The embedding and approximation for classes of functions with a dominant mixed difference

We obtain a criterion for embedding the class SH _p ^Ω into that SB _q,? ^Ω* (1 < p ≤ q < ∞). We also determine the exact order of the best approximations of functions from classes SB _p,? ^Ω by trigonometric polynomials whose harmonics belong to sets generated by level surfaces of the majorant Λ (t).     

Embedding theorems of different metrics for classes of functions with dominant mixed modulus of smoothness

Embedding theorems in a mixed norm are proved in the paper for classes of functions with a dominant mixed modulus of smoothness being extensions of well-known classes of Nikol’skii.     

Traces of functions with majorizable derivatives

We study the limiting values (y→+0) of functionsf (x, y), x ε R_n, y > 0, for which ¦?f/?y¦≤M?(y), ¦?f/?x_k¦≤Mψk(y), M=M [f], in the case of arbitrary weight functions. It is shown that the space of traces can be described as the set of all functionsf (x, 0) which satisfy a Lipschitz condition in some metricω(x, x) associated with the weights.     

Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains

For functions from the Sobolev space Hs(Ω), , definitions of non-unique generalized and unique canonical co-normal derivative are considered, which are related to possible extensions of a partial differential operator and its right-hand side from the domain Ω, where they are prescribed, to the domain boundary, where they are not. Revision of the boundary value problem settings, which makes them insensitive to the generalized co-normal derivative inherent non-uniqueness are given. It is shown, that the canonical co-normal derivatives, although defined on a more narrow function class than the generalized ones, are continuous extensions of the classical co-normal derivatives. Some new results about trace operator estimates and Sobolev spaces characterizations, are also presented.     

Chern classes of bundles on blown-up surfaces

Consider the blow upπ [Xtilde]→ Xof a complex surface Xat a point. Let [Etilde] be a holomorphic bundle on [Xtilde]whose restriction to the exceptional divisor is O(j)n ? O(-j), and define E+(π_* E*)^vv. Friedman and Morgan gave the following bounds for the second Chern classes j≤ c ₂([Etilde])- c ₂(E) ≤ j ²We show that these bounds are sharp.     

Flux-splitting schemes for parabolic equations with mixed derivatives

Difference schemes of required quality are often difficult to construct as applied to boundary value problems for parabolic equations with mixed derivatives. Specifically, difficulties arise in the design of monotone difference schemes and unconditionally stable locally one-dimensional splitting schemes. In parabolic problems, certain opportunities are offered by restating the problem in question so that the quantities to be determined are fluxes (directional derivatives). The original problem is then rewritten as a boundary value one for a system of equations in flux variables. Weighted schemes for parabolic equations in flux coordinates are examined. Unconditionally stable locally one-dimensional flux schemes that are first- and second-order accurate in time are constructed for a parabolic equation without mixed derivatives. A feature of systems in flux variables for equations with mixed derivatives is that the terms with time derivatives are coupled with each other.     

Nonlinear equations with mixed derivatives

The nonlinear hyperbolic equation ∂²u(x, y)/∂x ∂y + g(x, y)f(u(x, y)) = 0 with u(x, 0) = φ(x) and u(0, y) = Ψ(y), considered by [1.], 31–45) under appropriate smoothness conditions, is solvable by the author's decomposition method (“Stochastic Systems,” Academic Press, 1983 and “Nonlinear Stochastic Operator Equations,” Academic Press, 1986).     

Optimal quadrature formulas for classes of functions with bounded mixed difference

Translated from Matematicheskie Zametki, Vol. 49, No. 1, pp. 149–151, January, 1991.     

Sharp upper bounds of norms of functions and their derivatives on classes of functions with given comparison function

For arbitrary [α, β] ⊂ R and p > 0, we solve the extremal problem

Optimal recovery on the classes of functions with bounded mixed derivative

Temlyakov considered the optimal recovery on the classes of functions with bounded mixed derivative in the Lp metrics and gave the upper estimates of the optimal recovery errors. In this paper, we determine the asymptotic orders of the optimal recovery in Sobolev spaces by standard information, i.e., function values, and give the nearly optimal algorithms which attain the asymptotic orders of the optimal recovery.     

Approximation on anisotropic Besov classes with mixed norms by standard information

This article considers the approximation problem on periodic functions of anisotropic Besov classes with mixed norms using standard information. The asymptotic decay rates of the best algorithms in the worst-case setting are determined. An interpolating algorithm that attains this decay rate is given as well.     

Chern classes of reflexive sheaves on normal surfaces

We define Chern classes of reflexive sheaves using Wahl's relative local Chern classes of vector bundles. The main result of the paper bounds contributions of singularities of a sheaf to the Riemann–Roch formula. Using it we are able to prove inequality in Wahl's conjecture on relative asymptotic RR formula for rank 2 vector bundles. Moreover, we prove that if Wahl's conjecture is true for a singularity then it is true for any its quotient. This implies Wahl's conjecture for quotient singularities and for quotients of cones over elliptic curves. Received March 2, 1998; in final form March 24, 1999 / Published online September 14, 2000     

Power function on stationary classes

We show that under certain large cardinal requirements there is a generic extension in which the power function behaves differently on different stationary classes. We achieve this by doing an Easton support iteration of the Radin on extenders forcing.     

Quadrature formulas for classes of functions with bounded mixed derivative or difference

Quadrature formulas are considered for classes of smooth functions W _p ^r , B _p ^r , θ with bounded mixed derivative or difference. For the classes of functions indicated above, the result that quadrature formulas constructed with the help of number-theoretic methods are optimal (in the sense of order) is proved, and the optimal order of the error estimates is obtained. Project supported by the National Natural Science Foundation of China and the Doctoral Program Foundation of the State Education Commission of China.