Factoriality of complete intersections in ℙ<Superscript>5</Superscript>

Let X be a complete intersection of two hypersurfaces F _n and F _k in ℙ⁵ of degree n and k, respectively, with n ≥ k, such that the singularities of X are nodal and F _k is smooth. We prove that if the threefold X has at most (n + k − 2)(n − 1) − 1 singular points, then it is factorial.     

Accuracy of approximation in the Poisson theorem in terms of the χ<Superscript>2</Superscript>-distance

We study the limit behavior of the χ²-distance between the distributions of the nth partial sum of independent not necessarily identically distributed Bernoulli random variables and the accompanying Poisson law. As a consequence in the i.i.d. case we make the multiplicative constant preciser in the available upper bound for the rate of convergence in the Poisson limit theorem.     

Complete surfaces of constant curvature in H<Superscript>2</Superscript> × R and S<Superscript>2</Superscript> × R

We study isometric immersions of surfaces of constant curvature into the homogeneous spaces and . In particular, we prove that there exists a unique isometric immersion from the standard 2-sphere of constant curvature c > 0 into and a unique one into when c > 1, up to isometries of the ambient space. Moreover, we show that the hyperbolic plane of constant curvature c < −1 cannot be isometrically immersed into or . J.A. Aledo was partially supported by Ministerio de Education y Ciencia Grant No. MTM2004-02746 and Junta de Comunidades de Castilla-La Mancha, grant no. PAI-05-034. J.M. Espinar and J.A. Gálvez were partially supported by Ministerio de Education y Ciencia grant no. MTM2004-02746 and Junta de Andalucía Grant No. FQM325.     

The global dynamics of a class of vector fields in ℝ<Superscript>3</Superscript>

In this paper, we find a bridge connecting a class of vector fields in ℝ³ with the planar vector fields and give a criterion of the existence of closed orbits, heteroclinic orbits and homoclinic orbits of a class of vector fields in ℝ³. All the possible nonwandering sets of this class of vector fields fall into three classes: (i) singularities; (ii) closed orbits; (iii) graphs of unions of singularities and the trajectories connecting them. The necessary and sufficient conditions for the boundedness of the vector fields are also obtained.     

Fundamental solution of rotating generalized stokes problem in ℝ<Superscript>3</Superscript>

In this paper,the fundamental solution of rotating generalized Stokes problem in R 3 is established.To obtain it,some fundamental solutions of other problems also are established,such as generalized Laplace problem,generalized Stokes problem and rotating Stokes problem.     

Equivalence of Self-similar and Pseudo-self-similar Tiling Spaces in ℝ<Superscript>2</Superscript>

Given a tiling T, one may form a related tiling, called the derived Voronoi tiling of T, based on a patch of tiles in T. Similarly, for a tiling space X, one can identify a patch which appears regularly in all tilings in X, and form a derived Voronoi space of tilings, based on that patch.     

Moving Object in ℝ<Superscript>2</Superscript> and a Group of Observers

We formulate an extremal problem of constructing a trajectory of a moving object that is farthest from a group of observers with fixed visibility cones. Under some constraints on the arrangement of the observers, we give a characterization and a method of construction of an optimal trajectory.     

On one class of Lipschitz vector fields in ℝ<Superscript>3</Superscript>

Under consideration are some continuous metric functions induced by one class of Lipschitz vector fields in ℝ³. These functions are showed to be quasimetrics within the domain of definition of the vector fields. We prove some analogs of the Rashevsky-Chow Theorem and the Ball-Box Theorem under some restriction on the class of vector fields. The methods of proofs do not use the existence of the nilpotent tangent cone.     

Characterization of convex domains in ℂ<Superscript>2</Superscript> with non-compact automorphism group

In the field of several complex variables, the Greene-Krantz Conjecture, whose consequences would be far reaching, has yet to be proven. The conjecture is as follows: Let D be a smoothly bounded domain in ?ⁿ. Suppose there exists {g _j} ? Aut(D) such that {g _j(z)} accumulates at a boundary point p ∈ ?D for some z ∈ D. Then ?D is of finite type at p. In this paper, we prove the following result, yielding further evidence to the probable veracity of this important conjecture: Let D be a bounded convex domain in ?² with C ² boundary. Suppose that there is a sequence {g _j} ? Aut(D) such that {g _j(z)} accumulates at a boundary point for some point z ∈ D. Then if p ∈ ?D is such an orbit accumulation point, ?D contains no non-trivial analytic variety passing through p.     

Polysubharmonic Functions Near Infinity in ℜ<Superscript>n</Superscript>

For polyharmonic functions u of finite order defined outside a compact set in ⁿ, representations by means of -subharmonic functions defined on the whole of ⁿ are given. As a consequence, certain analogues of the classical Bôcher theorem are proved for u. Some of these results are extended to the case of polysubharmonic functions defined near infinity in ⁿ. Mathematics Subject Classification (2000) 31B30.     

Minimal cubature formulas of degree 3 for a torus in ℝ<Superscript>3</Superscript>

We construct minimal cubature formulas of degree 3 for a torus in R³. The cases of a degenerate torus with radius r = 1 and a torus with arbitrary radius r > 1 are considered separately.     

On the definition of the Monge-Ampère operator in ℂ<Superscript>2</Superscript>

We show that if u is a plurisubharmonic function defined on an open subset of ² then the Monge-Ampère measure (dd^cu)² can be well defined if and only if u belongs to the Sobolev space W^1,2_loc().Partially supported by KBN Grant #2 P03A 028 19     

Biflatness of ℓ<Superscript>1</Superscript>-Semilattice Algebras

We show that if L is a semilattice then the ℓ¹-convolution algebra of L is biflat precisely when L is "uniformly locally finite". Our proof technique shows in passing that if this convolution algebra is biflat then it is isomorphic as a Banach algebra to the Banach space ℓ¹(L) equipped with pointwise multiplication. At the end we sketch how these techniques may be extended to prove an analogous characterisation of biflatness for Clifford semigroup algebras.     

On the uniqueness of generalized and quasi-regular solutions to equations of mixed type in ℝ<Superscript>3</Superscript>

Some three-dimensional (3D) problems for mixed type equations of first and second kind are studied. For equation of Tricomi type, they are 3D analogs of the Darboux (or Cauchy-Goursat) plane problem. Such type problems for a class of hyperbolic and weakly hyperbolic equations as well as for some hyperbolic-elliptic equations are formulated by M. Protter in 1952. In contrast to the well-posedness of the Darboux problem in the 2D case, the new 3D problems are strongly ill-posed. A similar statement of 3D problem for Keldysh-type equations is also given. For mixed type equations of Tricomi and Keldysh type, we introduce the notion of generalized or quasi-regular solutions and find sufficient conditions for the uniqueness of such solutions to the Protter’s problems. The dependence of lower order terms is also studied.     

On One Family of Minimal Tori in ℝ<Superscript>3</Superscript> with Planar Embedded Ends

We construct new examples of complete minimal tori in the three-dimensional Euclidean space with an arbitrary even number n ≥ 6 of planar embedded ends.     

Weierstrass representation for discrete isotropic surfaces in ℝ<Superscript>2,1</Superscript>, ℝ<Superscript>3,1</Superscript>, and ℝ<Superscript>2,2</Superscript>

Using an integrable discrete Dirac operator, we construct a discrete version of the Weierstrass representation for hyperbolic surfaces parameterized along isotropic directions in ℝ^2,1, ℝ^3,1, and ℝ^2,2. The corresponding discrete surfaces have isotropic edges. We show that any discrete surface satisfying a general monotonicity condition and having isotropic edges admits such a representation.     

Isotropic Lagrangian submanifolds in the homogeneous nearly Kähler S<Superscript>3</Superscript> × S<Superscript>3</Superscript>

We show that isotropic Lagrangian submanifolds in a 6-dimensional strict nearly Kähler manifold are totally geodesic. Moreover, under some weaker conditions, a complete classification of the J-isotropic Lagrangian submanifolds in the homogeneous nearly Kähler S³ × S³ is also obtained. Here, a Lagrangian submanifold is called J-isotropic, if there exists a function λ, such that g((?h)(v, v, v), Jv) = λ holds for all unit tangent vector v.     

On the Nitsche conjecture for harmonic mappings in ℝ<Superscript>2</Superscript> and ℝ<Superscript>3</Superscript>

We give the new inequality related to the J. C. C. Nitsche conjecture (see [6]). Moreover, we consider the two- and three-dimensional case. LetA(r, 1)={z:r<|z|<1}. Nitsche's conjecture states that if there exists a univalent harmonic mapping from an annulusA(r, 1), to an annulusA(s, 1), thens is at most 2r/(r ²+1).Lyzzaik's result states thats<t wheret is the length of the Grötzsch's ring domain associated withA(r, 1) (see [5]). Weitsman's result states thats≤1/(1+1/2(r logr)²) (see [8]).Our result for two-dimensional space states thats≤1/(1+1/2 log² r) which improves Weitsman's bound for allr, and Lyzzaik's bound forr close to 1. For three-dimensional space the result states thats≤1/(r?logr).