Synthetic studies toward swinhoeisterol A,a novel steroid with an unusual carbon skeleton

Swinhoeisterol A is a novel steroid with unusual 6/6/5/7 tetracyclic skeleton. The model compound with BCD rings is constructed by Friedel–Crafts acylation and an oxidative dearomatization as key steps.     

Phytoecdysteroids of Plants of the Silene Genus. 2-Deoxyecdysterone-25-Acetate from Silene wallichiana

The new ecdysteroid 2-deoxyecdysterone-25-acetate was isolated from roots of Silene wallichiana Klotzsch.     

Explicit local time-stepping methods for Maxwell’s equations

Explicit local time-stepping methods are derived for time dependent Maxwell equations in conducting and non-conducting media. By using smaller time steps precisely where smaller elements in the mesh are located, these methods overcome the bottleneck caused by local mesh refinement in explicit time integrators. When combined with a finite element discretisation in space with an essentially diagonal mass matrix, the resulting discrete time-marching schemes are fully explicit and thus inherently parallel. In a non-conducting source-free medium they also conserve a discrete energy, which provides a rigorous criterion for stability. Starting from the standard leap-frog scheme, local time-stepping methods of arbitrarily high accuracy are derived for non-conducting media. Numerical experiments with a discontinuous Galerkin discretisation in space validate the theory and illustrate the usefulness of the proposed time integration schemes.     

Positive-definite functions on infinite-dimensional groups

This paper concerns positive-definite functions on infinite-dimensional groups G. Our main results are as follows: first, we claim that if G has a σ-finite measure μ on the Borel field whose right admissible shifts form a dense subgroup G ₀, a unique (up to equivalence) unitary representation (H, T) with a cyclic vector corresponds to through a method similar to that used for the G–N–S construction. Second, we show that the result remains true, even if we go to the inductive limits of such groups, and we derive two kinds of theorems, those taking either G or G ₀ as a central object. Finally, we proceed to an important example of infinite-dimensional groups, the group of diffeomorphisms on smooth manifolds M, and see that the correspondence between positive-definite functions and unitary representations holds for under a fairy mild condition. For a technical reason, we impose condition (c) in Sect. 2 on the measure space throughout this paper. It is also a weak condition, and it is satified, if G is separable, or if μ is Radon. This research was partially supported by a Grant-in-Aid for Scientific Research (No.18540184), Japan Socieity of the Promotion of Science.     

Compactness of sequences of two-dimensional energies with a zero-order term. Application to three-dimensional nonlocal effects

In this paper we study the limit, in the sense of the Γ-convergence, of sequences of two-dimensional energies of the type , where A _n is a symmetric positive definite matrix-valued function and μ _n is a nonnegative Borel measure (which can take infinite values on compact sets). Under the sole equicoerciveness of A _n we prove that the limit energy belongs to the same class, i.e. its reads as , where is a diffusion independent of μ _n and μ is a nonnegative Borel measure which does depend on . This compactness result extends in dimension two the ones of [11,23] in which A _n is assumed to be uniformly bounded. It is also based on the compactness result of [7] obtained for sequences of two-dimensional diffusions (without zero-order term). Our result does not hold in dimension three or greater, since nonlocal effects may appear. However, restricting ourselves to three-dimensional diffusions with matrix-valued functions only depending on two coordinates, the previous two-dimensional result provides a new approach of the nonlocal effects. So, in the periodic case we obtain an explicit formula for the limit energy specifying the kernel of the nonlocal term.     

On the Cauchy problem for the fast diffusion equation

For , the author studies the existence of a kind of weak solution to the Cauchy problem

     

Nonlinear diffusion in type-II superconductors

In this paper we study a nonlinear evolution equation _∂t(σ(|E|)E)+∇×∇×E=F${\mathrm{\partial }}_t(\sigma (\left|\mathit{E}\right|)\mathit{E})+\nabla \times \nabla \times \mathit{E}=\mathit{F}$ in a bounded domain subject to appropriate initial and boundary conditions. This governs the evolution of the electric field E$\mathit{E}$ in a conductive medium under the influence of a force F$\mathit{F}$. It is an approximation of Bean's critical-state model for type-II superconductors. We design a nonlinear numerical scheme for the time discretization. We prove the convergence of the proposed method. The proof is based on a generalization of div–curl lemma for transient problems. We also derive some error estimates for the approximate solution.     

PRME AND k-PRIME IDEALS IN Ω-GROUPS

Abstract

In this paper we define two concepts of prime ideals for Ω-groups. The first generalizes the definitions of prime ideal in rings, nearrings, Γ-rings, associative algebras and Lie algebras. The second generalizes a concept defined for groups by ??ukin ([21]). We show that both lead to radicals in the sense of Hoehnke ([10]). Furthermore in the case of rings, Γ-rings, abelian zero-symmetric nearrings and cubic rings these two definitions coincide, thus obtaining a new characterization for the prime ideal. Zero-symmetric Ω-groups are defined analogously to the nearring case and a new characterization in term of ideals is given.