Isoperimetric inequalities for quermassintegrals

We study the first eigenfunction φ 1 of the Dirichlet Laplacian on a convex domain in Euclidean space. Elementary properties of Bessel functions yield that $$\left\| {\phi _1 } \right\|_\infty /\left\| {\phi _1 } \right\|_2 \to \infty$$ if D is a sector in Euclidean plane with area 1 and the angle tends to 0. We aim to characterize those domains D such that $$(vol(D))^{1/2} \left\| {\phi _1 } \right\|_\infty /\left\| {\phi _1 } \right\|_2$$ is large in terms of the ratio of the first eigenvalue of D and the infimum of the first eigenvalues of all subdomains D≈ of D with given volume.

In 1913, E. Helly[i] has established: When C 1 , ..., C m are convex domains in an n-dimensional Euclidean space and every set of n+I of these domains have a common point, then there exists a point which is common to all the domains. Recently[6] this theorem has obtained an enhanced importance because of the numerous applications which it affords.

A bosonic Laplacian is a conformally invariant second‐order differential operator acting on smooth functions defined on domains in Euclidean space and taking values in higher‐order irreducible representations of the special orthogonal group. In this paper, we firstly introduce the motivation for study of the generalized Maxwell operators and bosonic Laplacians (also known as the higher spin Laplace operators). Then, with the help of connections between Rarita–Schwinger type operators and bosonic Laplacians, we solve Poisson's equation for bosonic Laplacians. A representation formula for bounded solutions to Poisson's equation in Euclidean space is also provided. In the end, we provide Green's formulas for bosonic Laplacians in scalar‐valued and Clifford‐valued cases, respectively. These formulas reveal that bosonic Laplacians are self‐adjoint with respect to a given L2 inner product on certain compact supported function spaces.

Using the first exit time for Brownian motion from a smoothly bounded domain in Euclidean space, we define two natural functionals on the space of embedded, compact, oriented, unparametrized hypersurfaces in Euclidean space. We develop explicit formulas for the first variation of each of the functionals and characterize the critical points.

The purpose of this paper is to investigate the topological and analytical restrictions on a domain D in euclidean n-space R n on which an infinite discrete quasiconformal group can act. We will see that the restrictions are indeed severe, unlike the case of a discrete group of topological or differentiable homeomorphisms.KeywordsUnit BallLimit PointUniversal CoverTangent PlaneQuasiconformal MappingThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

In this paper we consider Monge–Ampère equations on compact Hessian manifolds, or equivalently Monge–Ampère equations on certain unbounded convex domains in Euclidean space, with a periodicity constraint given by the action of an affine group. In the case where the affine group action is volume preserving, i.e., when the manifold is special, the solvability of the corresponding Monge–Ampère equation was first established by Cheng and Yau using the continuity method. In the general case we set up a variational framework involving certain dual manifolds and a generalization of the classical Legendre transform. We give existence and uniqueness results and elaborate on connections to optimal transport and quasi-periodic tilings of convex domains.

97/04852 Coal pulsating combustion model: Xu, W. Ranshao Kexue Yu Jishu., 1996, 2, (3), 260–269. (In Chinese)

In this talk, we discuss the insulated conductivity problem with multiple inclusions embedded in a bounded domain in n-dimensional Euclidean space. The gradient of a solution may blow up as two inclusions approach each other. The optimal blow up rate was known in dimension n=2. It was not known whether the established upper bound of the blow up rates in higher dimensions were optimal. We answer this question by improving the previously known upper bound of the blow up rates in dimension n > 2. This is a joint work with Zhuolun Yang.

CANM, a program for numerical solution of a system of nonlinear equations using the continuous analog of Newton's method

98/03268 Reopening of pulverized coal injection in blast furnace after a pause

99/00738 On the combustion of heat pump with solar steam turbine

We present here the fundamentals of a theory of domains that offers unifying techniques and terminology for a number of different fields. Using direct, geometric methods, we develop integration over p-dimensional domains in n-dimensional Euclidean space ⩄n, replacing the method of parametrization of a domain with the method of approximation in Banach spaces. We prove basic results needed for a theory of integration — continuity of the integral as a function of its domain and integrand (Corollary 4.8) and a generalization of Stokes’s theorem (Corollary 4.14).

99/00035 ESR study of the macromolecular structure of copper-porphyrin-labeled coal

This note addresses a problem of Dvoretzky concerning the harmonic measure of the set of boundary points of a domain in Euclidean space that are of angular type.

Let M be a domain in an Euclidean space and let Γ be a pseudo group of transformations of M. We say that Γ is a Cartan pseudo group [1, 2] if the following conditions are satisfied:1)There exists a domain M′ and a projection ρ: M-→M′, such that the orbits of Γ are the fibers of the projection ρ.