Projective Generation of Curves in Polynomial Extensions of an Affine Domain (II)

Let Abe an affine domain of dimension nover an algebraically closed field kof characteristic 0. Let I A[T]be a local complete intersection ideal of height nsuch that I/I2 is generated by n elements. It is proved that there exists a projective A[T]module Pof rank nsuch that Iis a surjective image of P.     

Patterns of electro-convection in planar-periodic nematic cells

Nematic liquid crystals are anisotropic fluids that can undergo electro-convective instabilities. These instabilities have been extensively studied for nematics in unidirectional planar alignment. We investigate the electro-convective instability patterns in planar-periodic cells, wherein, in the quiescent situation the nematic undergoes static twist deformations separated by defect lines. We find two regimes that depend on the sample thickness. For large thickness, electro-convective patterns are not affected by the twist, and the samples behave as they were in planar unidirectional alignment. For small thickness, either the convective motion vertically shifts the defect lines or the twist affects the electro-convective domains and resultant curved patterns are observed.     

Reinhardt域上一类推广的Roper-Suffridge算子

研究一类推广的Roper-Suffridge算子F(z)=(f(z_1)+f′(z_1)∑_(k=2)~nakz_k~pk,f′(z)1)(~1/p2)z_2,…,f′(z_1)~(1/pn)z_n)′,证明该算子在复欧氏空间中的Reinhardt域Ω_(n,p2,%…,pn)={z=(z_1,…,z_n)∈C~n:|z_|~2+∑_(k=2)~n|zk|~(pk)1,Pk∈N~+,k=2,…,n}上分别保持α次的殆β型螺形性,α次的β型螺形性及强β型螺形性.     

Canonical Metrics on Generalized Cartan-Hartogs Domains

In this paper, the author considers a class of bounded pseudoconvex domains,i.e., the generalized Cartan-Hartogs domains Ω(μ, m). The first result is that the natural Khler metric g~(Ω(μ,m)) of Ω(μ, m) is extremal if and only if its scalar curvature is a constant. The second result is that the Bergman metric, the Ka¨hler-Einstein metric, the Carathéodary metric, and the Koboyashi metric are equivalent for Ω(μ, m).     

Eigenvalue problems of nonhomogeneous semilinear elliptic equations in Esteban–Lions domains with holes

In this article, we consider the following eigenvalue problems

('∗

λ' render=n">

where λ>0, N2 and is the upper semi-strip domain with a hole in . Under some suitable conditions on f and h, we show that there exists a positive constant λ^* such that Eq. (*)_λ has at least two solutions if λ(0,λ^*), a unique positive solution if λ=λ^*, and no positive solution if λ>λ^*. We also obtain some further properties of the positive solutions of (*)_λ.     

Irredundant intersections of valuation overrings of two-dimensional Noetherian domains

Let D be a two-dimensional Noetherian domain, let R be an overring of D, and let Σ and Γ be collections of valuation overrings of D. We consider circumstances under which (_VΣV)∩R=(_WΓW)∩R implies that Σ=Γ. We show that if R is integrally closed, these representations are “strongly” irredundant, and every member of ΣΓ has Krull dimension 2, then Σ=Γ. If in addition Σ and Γ are Noetherian subspaces of the Zariski–Riemann space of the quotient field of D (e.g. if Σ and Γ have finite character), then the restriction that the members of ΣΓ have Krull dimension 2 can be omitted. An example shows that these results do not extend to overrings of three-dimensional Noetherian domains.     

Pointwise estimates for relative fundamental solutions of heat equations in $${\mathbb{R} \times \mathbb{C}}$$

Let be a subharmonic, nonharmonic polynomial and a parameter. Define , a closed, densely defined operator on . If and , we solve the heat equations , u(0,z) = f(z) and , . We write the solutions via heat semigroups and show that the solutions can be written as integrals against distributional kernels. We prove that the kernels are C ^∞ off of the diagonal {(s, z, w) : s = 0 and z = w} and find pointwise bounds for the kernels and their derivatives.      

Singular Integrals and Potential Theory

In this paper I consider a class of non-standard singular integrals motivated by potential theoretic and probabilistic considerations. The probabilistic applications, which are by far the most interesting part of this circle of ideas, are only outlined in Section 1.5: They give the best approximation of the solution of the classical Dirichlet problem in a Lipschitz domain by the corresponding solution by finite differences. The potential theoretic estimate needed for this gives rise to a natural duality between the L _p functions on the boundary ∂Ω and a class of functions A on Ω that was first considered by Dahlberg. The actual duality is given by ∫_Ω S f(x)A(x)dx = (f, A) where S f(x) = ∫_∂Ω |x − y|¹⁻ⁿ f(y)dy is the Newtonian potential. We can identify the upper half Lipschitz space with in the obvious way and express for an appropriate kernel K. It is the boundedness properties of the above (for , ) that is the essential part of this work. This relates with more classical (but still “rough”) singular integrals that have been considered by Christ and Journé. Lecture held in the Seminario Matematico e Fisico on March 14, 2005 Received: April 2007     

Domain Analysis and Motif Matcher (DAMM): A Program to Predict Selectivity Determinants in Monosiga brevicollis PDZ Domains Using Human PDZ Data

Choanoflagellates are single-celled eukaryotes with complex signaling pathways. They are considered the closest non-metazoan ancestors to mammals and other metazoans and form multicellular-like states called rosettes. The choanoflagellate Monosiga brevicollis contains over 150 PDZ domains, an important peptide-binding domain in all three domains of life (Archaea, Bacteria, and Eukarya). Therefore, an understanding of PDZ domain signaling pathways in choanoflagellates may provide insight into the origins of multicellularity. PDZ domains recognize the C-terminus of target proteins and regulate signaling and trafficking pathways, as well as cellular adhesion. Here, we developed a computational software suite, Domain Analysis and Motif Matcher (DAMM), that analyzes peptide-binding cleft sequence identity as compared with human PDZ domains and that can be used in combination with literature searches of known human PDZ-interacting sequences to predict target specificity in choanoflagellate PDZ domains. We used this program, protein biochemistry, fluorescence polarization, and structural analyses to characterize the specificity of A9UPE9_MONBE, a M. brevicollis PDZ domain-containing protein with no homology to any metazoan protein, finding that its PDZ domain is most similar to those of the DLG family. We then identified two endogenous sequences that bind A9UPE9 PDZ with <100 μM affinity, a value commonly considered the threshold for cellular PDZ–peptide interactions. Taken together, this approach can be used to predict cellular targets of previously uncharacterized PDZ domains in choanoflagellates and other organisms. Our data contribute to investigations into choanoflagellate signaling and how it informs metazoan evolution.     

Approximation of Metric Spaces by Partial Metric Spaces

Partial metrics are generalised metrics with non-zero self-distances. We slightly generalise Matthews' original definition of partial metrics, yielding a notion of weak partial metric. After considering weak partial metric spaces in general, we introduce a weak partial metric on the poset of formal balls of a metric space. This weak partial metric can be used to construct the completion of classical metric spaces from the domain-theoretic rounded ideal completion.