Special Lagrangian Submanifolds with Isolated Conical Singularities. II. Moduli spaces

This is the second in a series of five papers studying special Lagrangiansubmanifolds (SLV m-folds) X in (almost) Calabi–Yau m-folds M with singularities x ₁ , ..., x _n locally modelled on specialLagrangian cones C ₁, ..., C _n in ^m with isolated singularities at 0.Readers are advised to begin with Paper V.This paper studies the deformation theory of compact SL m-folds X in Mwith conical singularities. We define the moduli space _X of deformations of X in M, and construct a natural topology on it. Then we show that _X is locally homeomorphic to the zeroes of a smooth map : _{X X} between finite-dimensional vector spaces.Here the infinitesimal deformation space _X depends only on the topology of X, and the obstruction space _X only on the cones C ₁, ..., C _n at x ₁, ..., x _n . If the cones C _i are stable then _X is zero, and _X is a smooth manifold. We also extend our results to families of almost Calabi–Yau structures on M.     

Four-Dimensional Terminal Gorenstein Quotient Singularities

In the present paper, we classify the finite subgroups such that the quotient by the action G has only isolated terminal Gorenstein singularities.     

Special Lagrangian Submanifolds with Isolated Conical Singularities. III. Desingularization,The Unobstructed Case

This is the third in a series of five papers studying special Lagrangian submanifolds(SLV m-folds) X in (almost) Calabi–Yau m-folds M with singularities x ₁, ..., x _n locally modelled on special Lagrangian cones C ₁, ..., C _n in ^m with isolated singularities at 0. Readers are advised to begin with Paper V.This paper and Paper IV construct desingularizations of X, realizing X as a limitof a family of compact, nonsingular SL m-folds ^t in M for small t > 0.Suppose L ₁, ..., L _n are Asymptotically Conical SL m-folds in ^m , with L _i asymptotic to the cone C _i at infinity. We shrink L _i by a small t > 0, and glue tL _i into X at x _i for i= 1, ..., nto get a 1-parameter family of compact, nonsingular Lagrangian m-folds N ^t for small t > 0.Then we show using analysis that when t is sufficiently small we can deform N ^t to a compact,nonsingular special Lagrangian m-fold ^t , via a small Hamiltonian deformation.This ^t depends smoothly on t, and as t 0 it converges to the singular SL m-fold X, in the sense of currents.This paper studies the simpler cases, where by topological conditions on X and L _i we avoid various obstructions to existence of ^t . Paper IV will consider more complex cases when these obstructions are nontrivial, and also desingularization in families of almost Calabi–Yau m-folds.     

Convex Subgroups of Partially Right-Ordered Groups

We study into the question of whether a partial order can be induced from a partially right-ordered group onto a space of right cosets of w.r.t. some subgroup of . Examples are constructed showing that the condition of being convex for in is insufficient for this. A necessary and sufficient condition (in terms of a subgroup and a positive cone of ) is specified under which an order of can be induced onto . Sufficient conditions are also given. We establish properties of the class of partially right-ordered groups for which is partially ordered for every convex subgroup , and properties of the class of groups such that is partially ordered for every partial right order on and every subgroup that is convex under .     

Transversal Twistor Spaces of Foliations

The transversal twistor space of a foliation of an even codimension is the bundle of the complex structures of the fibers of the transversalbundle of . On there exists a foliation by covering spaces of the leaves of , and any Bottconnection of produces an ordered pair of transversal almost complex structures of . The existence of a Bott connection which yields a structure ₁ that is projectable to the space of leaves isequivalent to the fact that is a transversallyprojective foliation. A Bott connection which yields a projectablestructure ₂ exists iff isa transversally projective foliation which satisfies a supplementarycohomological condition, and, in this case, ₁is projectable as well. ₂ is never integrable.The essential integrability condition of ₁ isthe flatness of the transversal projective structure of .     

Topological and bornological characterisations of ideals in von Neumann algebras: I

Suppose is a von Neumann algebra on a Hilbert space and is any ideal in . We determine a topology on , for which the members of that are to norm continuous are exactly those in ; and a bornology on such that the elements of which map the unit ball to an element of , equivalently those members of that are norm to bounded, are exactly those in . This is achieved via analogues of the notions of injectivity and surjectivity in the theory of operator ideals on Banach spaces.     

Absolute Continuity of the Spectrum of a Periodic Schrödinger Operator

We prove the absolute continuity of the spectrum of the Schrödinger operator in , , with periodic (with a common period lattice ) scalar and vector potentials for which either , , or the Fourier series of the vector potential converges absolutely, , where is an elementary cell of the lattice , for , and for , and the value of is sufficiently small, where and otherwise, , and .     

Trajectory and Global Attractors of Three-Dimensional Navier--Stokes Systems

We construct the trajectory attractor of a three-dimensional Navier--Stokes system with exciting force . The set consists of a class of solutions to this system which are bounded in , defined on the positive semi-infinite interval of the time axis, and can be extended to the entire time axis so that they still remain bounded-in- solutions of the Navier--Stokes system. In this case any family of bounded-in- solutions of this system comes arbitrary close to the trajectory attractor . We prove that the solutions are continuous in t if they are treated in the space of functions ranging in . The restriction of the trajectory attractor to , , is called the global attractor of the Navier--Stokes system. We prove that the global attractor thus defined possesses properties typical of well-known global attractors of evolution equations. We also prove that as the trajectory attractors and the global attractors of the -order Galerkin approximations of the Navier--Stokes system converge to the trajectory and global attractors and , respectively. Similar problems are studied for the cases of an exciting force of the form depending on time and of an external force rapidly oscillating with respect to the spatial variables or with respect to time .     

Joint Approximations of Distributions in Banach Spaces

For a given homogeneous elliptic partial differential operator with constant complex coefficients, two Banach spaces and of distributions in , and compact sets and in , we study joint approximations in the norms of the spaces and (the spaces of Whitney jet-distributions) by the solutions of the equation in neighborhoods of the set . We obtain a localization theorem, which, under certain conditions, allows one to reduce the above-cited approximation problem to the corresponding separate problems in each of the spaces.     

Estimates of Marcinkiewicz Integrals with Bounded Homogeneous Kernels of Degree Zero

Under the cancellation property and a certain Dini-type condition on kernels, we prove that Marcinkiewicz integrals with kernels which are homogeneous functions of degree zero, are bounded from to , from to , and from to for .     

Basic subgroups in abelian group rings

Suppose is a commutative ring with identity of prime characteristic and is an arbitrary abelian -group. In the present paper, a basic subgroup and a lower basic subgroup of the -component and of the factor-group of the unit group in the modular group algebra are established, in the case when is weakly perfect. Moreover, a lower basic subgroup and a basic subgroup of the normed -component and of the quotient group are given when is perfect and is arbitrary whose is -divisible. These results extend and generalize a result due to Nachev (1996) published in Houston J. Math., when the ring is perfect and is -primary. Some other applications in this direction are also obtained for the direct factor problem and for a kind of an arbitrary basic subgroup.     

The Ideal Structure of Toeplitz Algebras

We investigate the ideal structure of the Toeplitz algebra of a totally ordered abelian group . We show that the primitive ideals of are parametrised by the disjoint union of the duals of the order ideals of , and identify the hull-kernel topology on when the chain of orderideals in is isomorphic to a subset of      

Application of Upper Hemi-Continuous Operators on Generalized Bi-quasi-variational Inequalities in Locally Convex Topological Vector Spaces

Let and be Hausdorff topological vector spaces over the field , let be a bilinear functional, and let be a non-empty subset of . Given a set-valued map and two set-valued maps , the generalized bi-quasi-variational inequality (GBQVI) problem is to find a point and a point such that and for all and for all or to find a point a point and a point such that and for all . The generalized bi-quasi-variational inequality was introduced first by Shih and Tan [8] in 1989. In this paper we shall obtain some existence theorems of generalized bi-quasi-variational inequalities as application of upper hemi-continuous operators [4] in locally convex topological vector spaces on compact sets.     

Commutativity of the Arens product in lattice ordered algebras

Let be an Abelian Archimedean lattice ordered algebra. The order bidual furnished with the Arens product is again a lattice ordered algebra. We show that the order continuous order bidual is Abelian. This solves an open problem and improves a result of Scheffold, who proved it for the case of normed lattice ordered algebras. The proof is based on the up-down-up approximation of positive elements in the order continuous order bidual by elements in the canonical image of in Components of positive elements in are characterized and the result is applied to the Arens product of -and almost -algebras.     

On the Best Ranges for A p + and RH r +

In this paper we study the relationship between one-sided reverse Holder classes and the classes. We find the best possible range of to which an weight belongs, in terms of the constant. Conversely, we also find the best range of to which a weight belongs, in terms of the constant. Similar problems for , and are solved using factorization.     

Bernstein-Type Theorems and Uniqueness Theorems

Let be an entire function of finite type with respect to finite order and let be a subset of an open cone in a certain n-dimensional subspace (the smaller , the sparser ). We assume that this cone contains a ray 0} \right\}$$ " align="middle" border="0"> . It is shown that the radial indicator of at any point may be evaluated in terms of function values at points of the discrete subset . Moreover, if tends to zero fast enough as over , then this function vanishes identically. To prove these results, a special approximation technique is developed. In the last part of the paper, it is proved that, under certain conditions on and , which are close to exact conditions, the function bounded on is bounded on the ray.     

A Version of the Ruh--Vilms Theorem for Surfaces of Constant Mean Curvature in S 3

We study a version of the Gauss map for a surface immersed in and prove an analog of the Ruh--Vilms theorem which states that this map is harmonic iff has a constant mean curvature. As a corollary, we conclude that an embedded flat torus with constant mean curvature is a spherical Delonay surface.     

Spin Models and Strongly Hyper-Self-Dual Bose-Mesner Algebras

We introduce the notion of hyper-self-duality for Bose-Mesner algebras as a strengthening of formal self-duality. Let denote a Bose-Mesner algebra on a finite nonempty set X. Fix p X, and let and denote respectively the dual Bose-Mesner algebra and the Terwilliger algebra of with respect to p. By a hyper-duality of , we mean an automorphism of such that for all ; and is a duality of . is said to be hyper-self-dual whenever there exists a hyper-duality of . We say that is strongly hyper-self-dual whenever there exists a hyper-duality of which can be expressed as conjugation by an invertible element of . We show that Bose-Mesner algebras which support a spin model are strongly hyper-self-dual, and we characterize strong hyper-self-duality via the module structure of the associated Terwilliger algebra.     

Some New Extrapolation Estimates for the Scale of Spaces

Using new extrapolation estimates for the - and -functionals of couples of limit spaces of the -scale , we introduce a class of extrapolation functors. A characterization of this class via the real interpolation method permits one to obtain new equivalent expressions for the norms in symmetric spaces close to and , which depend only on the -norms of a function.     

Imprimitive Q-polynomial Association Schemes

It is well known that imprimitive P-polynomial association schemes with are either bipartite or antipodal, i.e., intersection numbers satisfy either for all for all . In this paper, we show that imprimitive -polynomial association schemes with are either dual bipartite or dual antipodal, i.e., dual intersection numbers satisfy either .