We develop a general L–framework for deriving continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations with the aid of the Chen-Perthame kinetic approach [9]. We apply our L–framework to establish an explicit estimate for continuous dependence on the nonlinearities and an optimal error estimate for the vanishing anisotropic viscosity method, without t...

We develop a general L1–framework for deriving continuous dependence and error estimates for quasilinear anisotropic degenerate parabolic equations with the aid of the Chen-Perthame kinetic approach [9]. We apply our L1–framework to establish an explicit estimate for continuous dependence on the nonlinearities and an optimal error estimate for the vanishing anisotropic viscosity method, without...

We analyze the problem of finding a point strictly interior to a bounded, fully dimensional set from a finite dimensional Hilbert space. We generalize the results obtained for the LP, SDP and SOCP cases. The cuts added by our algorithm are central and conic. In our analysis, we find an upper bound for the number of Newton steps required to compute an approximate analytic center. Also, we provid...

The strongly bounded Turing reducibilities r = cl (computable Lipschitz reducibility) and r = ibT (identity bounded Turing reducibility) are defined in terms of Turing reductions where the use function is bounded by the identity function up to an additive constant and the identity function, respectively. We show that, for r = ibT, cl, every computably enumerable (c.e.) r-degree a > 0 has the an...

In this paper we prove the Pohozaev identity for the semilinear Dirichlet problem (−∆)u = f(u) in Ω, u ≡ 0 in R\Ω. Here, s ∈ (0, 1), (−∆) is the fractional Laplacian in R, and Ω is a bounded C domain. To establish the identity we use, among other things, that if u is a bounded solution then u/δ|Ω is C up to the boundary ∂Ω, where δ(x) = dist(x, ∂Ω). In the fractional Pohozaev identity, the func...

Let A be a Banach algebra, not necessarily unital, and let B be a closed subalgebra of A. We establish a connection between the Banach cyclic cohomology group HC(A) of A and the Banach B-relative cyclic cohomology group HCnB(A) of A. We prove that, for a Banach algebra A with a bounded approximate identity and an amenable closed subalgebra B of A, up to topological isomorphism, HC(A) = HCnB(A) ...

A homogeneous boundary condition is constructed for the parabolic equation (∂t + I − Δ)u = f in an arbitrary cylindrical domain Ω×R (Ω ⊂ Rn being a bounded domain, I and Δ being the identity operator and the Laplacian) which generates an initial-boundary value problem with an explicit formula of the solution u. In the paper, the result is obtained not just for the operator ∂t + I −Δ, but also f...

Let G be a locally compact group, and let R(G) denote the ring of subsets of G generated by the left cosets of open subsets of G. The Cohen–Host idempotent theorem asserts that a set lies in R(G) if and only if its indicator function is a coefficient function of a unitary representation of G on some Hilbert space. We prove related results for representations of G on certain Banach spaces. We ap...