the law of the iterated logarithm for the oscillation of functions in h _ν ^∞ along vertical lines.     

Hájek-Šidák approach to the asymptotic distribution of multivariate rank order statistics

Let X_α = (X_1α,…, X_pα), 1 ≤ α ≤ N_ν, ν ≥ 1 be N_ν independent observations from a density function f(x) where x ∈ R^p, the p-dimensional real space. Let R_νjα denote the rank of X_jα in the ordered array of X_j1 ,…, X_jNν; 1 ≤ j ≤ p and consider the multivariate rank order statistics

$\text{T}{}_{\mathrm{vj}}\text{=}\text{∑}\text{α = 1}\text{N}{}_{\mathrm{v}}\text{C}{}_{\mathrm{y\alpha }}\text{a}{}_{\mathrm{vj}}\text{(R}{}_{\mathrm{vj\alpha }}\text{),}$

where the constants, c_να, 1 ≤ α ≤ N_ν satisfy the Noether condition and the scores, a_νj(α), 1 ≤ j ≤ p, 1 ≤ α ≤ N_ν converge as ν → ∞, for each j, in quadratic mean to a nonconstant, square integrable function π_j(u), 0 < u < 1. Under the hypothesis of randomness, the joint asymptotic conditional and uncoditional normality of the statistics T_νj, 1 ≤ j ≤ p is established. Further, under mild conditions on the underlying density functions and assuming contiguous location shift alternatives, the joint asymptotic normality of these statistics is also established.     

Convergent expansions of the Bessel functions in terms of elementary functions

We consider the Bessel functions J _ν (z) and Y _ν (z) for R ν > ?1/2 and R z ≥ 0. We derive a convergent expansion of J _ν (z) in terms of the derivatives of \((\sin z)/z\), and a convergent expansion of Y _ν (z) in terms of derivatives of \((1-\cos z)/z\), derivatives of (1 ? e ^?z )/z and Γ(2ν, z). Both expansions hold uniformly in z in any fixed horizontal strip and are accompanied by error bounds. The accuracy of the approximations is illustrated with some numerical experiments.     

The block grade of a block Krylov space

The aim of the paper is to compile and compare basic theoretical facts on Krylov subspaces and block Krylov subspaces. Many Krylov (sub)space methods for solving a linear system Ax=b have the property that in exact computer arithmetic the true solution is found after ν iterations, where ν is the dimension of the largest Krylov subspace generated by A from r₀, the residual of the initial approximation x₀. This dimension is called the grade of r₀ with respect to A. Though the structure of block Krylov subspaces is more complicated than that of ordinary Krylov subspaces, we introduce here a block grade for which an analogous statement holds when block Krylov space methods are applied to linear systems with multiple, say s, right-hand sides. In this case, the s initial residuals are bundled into a matrix R₀ with s columns. The possibility of linear dependence among columns of the block Krylov matrix , which in practical algorithms calls for the deletion (or, deflation) of some columns, requires extra care. Relations between grade and block grade are also established, as well as relations to the corresponding notions of a minimal polynomial and its companion matrix.     

Directional K- Functionals Claudia Cottin

When considering approximation of continuous periodic functions f: R ^d → R by blending-type approximants which depend on directions ξ₁,…,ξ_ν ∈ R ^d directional moduli of smoothness (1) are appropriate measures of smoothness of /. In this paper, we introduce equivalent directional K- functionals. As an application, we obtain a result on the degree of approximation by certain trigonometric blending functions.     

Differential polynomial rings which are generalized Asano prime rings

Let R[x; δ] be a differential polynomial ring over a prime Goldie ring R in an indeterminate x, where δ is a derivation of R. In this paper, we describe explicitly the group of δ-stable v-R-ideals and using this results, we show that R[x; δ] is a generalized Asano prime ring if and only if R is a δ-generalized Asano prime ring.     

Equazioni monodimensionali di un gas viscoso barotropico con una perturbazione poco regolare

The equations for a barotropic viscous gas in one space dimensiondν=(μ(?ν_ε)_ε?p _ε)dt+dG,? _t +?²ν_ε=0,p=?^γ with a perturbationdG are considered under the assumption thatG is only a function of bounded variation inL ²(Θ) orH ₀ ¹ (Θ) (Θ=]0, α[) and the esistence and the uniqueness of the global solution in a class of solutions of «strong type» as well as in a class of solutions of «weak type» are proved. This result constitutes a generalization of the result of Kazhikhov [8] and that of Shelukhin [10] and contains preliminary considerations for the corrisponding stochastic equations.     

On some combinatorial questions in finite-dimensional spaces

Bounds are given for supinf∥Σ^p_i=1ν_i∥, where sup is taken over all set systems V₁,…, V_p of Rⁿ with 0∈∩^p_i=1convV_i and sup_v∈Vi∥ν∥?1 for i=1,…, p, and inf is taken over all possible choices ν_i∈V_i for i=1,…, p. Another similar problem is considered. The bounds are sharp.     

The matrix geometric mean of parameterized, weighted arithmetic and harmonic means

We define a new family of matrix means {L_μ(ω;A)}_μ∈R where ω and A vary over all positive probability vectors in R^m and m-tuples of positive definite matrices resp. Each of these means interpolates between the weighted harmonic mean (μ=-∞) and the arithmetic mean of the same weight (μ=∞) with L_μ≤L_ν for μ≤ν. Each has a variational characterization as the unique minimizer of the weighted sum for the symmetrized, parameterized Kullback-Leibler divergence. Furthermore, each can be realized as the common limit of the mean iteration by arithmetic and harmonic means (in the unparameterized case), or, more generally, the arithmetic and resolvent means. Other basic typical properties for a multivariable mean are derived.     

Fully discrete finite element scheme for Maxwell's equations with non-linear boundary condition

We study a full Maxwell's system accompanied with a non-linear degenerate boundary condition, which represents a generalization of the classical Silver-Müller condition for a non-perfect conductor. The relationship between the normal components of electric E and magnetic H field obeys the following power law ν×H=ν×(|E×ν|^α−1E×ν) for some α∈(0,1]. We establish the existence and uniqueness of a weak solution in a suitable function spaces under the minimal regularity assumptions on the boundary Γ and the initial data E₀ and H₀. We design a non-linear time discrete approximation scheme and prove convergence of the approximations to a weak solution. We also derive the error estimates for the time discretization. As a next step we study the fully discrete problem using curl-conforming edge elements and derive the corresponding error estimates. Finally we present some numerical experiments.     

Bounds for the first positive zero of a mixed Bessel function

The present article is concerned with lower and upper bounds of the first positive zero of the function H_ν(z, α) = αJ_ν(z) + zJ′_ν(z), Where J_gn(z) is the ordinary Bessel function of order ν > −1 and J′_ν(z) is the derivative of J_ν(z). A lower bound found here improves and extends the range of validity of the order ν, of a lower bound found in a previous work [8]. Also, two upper bounds given here improve a previously known upper bound [8]. In the particular case α = 0, these bounds lead to lower and upper bounds for the first positive zero j′_ν,1 of J′_ν(z) which improve well-known bounds in the literature.