A Canonical Compatible Metric for Geometric Structures on Nilmanifolds

Let (N, γ) be a nilpotent Lie group endowed with an invariant geometric structure (cf. symplectic, complex, hypercomplex or any of their ‘almost’ versions). We define a left invariant Riemannian metric on N compatible with γ to be minimal, if it minimizes the norm of the invariant part of the Ricci tensor among all compatible metrics with the same scalar curvature. We prove that minimal metrics (if any) are unique up to isometry and scaling, they develop soliton solutions for the ‘invariant Ricci’ flow and are characterized as the critical points of a natural variational problem. The uniqueness allows us to distinguish two geometric structures with Riemannian data, giving rise to a great deal of invariants.Our approach proposes to vary Lie brackets rather than inner products; our tool is the moment map for the action of a reductive Lie group on the algebraic variety of all Lie algebras, which we show to coincide in this setting with the Ricci operator. This gives us the possibility to use strong results from geometric invariant theory.Communicated by: Nigel Hitchin (Oxford) Mathematics Subject Classifications (2000): Primary: 53D05, 53D55; Secondary: 22E25, 53D20, 14L24, 53C30.     

Jensen's inequality for spectral order and submajorization

Let A be a C^∗-algebra and be a positive unital map. Then, for a convex function defined on some open interval and a self-adjoint element a∈A whose spectrum lies in I, we obtain a Jensen's-type inequality f(?(a))??(f(a)) where ? denotes an operator preorder (usual order, spectral preorder, majorization) and depends on the class of convex functions considered, i.e., monotone convex or arbitrary convex functions. Some extensions of Jensen's-type inequalities to the multi-variable case are considered.     

Illuminating high-dimensional convex sets

A setL of points in thed-spaceE ^d is said toilluminate a familyF={S ₁, ...,S _n } ofn disjoint compact sets inE ^d if for every setS _i inF and every pointx in the boundary ofS _i there is a pointv inL such thatv illuminatesx, i.e. the line segment joiningv tox intersects the union of the elements ofF in exactly {x}.The problem we treat is the size of a setS needed to illuminate a familyF={S ₁, ...,S _n } ofn disjoint compact sets inE ^d . We also treat the problem of putting these convex sets in mutually disjoint convex polytopes, each one having at most a certain number of facets.     

Scheduling tasks with communication delays on parallel processors

LetP={v ₁,...,v _n } be a set ofn jobs to be executed on a set ofm identical machines. In many instances of scheduling problems, if a jobv _i has to be executed before the jobv _j and both jobs are to be executed on different machines, some sort of information exchange has to take place between the machines executing them. The time it takes for this exchange of information is called a communication delay.In this paper we give anO(n) algorithm to find an optimal scheduling with communication delays when the number of machines is not limited and the precedence constraints on the jobs form a tree.     

A biased random-key genetic algorithm with forward-backward improvement for the resource constrained project scheduling problem

This paper presents a biased random-key genetic algorithm for the resource constrained project scheduling problem. The chromosome representation of the problem is based on random keys. Active schedules are constructed using a priority-rule heuristic in which the priorities of the activities are defined by the genetic algorithm. A forward-backward improvement procedure is applied to all solutions. The chromosomes supplied by the genetic algorithm are adjusted to reflect the solutions obtained by the improvement procedure. The heuristic is tested on a set of standard problems taken from the literature and compared with other approaches. The computational results validate the effectiveness of the proposed algorithm.     

Electrical turbulence as a result of the critical curvature for propagation in cardiac tissue

In cardiac tissue, the propagation of electrical excitation waves is dependent on the active properties of the cell membrane (ionic channels) and the passive electrical properties of cardiac tissue (passive membrane properties, distribution of gap junctions, and cell shapes). Initiation of cardiac arrhythmias is usually associated with heterogeneities in the active and/or passive properties of cardiac tissue. However, as a result of the effect of wave front geometry (curvature) on propagation of cardiac waves, inexcitable anatomical obstacles, like veins and arteries, may cause the formation of self-sustained vortices and uncontrolled high-frequency excitation in normal homogeneous myocardium. (c) 1998 American Institute of Physics.     

Sobolev estimates for (pseudo)‐differential operators and applications

In this work we show that if is a linear differential operator of order ν with smooth complex coefficients in from a complex vector space E to a complex vector space F, the Sobolev a priori estimate holds locally at any point if and only if is elliptic and the constant coefficient homogeneous operator is canceling in the sense of Van Schaftingen for every which means that Here is the homogeneous part of order ν of and is the principal symbol of . This result implies and unifies the proofs of several estimates for complexes and pseudo‐complexes of operators of order one or higher proved recently by other methods as well as it extends —in the local setup— the characterization of Van Schaftingen to operators with variable coefficients.     

Noncommutative Jordan C*-algebras

We introduce noncommutative JB*-algebras which generalize both B*-algebras and JB*-algebras and set up the bases for a representation theory of noncommutative JB*-algebras. To this end we define noncommutative JB*-factors and study the factor representations of a noncommutative JB*-algebra. The particular case of alternative B*-factors is discussed in detail and a Gelfand-Naimark theorem for alternative B*-algebras is given.     

Computation of Almost Split Sequences with Applications to Relatively Projective and Prinjective Modules

Let be an Artin algebra, let mod be the category of finitely generated -modules, and let Amod be a contravariantly finite and extension closed subcategory. For an indecomposable and not Ext-projective module CA, we compute the almost split sequence 0ABC0 in A from the almost split sequence 0DTrCEC0 in mod. Since the computation is particularly simple if the minimal right A-approximation of DTrC is indecomposable for all indecomposable and not Ext-projective CA, we manufacture subcategories A with the desired property using orthogonal subcategories. The method of orthogonal subcategories is applied to compute almost split sequences for relatively projective and prinjective modules.