Pseudoconvexity and Analytic Discs

The purpose of this paper is to investigate the geometry of pseudoconvex hypersurfaces and their interrelation with analytic discs. The method of analytic discs is a very powerful tool in the study of pseudoconvexity. One way to use this tool is via the so-called type function introduced by Dwilewicz and Hill for arbitrary Cauchy–Riemann manifolds. In the hypersurface case, and moreover, under the assumption of pseudoconvexity, it has additional properties which are given in this paper. As an application, boundary estimates of plurisubharmonic functions are given.     

Analytic Discs and Extension of CR Functions

Let M be a manifold of X = C ⁿ , A a small analytic disc attached to M, z ^o a point of A where A is tangent to M, z ¹ another point of A where M extends to a germ of manifold M ₁ with boundary M. We prove that CR functions on M which extend to M ₁ at z ¹ also extend at z ^o to a new manifold M ₂. The directions M ₁ and M ₂ point to, are related by a sort of connection associated to A which is dual to the connection obtained by attaching 'partial analytic lifts' of A to the co-normal bundle to M in X.     

Difference Equation for Functions Analytic Outside Two Squares

We consider a totality of two squares built on primitive periods 1 and i and “sufficiently close to each other“. In a vicinity of this set we investigate four-element difference equation with constant coefficients, whose linear shifts are generating transforms of the corresponding doubly periodic group and the inverse transforms. We seek a solution in a class of functions, which are analytic outside this set and vanish at infinity. The equation is applicable to the moments problem for entire functions of exponential type.     

On Analytic Interpolation Manifolds in Boundaries of Weakly Pseudoconvex Domains

Let Ω be a bounded, weakly pseudoconvex domain in C ⁿ , n ≤ 2, with real-analytic boundary. A real-analytic submanifold M ? ?Ω is called an analytic interpolation manifold if every real-analytic function on M extends to a function belonging to (Ω¯). We provide sufficient conditions for M to be an analytic interpolation manifold. We give examples showing that neither of these conditions can be relaxed, as well as examples of analytic interpolation manifolds lying entirely within the set of weakly pseudoconvex points of ?Ω.     

Link Cobordism and the Intersection of Slice Discs

We work in the smooth category. An (oriented) (ordered) m-component n-(dimensional) link isa smooth oriented submanifold L = {K₁, ..., K_m} of Sⁿ⁺² whichis the ordered disjoint union of m manifolds, each PL-homeomorphicto the standard n-sphere. If m = 1, then L is called a knot. We say that m-component n-dimensional links L₀ and L₁ are (link-)concordantor (link-)cobordant if there is a smooth oriented submanifoldC = {C₁, ..., C_m} of Sⁿ⁺² x [0, 1] which meets the boundarytransversely in C, is PL-homeomorphic to L₀ x [0, 1], and meetsSⁿ⁺² x {l} in L_l (l = 0, 1). If m = 1, then we say that n-knotsL₀ and L_l are (knot-)concordant or (knot-)cobordant. Then wecall C a concordance-cylinder of the two n-knots L₀ and L_l. If an n-link L is concordant to the trivial link, then we callL a slice link. If an n-link L = {K₁, ..., K_m} Sⁿ⁺² = Bⁿ⁺³ Bⁿ⁺³ is slice,then there is a disjoint union of (n + 1)-discs in Bⁿ⁺³ such that is called a set of slice discs for L. If m = 1, then is called a slice disc for the knotL. 1991 Mathematics Subject Classification 57M25, 57Q45.     

Multipulse Orbits in the Motion of Flexible Spinning Discs

Summary. The global dynamics of flexible spinning discs are studied. The discs studied are parametrically excited in their spin rate, and have imperfections that cause symmetry-breaking. After determining the equations of motion in a suitable form, the energy-phase method is employed to show the existence of chaotic dynamics by identifying multipulse jumping orbits in the perturbed phase space. We provide restrictions on the damping, forcing, and symmetry-breaking parameters in order for these complicated dynamics to occur. The dissipative version of the energy-phase method predicts a wider range of values for which chaotic dynamics occurs than the traditional Melnikov method. The results are then discussed in terms of the physical motion of the spinning disc system. The multipulse orbits are manifested in the physical system as a shifting between two different nodal configurations of the disc. When the motion is chaotic, an observer will see a random jumping between the two nodal configurations of the disc. Received February 7, 2000; accepted November 18, 2001     

Upper Semicontinuous Valuations on the Space of Convex Discs

We show that every rigid motion invariant and upper semicontinuous valuation on the space of convex discs is a linear combination of the Euler characteristic, the length, the area, and a suitable curvature integral of the convex disc.     

Hanoi塔问题的一个公式解

Hanoi塔问题自提出以来已有一百多年的历史.其间,这一问题吸引了许多的研究者.正如H.A.Simon所指出的,Hanoi塔问题对于认知科学就象大肠杆菌对现代基因学那样,是一个无价的研究标本.事实上,它已成为组合数学,人工智能,计算机科学以及规划等中的递归问题的典型例子,并由此产生了各种各样成熟的算法.回顾这些结果,我们提出一个基本问题能否对Hanoi塔问题给出一个公式解?本文就此给出了一个肯定的回答.在我们的研究中,图论将是一个有力的工具     

On the Density of 2-Saturated Lattice Packings of Discs

 In a recent paper of G. Fejes Tóth, G. Kuperberg and W. Kuperberg [1] a conjecture has been published concerning the greatest lower bound of the density of a 2-saturated packing of unit discs in the plane. (A packing of unit discs is said to be 2-saturated if none of the discs could be replaced by two other ones of the same size to generate a new packing. A packing of the unit disc is a lattice packing if the centers form a point lattice.) In the present note we study this problem for lattice packings, however, in a more general form in which the removed unit disc is replaced by two discs of radius r. A corollary of our results supports the above conjecture proving that a lattice packing cannot be 2-saturated except if its density is larger than the conjectured bound. (Received 6 December 2000; in revised form March 29, 2001)     

Compact Packings of the Plane with Two Sizes of Discs

We consider packings of the plane using discs of radius 1 and r. A packing is compact if every disc D is tangent to a sequence of discs D₁, D₂, ..., D_n such that D_i is tangent to D_i+1. We prove that there are only nine values of r with r < 1 for which such packings are possible. For each of the nine values we describe the possible compact packings.     

On the Continuability of Multivalued Analytic Functions to an Analytic Subset

In the paper, it is shown that a germ of a many-valued analytic function can be continued analytically along the branching set at least until the topology of this set is changed. This result is needed to construct the many-dimensional topological version of Galois theory. The proof heavily uses Whitney's stratification.     

Fluctuations near the Boundaries in the Six-Vertex Model

Horizontal-arrow fluctuations near the boundaries in the six-vertex model with domain-wall boundary conditions are considered. For these fluctuations, a representation in terms of the standard objects of the theory of orthogonal polynomials is obtained. This representation is used for the study of the large N limit. Bibliography: 18 titles.     

On the Attached Prime Ideals of Local Cohomology Modules

In this article, we shall prove some new properties about attached prime ideals over local cohomology modules. Also we generalize some of the results of [2 Brodmann , M. P. , Sharp , R. Y. ( 1998 ). Local Cohomology; An Algebraic Introduction with Geometric Applications . Cambridge : Cambridge University Press .[Crossref] , [Google Scholar]].     

Analytic subgroups of the reals

We prove that every analytic proper subgroup of the reals can be covered by an null set. We also construct a proper Borel subgroup of the reals that cannot be covered by countably many sets such that is nowhere dense for every

     

Determining the Optimum Stratum Boundaries Using Mathematical Programming

The method of choosing the best boundaries that make strata internally homogeneous, given some sample allocation, is known as optimum stratification. In order to make the strata internally homogeneous, the strata are constructed in such a way that the strata variances should be as small as possible for the characteristic under study. In this paper the problem of determining optimum strata boundaries (OSB) is discussed when strata are formed based on a single auxiliary variable with a varying measurement cost per units across strata. The auxiliary variable considered in the problem is a size variable that holds a common model for a whole population. The OSB are achieved effectively by assuming a suitable distribution of the auxiliary variable and creating strata by cutting the range of the distribution at optimum points. The problem of finding the OSB, which minimizes the variance of the estimated population mean under a weighted stratified balanced sampling, is formulated as a mathematical programming problem (MPP). Treating the formulated MPP as a multistage decision problem, a solution procedure using dynamic programming technique is developed. A numerical example using a hospital population data is presented to illustrate the computational details of the solution procedure. A software program coded in JAVA is written to carry out the computation. The distribution of the auxiliary variable in this example is considered to be continuous with an exponential density function.     

On the Initial-boundary-value Problem for the Extensible Beam with Attached Load

When one end of an extensible beam whose ends are a fixed distance apart, is hinged while to other end a load is attached, the mathematical model describing the vibrations of the beam contains a non-linearity both in the partial differential equation as well as in the dynamical boundary condition. We introduce weak damping and consider the resulting boundary-value problem within the framework of the theories of B-evolutions and fractional powers of a closed pair of operators. This approach enables us to construct a unique weak solution which exhibits exponential decay by employing Faedo–Galerkin approximations.     

S-adic L-Functions Attached to the Symmetric Square of a Newform

In order to apply the ideas of Iwasawa theory to the symmetricsquare of a newform, we need to be able to define non-archimedeananalogues of its complex L-series. The interpolated p-adic L-functionis closely connected via a "Main Conjecture" with certain Selmergroups over the cyclotomic Z_p-extension of Q. In the p-ordinarycase these functions are well understood. In this article we extend the interpolation to an arbitraryset S of good primes (not necessarily satisfying ordinarityconditions). The corresponding S-adic functions can be characterisedin terms of certain admissibility criteria. We also allow interpolationat particular primes dividing the level of the newform. One interesting application is to the symmetric square of amodular elliptic curve E defined over Q. Our constructions yieldp-adic L-functions at all primes of stable or semi-stable reduction.If p is ordinary or multiplicative the corresponding analyticfunction is bounded; if p is supersingular our function behaveslike log²(1 + T). 1991 Mathematics Subject Classification: 11F67,11F66, 11F33, 11F30