Betti numbers of modules of essentially monomial type

Let be a Noetherian local ring. In this paper we supply formulae for computing the ranks of syzygy and Betti numbers of -modules of essentially monomial type. These modules are defined with respect to various -regular sequences. For example, finite length modules of monomial type over regular local rings of dimension are modules of essentially monomial type with respect to -regular sequences of length . If a module is of essentially monomial type with respect to an -regular sequence of length , then the rank of its -th syzygy is at least and its -th Betti number is at least .

     

An Indefinite Kahler Metric on the Space of Oriented Lines

The total space of the tangent bundle of a Kähler manifoldadmits a canonical Kähler structure. Parallel translationidentifies the space T of oriented affine lines in R³ with thetangent bundle of S². Thus the round metric on S² induces aKähler structure on T which turns out to have a metricof neutral signature. It is shown that the identity componentof the isometry group of this metric is isomorphic to the identitycomponent of the isometry group of the Euclidean metric on R³. The geodesics of this metric are either planes or helicoidsin R³. The signature of the metric induced on a surface inT is determined by the degree of twisting of the associatedline congruence in R³, and it is shown that, for Lagrangian,the metric is either Lorentz or totally null. For such surfacesit is proved that the Keller–Maslov index counts the numberof isolated complex points of J inside a closed curve on .     

An Integrability Condition for Monge-Ampere Equations on a Kahler Manifold

The symmetry group of the Monge-Ampère equation on a Kähler manifold is determined and an integrability condition on the solution is derived as a conservation law.     

Reduced products which are not saturated

Summary IfT is a complete theory of Boolean algebra, then we writeA ⊲_T B to denote that for every cardinal κ and every κ-regular filter over a setI such that the Boolean algebra 2 _F ^I of all subsets ofI reduced byF is a model ofT, the reduced powerA _F ^I isK ⁺-saturated wheneverB _F ^I isK ⁺-saturated. The relation ⊲_T generalizes the relation ◃ introduced by Keisler. As in the case of Keisler's ◃ it happens that ⊲_T’s are relations between complete theories, i.e. ifA≡B thenA ⊲_T B andB ⊲_T A. In this paper some examples of theories which are maximal (minimal) with respect to ⊲_T’s are provided and the relations ⊲_T are compared with each other. Presented by J. Mycielski     

The Normal Holonomy Group of Kahler Submanifolds

We study the (restricted) holonomy group Hol() of the normalconnection (shortened to normal holonomy group) of a Kählersubmanifold of a complex space form. We prove that if the normalholonomy group acts irreducibly on the normal space then itis linear isomorphic to the holonomy group of an irreducibleHermitian symmetric space. In particular, it is a compact groupand the complex structure J belongs to its Lie algebra. We prove that the normal holonomy group acts irreducibly ifthe submanifold is full (that is, it is not contained in a totallygeodesic proper Kähler submanifold) and the second fundamentalform at some point has no kernel. For example, a Kähler–Einsteinsubmanifold of CPⁿ has this property. We define a new invariant µ of a Kähler submanifoldof a complex space form. For non-full submanifolds, the invariantµ measures the deviation of J from belonging to the normalholonomy algebra. For a Kähler–Einstein submanifold,the invariant µ is a rational function of the Einsteinconstant. By using the invariant µ, we prove that thenormal holonomy group of a not necessarily full Kähler–Einsteinsubmanifold of CPⁿ is compact, and we give a list of possibleholonomy groups. The approach is based on a definition of the holonomy algebrahol(P) of an arbitrary curvature tensor field P on a vectorbundle with a connection and on a De Rham type decompositiontheorem for hol(P). 2000 Mathematics Subject Classification53C40 (primary), 53B25 (secondary).     

Minimal Immersions of Kahler Manifolds into Euclidean Spaces

It is proved here that a minimal isometric immersion of a Kähler-Einsteinor homogeneous Kähler-manifold into an Euclidean spacemust be totally geodesic. As an application, it is shown thatan open subset of the real hyperbolic plane RH² cannot be minimallyimmersed into the Euclidean space. As another application, aproof is given that if an irreducible Kähler manifold isminimally immersed in a Euclidean space, then its restrictedholonomy group must be U(n), where n = dim_CM. 2000 MathematicsSubject Classification 53B25 (primary); 53C42 (secondary).     

Cauchy problem for the essentially infinite-dimensional heat equation on a surface in a Hilbert space

It is proved that the Cauchy problem for a simple parabolic equation with essentially infinite-dimensional coefficients on bounded level surfaces of smooth functions in a Hilbert space is uniformly well posed.Published in Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 6, pp. 737–746, June, 1995.This work was supported by the Ukrainian State Committee on Science and Technology and (partially) by the International Science Foundation, Grant No. U44000.     

On almost contact metric hypersurface of a Kahler manifold

We have studied the normality of an almost contact metric hypersurface of a Kahler manifold. Quasi-umbilical and umbilical properties have also been studied.     

Linear waves at the free surface of a saturated porous media

The principal aim of this paper is to study the propagation of linear waves at the free surface of a saturated porous media. This problem is formulated as an eigenvalue problem with complex eigenvalues, and the solution is given in term of an orthogonal eigenfunction expansion, whose completeness has been taken for granted in the literature regarding the problem as a classical Sturm-Liouville problem, which is not the case due to the complex nature of the eigenvalues. The main purpose of the present work is to prove the completeness of the eigenfunctions for all possible physical values of the parameters involved, even for some values of the parameters, where previous numerical works have found abnormal behavior of the eigenvalues. In those cases if we mistakenly consider the problem as a Sturm-Liouville one, as has been done before, the eigenfunction expansion will not hold, but indeed we will prove that it does.On leave from Institute de Mecánica de los Fluidos, Universidad Central de Venezuela, Caracas, Venezuela.     

Galois closure of essentially finite morphisms

Let X be a reduced connected k-scheme pointed at a rational point x∈X(k). By using tannakian techniques we construct the Galois closure of an essentially finite k-morphism f:Y→X satisfying the condition H⁰(Y,O_Y)=k; this Galois closure is a torsor dominating f by an X-morphism and universal for this property. Moreover, we show that is a torsor under some finite group scheme we describe. Furthermore we prove that the direct image of an essentially finite vector bundle over Y is still an essentially finite vector bundle over X. We develop for torsors and essentially finite morphisms a Galois correspondence similar to the usual one. As an application we show that for any pointed torsor f:Y→X under a finite group scheme satisfying the condition H⁰(Y,O_Y)=k, Y has a fundamental group scheme π₁(Y,y) fitting in a short exact sequence with π₁(X,x).     

Systems of essentially infinite-dimensional differential equations

We investigate systems of differential equations with essentially infinite-dimensional elliptic operators (of the Laplace–Lévy type). For nonlinear systems, we prove theorems on the existence and uniqueness of solutions. For a linear system, we give an explicit formula for the solution.     

Variants of Robinson's essentially undecidable theoryR

Cobham has observed that Raphael Robinson's well known essentially undecidable theoryR remains essentially undecidable if the fifth axiom scheme is omitted. We note that whether the resulting system is in a sense minimal essentially undecidable depends on what the basic constants are taken to be. We give an essentially undecidable theory based on three axiom schemes involving only multiplication and less than or equals.     

On the existence of a local-integral manifold of neural type for an essentially nonlinear system of differential equations

An essentially nonlinear system of differential equations (i.e., a system without linear terms) is considered. It is proved that there exists a local-integral manifold of neutral type (central manifold) near an equilibrium point. Coefficient conditions for logarithmic norms are used.     

An algorithm for the local exhaustive search for alternatives in an essentially non-linear eigenvalue problem

An algorithm for the local exhaustive search for alternatives, which enable one to reduce the number of auxiliary linear eigenvalue problems that must be solved is illustrated using the example of the problem of the supercritical behaviour of a longitudinally compressed rod at the interface of two dissimilar Winkler media. The basic principle of the algorithm consists of developing a qualitatively adequate natural form based on a complete exhaustive search for alternatives on a “widely spaced” grid with further subsequent doubling of the number of its nodes and exhaustively searching for alternatives only in the region of the roots of the natural form.     

The essentially free spectrum of a variety

We partially prove a conjecture from [MeSh] which says that the spectrum of almost free, essentially free non-free algebras in a variety is either empty or consists of the class of all successor cardinals. The author was supported by the NSERC. Prof. Mekler died on June 10, 1992. The author is supported by the United States-Israel Binational Science Foundation; publication 417. The author is supported by the Schweizer Nationalfonds.     

THEOREMS OF BARTH-LEFSCHETZ TYPE ON KAHLER MANIFOLDS WITH PARTIALLY POSITIVE CURVATURE

The author obtains the theorems of Barth-Lefschetz type on Kahler manifolds with partially positive bisectional curvature without the assumption of nonnegative bisectional curvature. Some applications of the results to holomorphic mappings are given.