On the action of the complex Monge-Ampère operator on piecewise linear functions

The action of the mixed complex Monge-Ampère operator (h ₁, …, h _k ) ? dd ^c h ₁ ∧ … ∧ dd ^c h _k on piecewise linear functions h _i is considered. The language of Monge-Ampère operators is used to transfer results on mixed volumes and tropical varieties to a broader context, which arises under the passage from polynomials to exponential sums. In particular, it is proved that the value of the Monge-Ampère operator depends only on the product of the functions h _i .     

Jackson-Stechkin type inequalities for special moduli of continuity and widths of function classes in the space L 2

We obtain sharp Jackson-Stechkin type inequalities for moduli of continuity of kth order Ω _k in which, instead of the shift operator T _h f, the Steklov operator S _h (f) is used. Similar smoothness characteristic of functions were studied earlier in papers of Abilov, Abilova, Kokilashvili, and others. For classes of functions defined by these characteristics, we calculate the exact values of certain n-widths.     

A highly accurate homogeneous scheme for solving the laplace equation on a rectangular parallelepiped with boundary values in C k, 1

In this paper, a homogeneous scheme with 26-point averaging operator for the solution of Dirichlet problem for Laplace??s equation on rectangular parallelepiped is analyzed. It is proved that the order of convergence is O(h ⁴), where h is the mesh step, when the boundary functions are from C ^{3, 1}, and the compatibility condition, which results from the Laplace equation, for the second order derivatives on the adjacent faces is satisfied on the edges. Futhermore, it is proved that the order of convergence is O(h ⁶(|lnh| + 1)), when the boundary functions are from C ^{5, 1}, and the compatibility condition for the fourth order derivatives is satisfied. These estimations can be used to justify different versions of domain decomposition methods.     

A new approach to numerical solution of fixed-point problems and its application to delay differential equations

In this paper we consider a certain approximation of fixed-points of a continuous operator A mapping the metric space into itself by means of finite dimensional ε(h)-fixed-points of A. These finite dimensional functions are obtained from functions defined on discrete space grid points (related to a parameter h→0) by applying suitably chosen extension operators p_h. A theorem specifying necessary and sufficient conditions for existence of fixed-points of A in terms of ε(h)-fixed-points of A is given. A corollary which follows the theorem yields an approximate method for a fixed-point problem and determines conditions for its convergence. An example of application of the obtained general results to numerical solving of boundary value problems for delay differential equations is provided.Numerical experiments carried out on three examples of boundary value problems for second order delay differential equations show that the proposed approach produces much more accurate results than many other numerical methods when applied to the same examples.     

Stochastic integral representations of second quantization operators

We give a necessary and sufficient condition for the second quantization operator Γ(h) of a bounded operator h on , or for its differential second quantization operator λ(h), to have a representation as a quantum stochastic integral. This condition is exactly that h writes as the sum of a Hilbert-Schmidt operator and a multiplication operator. We then explore several extensions of this result. We also examine the famous counterexample due to Journé and Meyer and explain its representability defect.     

On the unimprovability of full-memory strategies in the risk minimization problem

We continue the study of approximation properties of local exponential splines on a uniform grid with step h > 0 corresponding to a linear differential operator L with constant coefficients and real pairwise different roots of the characteristic polynomial (such splines were constructed by E.V. Strelkova and V.T. Shevaldin). We find order estimates as h → 0 for the error of approximation of certain Sobolev classes of functions by splines of the described type that are exact on the kernel of the operator L.     

Some multilevel methods on graded meshes

We consider Yserentant's hierarchical basis method and multilevel diagonal scaling method on a class of refined meshes used in the numerical approximation of boundary value problems on polygonal domains in the presence of singularities. We show, as in the uniform case, that the stiffness matrix of the first method has a condition number bounded by (ln(1/h))², where h is the meshsize of the triangulation. For the second method, we show that the condition number of the iteration operator is bounded by ln(1/h), which is worse than in the uniform case but better than the hierarchical basis method. As usual, we deduce that the condition number of the BPX iteration operator is bounded by ln(1/h). Finally, graded meshes fulfilling the general conditions are presented and numerical tests are given which confirm the theoretical bounds.     

Spectral gaps for periodic Schrödinger operators with hypersurface magnetic wells: Analysis near the bottom

We consider a periodic magnetic Schrödinger operator Hh, depending on the semiclassical parameter h>0, on a noncompact Riemannian manifold M such that H¹(M,R)=0 endowed with a properly discontinuous cocompact isometric action of a discrete group. We assume that there is no electric field and that the magnetic field has a periodic set of compact magnetic wells. We suppose that the magnetic field vanishes regularly on a hypersurface S. First, we prove upper and lower estimates for the bottom λ₀(Hh) of the spectrum of the operator Hh in L²(M). Then, assuming the existence of non-degenerate miniwells for the reduced spectral problem on S, we prove the existence of an arbitrarily large number of spectral gaps for the operator Hh in the region close to λ₀(Hh), as h→0. In this case, we also obtain upper estimates for the eigenvalues of the one-well problem.     

Properties of delta functions of a class of observables on white noise functionals

Let δa be the Dirac delta function at a∈R and (E)⊂(L²)⊂^∗(E) the canonical framework of white noise analysis over white noise space (E^∗,μ), where E^∗=S^∗(R). For h∈H=L²(R) with h≠0, denote by Mh the operator of multiplication by Wh=〈⋅,h〉 in (L²). In this paper, we first show that Mh is δa-composable. Thus the delta function δa(Mh) makes sense as a generalized operator, i.e. a continuous linear operator from (E) to ^∗(E). We then establish a formula showing an intimate connection between δa(Mh) as a generalized operator and δa(Wh) as a generalized functional. We also obtain the representation of δa(Mh) as a series of integral kernel operators. Finally we prove that δa(Mh) depends continuously on a∈R.     

Another homogeneous q-difference operator

In this paper we show the equivalence between Goldman-Rota q-binomial identity and its inverse. We may specialize the value of the parameters in the generating functions of Rogers-Szegö polynomials to obtain some classical results such as Euler identities and the relation between classical and homogeneous Rogers-Szegö polynomials. We give a new formula for the homogeneous Rogers-Szegö polynomials h_n(x,y|q). We introduce a q-difference operator θ_xy on functions in two variables which turn out to be suitable for dealing with the homogeneous form of the q-binomial identity. By using this operator, we got the identity obtained by Chen et al. [W.Y.C. Chen, A.M. Fu, B. Zhang, The homogeneous q-difference operator, Advances in Applied Mathematics 31 (2003) 659-668, Eq. (2.10)] which they used it to derive many important identities. We also obtain the q-Leibniz formula for this operator. Finally, we introduce a new polynomials s_n(x,y;b|q) and derive their generating function by using the new homogeneous q-shift operator L(bθ_xy).     

Quantum monodromy and semi-classical trace formulæ

We consider an h-pseudodifferential operator whose symbol has a closed Hamiltonian trajectory. There exists a Fourier integral operator which quantizes in a natural way the Poincaré map. With the help of this monodromy operator, we give a trace formula which leads to a new proof of the trace formula of Duistermaat–Guillemin and Gutzwiller.     

On Fourier Series in Generalized Eigenfunctions of a Discrete Sturm–Liouville Operator

For semicontinuous summation methods generated by Λ = {λ_n(h)} (n = 0, 1, 2,...; h > 0) of Fourier series in eigenfunctions of a discrete Sturm–Liouville operator of class B, some results on the uniform a.e. behavior of Λ-means are obtained. The results are based on strong- and weak-type estimates of maximal functions. As a consequence, some statements on the behavior of the summation methods generated by the exponential means λ_n(h) = exp(?u^α(n)h) are obtained. An application to a generalized heat equation is given.     

Multiresolution expansion, approximation order and quasiasymptotic behavior of tempered distributions

Multiresolution analysis of tempered distributions is studied through multiresolution analysis on the corresponding test function spaces Sr(R), r∈N₀. For a function h, which is smooth enough and of appropriate decay, it is shown that the derivatives of its projections to the corresponding spaces Vj, j∈Z, in a regular multiresolution analysis of L²(R), denoted by hj, multiplied by a polynomial weight converge in sup norm, i.e., hj→h in Sr(R) as j→∞. Analogous result for tempered distributions is obtained by duality arguments. The analysis of the approximation order of the projection operator within the framework of the theory of shift-invariant spaces gives a further refinement of the results. The order of approximation is measured with respect to the corresponding space of test functions. As an application, we give Abelian and Tauberian type theorems concerning the quasiasymptotic behavior of a tempered distribution at infinity.     

A posteriori error estimates for hp finite element solutions of convex optimal control problems

In this paper, we present a posteriori error analysis for hp finite element approximation of convex optimal control problems. We derive a new quasi-interpolation operator of Clément type and a new quasi-interpolation operator of Scott-Zhang type that preserves homogeneous boundary condition. The Scott-Zhang type quasi-interpolation is suitable for an application in bounding the errors in L²-norm. Then hp a posteriori error estimators are obtained for the coupled state and control approximations. Such estimators can be used to construct reliable adaptive finite elements for the control problems.     

On h-convexity

We introduce a class of h-convex functions which generalize convex, s-convex, Godunova-Levin functions and P-functions. Namely, the h-convex function is defined as a non-negative function which satisfies f(αx+(1−α)y)?h(α)f(x)+h(1−α)f(y), where h is a non-negative function, α∈(0,1) and x,y∈J. Some properties of h-convex functions are discussed. Also, the Schur-type inequality is given.     

Duality for max-separable problems

In this paper we propose a general duality theory for a class of so called ‘max-separable’ optimization problems. In such problems functions h:R ^k → R of the form h(x ₁, . . . , x _k ) =? max _j ? h _j (x _j ), occur both as objective functions and as constraint functions (h _j are assumed to be strictly increasing functions of one variable). As a result we obtain pairs of max-separable optimization problems, which possess both weak and strong duality property without a duality gap.     

Keller-Osserman type conditions for differential inequalities with gradient terms on the Heisenberg group

We study the qualitative behavior of non-negative entire solutions of differential inequalities with gradient terms on the Heisenberg group. We focus on two classes of inequalities: Δφu?f(u)l(|∇u|) and Δφu?f(u)−h(u)g(|∇u|), where f, l, h, g are non-negative continuous functions satisfying certain monotonicity properties. The operator Δφ, called the φ-Laplacian, generalizes the p-Laplace operator considered by various authors in this setting. We prove some Liouville theorems introducing two new Keller-Osserman type conditions, both extending the classical one which appeared long ago in the study of the prototype differential inequality Δu?f(u) in Rm. We show sharpness of our conditions when we specialize to the p-Laplacian. While proving these results we obtain a strong maximum principle for Δφ which, to the best of our knowledge, seems to be new. Our results continue to hold, with the obvious minor modifications, also for Euclidean space.     

Spectral shift function and resonances for slowly varying perturbations of periodic Schrödinger operators

We study the spectral shift function s(λ,h) and the resonances of the operator P(h)=-Δ+V(x)+W(hx). Here V is a periodic potential, W a decreasing perturbation and h a small positive constant. We give a representation of the derivative of s(λ,h) related to the resonances of P(h), and we obtain a Weyl-type asymptotics of s(λ,h). We establish an upper bound O(h^-n+1) for the number of the resonances of P(h) lying in a disk of radius h.     

Microlocal kernel of pseudodifferential operators at a hyperbolic fixed point

We study the microlocal kernel of h-pseudodifferential operators Oph(p)−z, where z belongs to some neighborhood of size O(h) of a critical value of its principal symbol p₀(x,ξ). We suppose that this critical value corresponds to a hyperbolic fixed point of the Hamiltonian flow Hp₀. First we describe propagation of singularities at such a hyperbolic fixed point, both in the analytic and in the C^∞ category. In both cases, we show that the null solution is the only element of this microlocal kernel which vanishes on the stable incoming manifold, but for energies z in some discrete set. For energies z out of this set, we build the element of the microlocal kernel with given data on the incoming manifold. We describe completely the operator which associate the value of this null solution on the outgoing manifold to the initial data on the incoming one. In particular it appears to be a semiclassical Fourier integral operator associated to some natural canonical relation.     

High accuracy analysis of the lowest order <Emphasis Type="Italic">H</Emphasis><Superscript>1</Superscript>-Galerkin mixed finite element method for nonlinear sine-Gordon equations

The lowest order H¹-Galerkin mixed finite element method (for short MFEM) is proposed for a class of nonlinear sine-Gordon equations with the simplest bilinear rectangular element and zero order Raviart-Thomas element. Base on the interpolation operator instead of the traditional Ritz projection operator which is an indispensable tool in the traditional FEM analysis, together with mean-value technique and high accuracy analysis, the superclose properties of order O(h²)/O(h² + τ²) in H¹-norm and H(div;Ω)-norm are deduced for the semi-discrete and the fully-discrete schemes, where h, τ denote the mesh size and the time step, respectively, which improve the results in the previous literature.