In this paper, the author presents a framework for getting a series of exactvacuum solutions to the Einstein equation. This procedure of resolution is based on acanonical form of the metric. According to this procedure, the Einstein equation can bereduced to some 2-dimensional Laplace-like equations or rotation and divergence equations,which are much convenient for the resolution.
In "Elements of small orders in K 2(F)" (Algebraic K-Theory, Lecture Notesin Math., 966, 1982, 1–6.), the author investigates elements of the form {a, Φn(a)} in theMilnor group K 2 F of a field F, where Φn(x) is the n-th cyclotomic polynomial. In thispaper, these elements are generalized. Applying the explicit formulas of Rosset and Tatefor the transfer homomorphism for K 2, the author proves some new results on elements ofsmall orders in K 2 F.
The authors derive a formula for the volume of a compact domain in a symmetricspace from normal sections through a special submanifold in the symmetric space. Thisformula generalizes the volume of classical domains as tubes or domains given as motionsalong the submanifold. Finally, some stereological considerations regarding this formulaare provided.
In this note, the authors resolve an evolutionary Wente’s problem associated to heat equation, where the special integrability of det∇u for u ∈ H 1(ℝ2,ℝ2) is used.
The authors consider proper holomorphic mappings between smoothly bounded pseudoconvex regions in complex 2-space, where the domain is of finite type and admits a transverse circle action. The main result is that the closure of each irreducible component of the branch locus of such a map intersects the boundary of the domain in the union of finitely many orbits of the group action.
Let F be a Hilbert filtration with respect to a Cohen-Macaulay module M.When G(F,M) and F K(F,M) have almost maximal depths, the Hilbert coefficients g i(F,M) is calculated. In the general case, an upper bound for g 2(F,M) is also given.
In the present paper, the full range Strichartz estimates for homogeneous Schrödinger equations with non-degenerate and non-smooth coefficients are proved. For inhomogeneous equation, the non-endpoint Strichartz estimates are also obtained.
In this paper, the authors discuss an inverse boundary problem for the axi-symmetric steady-state heat equation, which arises in monitoring the boundary corrosionfor the blast-furnace. Measure temperature at some locations are used to identify the shapeof the corrosion boundary.
The numerical inversion is complicated and consuming since the wear-line varies duringthe process and the boundary in the heat problem is not fixed. The authors suggest amethod that the unknown boundary can be represented by a given curve plus a smallperturbation, then the equation can be solved with fixed boundary, and a lot of computingtime will be saved.
A method is given to solve the inverse problem by minimizing the sum of the squaredresidual at the measuring locations, in which the direct problems are solved by axi-symmetric fundamental solution method.
The numerical results are in good agreement with test model data as well as industrialdata, even in severe corrosion case.
In this paper, the convergence of time-dependent Euler-Maxwell equations tocompressible Euler-Poisson equations in a torus via the non-relativistic limit is studied.The local existence of smooth solutions to both systems is proved by using energy estimates for first order symmetrizable hyperbolic systems. For well prepared initial data theconvergence of solutions is rigorously justified by an analysis of asymptotic expansions upto any order. The authors perform also an initial layer analysis for general initial data andprove the convergence of asymptotic expansions up to first order.
For a given polyhedron K ⊂ M, the notation R M(K) denotes a regular neigh-borhood of K in M. The authors study the following problem: find all pairs (m, k)such that if K is a compact k-polyhedron and M a PL m-manifold, then R M(f(K)) ≅ R M(g(K)) for each two homotopic PL embeddings f, g : K → M. It is proved that R S k+2 (S k) ≇ S k × D 2 for each k ≥ 2 and some PL sphere S k ⊂ S k +2 (even for any PLsphere S k ⊂ S k +2 having an isolated non-locally flat point with the singularity S k -1 ⊂ S k +1 such that π1(S k +1 – S k -1) ≇ ℤ).
In the present paper, the author studies the existence of sign-changing solutions for nonlinear elliptic equations, which have jumping nonlinearities, and may or may not be resonant with respect to Fučik spectrum, via linking methods under Cerami condition.