Codimension 3 Arithmetically Gorenstein Subschemes of projective NN-space

We study the lowest dimensional open case of the question whether every arithmetically Cohen--Macaulay subscheme of $\mathbb{P}^N$ is glicci, that is, whether every zero-scheme in $\mathbb{P}^3$ is glicci. We show that a set of $n \geq 56$ points in general position in \PP^3 admits no strictly descending Gorenstein liaison or biliaison. In order to prove this theorem, we establish a number of important results about arithmetically Gorenstein zero-schemes in $\mathbb{P}^3$.Comment: to appear in Annales de l'Institut Fourie

Deformations of Fuchsian Systems of Linear Differential Equations and the Schlesinger System

We consider holomorphic deformations of Fuchsian systems parameterized by the pole loci. It is well known that, in the case when the residue matrices are non-resonant, such a deformation is isomonodromic if and only if the residue matrices satisfy the Schlesinger system with respect to the parameter. Without the non-resonance condition this result fails: there exist non-Schlesinger isomonodromic deformations. In the present article we introduce the class of the so-called isoprincipal deformations of Fuchsian systems. Every isoprincipal deformation is also an isomonodromic one. In general, the class of the isomonodromic deformations is much richer than the class of the isoprincipal deformations, but in the non-resonant case these classes coincide. We prove that a deformation is isoprincipal if and only if the residue matrices satisfy the Schlesinger system. This theorem holds in the general case, without any assumptions on the spectra of the residue matrices of the deformation. An explicit example illustrating isomonodromic deformations, which are neither isoprincipal nor meromorphic with respect to the parameter, is also given

The Edge Electric Field of a Pyroelectric and its Applications

Following a change of temperature of a pyroelectric (PE), a depolarizing electric field appears both inside the PE, as well as outside its edges, the edge depolarizing electric field (EDEF). The EDEF extends outwards up to a distance of the order of magnitude of the PE width. The mapping and the strength of the EDEF have been calculated and analyzed for the case of a semi-infinite pyroelectric plate. This strong EDEF (104-105 V/cm), when penetrating into the surrounding medium, creates a variety of physical effects: inducing electrical current in a semiconductor and affecting its resistance, accelerating charged and neutral particles in vacuum or in a gas, generating electromagnetic waves, modifying optical characteristics by electrooptical and photoelasic effects, generating piezoelectric deformation and more. We show that these EDEF induced effects could serve as a basis for the development of various applications and devices.Comment: 27 pages including 13 figure

Optical alignment system Patent

Electro-optical/computer system for aligning large structural members and maintaining correct positio