On monoids over which all strongly flat cyclic right acts are projective

A new characterization of monoids over which all strongly flat cyclic right acts are projective (projective generators, free) is given. This research has been supported by the Estonian Science Foundation, Grant No. 930.     

Perfect monoids revisited

A new characterization of perfect monoids, i.e., monoids over which every act has a projective cover, is given. As was shown by Fountain [1] a monoid is perfect if and only if all strongly flat acts over it are projective. Using our new condition, an alternative version is given of a recent result, of Liu [7] describing monoids over which all strongly flat right acts are projective generators, or are free. This research has been supported by the Estonian Science Foundation, Grant No. 930. I would like to thank the School of Mathematical and Computational Sciences of the University of St. Andrews for excellent working conditions.     

A robust sequential quadratic programming method

The sequential quadratic programming method developed by Wilson, Han and Powell may fail if the quadratic programming subproblems become infeasible, or if the associated sequence of search directions is unbounded. This paper considers techniques which circumvent these difficulties by modifying the structure of the constraint region in the quadratic programming subproblems. Furthermore, questions concerning the occurrence of an unbounded sequence of multipliers and problem feasibility are also addressed.Work supported in part by the National Science Foundation under Grant No. DMS-8602399 and by the Air Force Office of Scientific Research under Grant No. ISSA-860080.Work supported in part by the National Science Foundation under Grant No. DMS-8602419.     

Complexes of rings

Homology group of complexes of finitely generated projective modules are shown to be torsion groups, and a simplified proof of the vanishing of the cohomology groupsn≧3 of inseparable extensions is given. This work has been sponsored by NSF Grant No. 1964 while the author was at Northwestern University and by Grant AF EOAR 63-70 of the European Office of Aerospace Research, U.S. Air Force.     

Localization on certain Grothendieck categories

Using categorical techniques we obtain some results on localization and colocalization theory in Grothendieck categories with a set of small projective generators. In particular, we give a sufficient condition for such category to be semiartinian. For semiartinian Grothendieck categories where every simple object has a projective cover, we obtain that every localizing subcategory is a TTF-class. In addition, some applications to semiperfect categories are obtained.     

Global dimension and left derived functors of Hom

It is well known that the right global dimension of a ring R is usually computed by the right derived functors of Hom and the left projective resolutions of right R-modules. In this paper, for a left coherent and right perfect ring R, we characterize the right global dimension of R, from another point of view, using the left derived functors of Hom and the right projective resolutions of right R-modules. It is shown that rD(R)≤n (n≥2) if and only if the gl right Proj-dim MR≤n - 2 if and only if Extn-1(N, M) = 0 for all right R-modules N and M if and only if every (n - 2)th Proj-cosyzygy of a right R-module has a projective envelope with the unique mapping property. It is also proved that rD(R)≤n (n≥1) if and only if every (n-1)th Proj-cosyzygy of a right R-module has an epic projective envelope if and only if every nth Vroj-cosyzygy of a right R-module is projective. As corollaries, the right hereditary rings and the rings R with rD(R)≤2 are characterized.     

Complete intersection dimension

A new homological invariant is introduced for a finite module over a commutative noetherian ring: its CI-dimension. In the local case, sharp quantitative and structural data are obtained for modules of finite CI-dimension, providing the first class of modules of (possibly) infinite projective dimension with a rich structure theory of free resolutions. The first author was partly supported by NSF Grant No. DMS-9102951.     

On Faith's correspondence theorem

A simple proof of an extension of Faith's correspondence theorem for projective modules is given for a Morita context (R, V, W, S) in whichVWV=V andWVW=W.Research supported by N.S.E.R.C. (Canada), Grant No. A 8075.     

On two numerical-analytic methods for the investigation of periodic solutions

We discuss two techniques useful in the investigation of periodic solutions of broad classes of non-linear non-autonomous ordinary differential equations, namely the trigonometric collocation and the method based upon periodic successive approximations. Supported in part by the Hungarian NFSR OTKA through Grant No. K68311.     

An efficient one—dimensional search procedure for barrier functions

In minimizing interior penalty functions, most of the computational time is spent on the one-dimensional search. This paper presents a method for performing this search on barrier functions which is significantly faster than current techniques. The method exploits the special structure of barrier functions. Comparative computational results are given for a set of six test problems.This research was partially supported by the National Aeronautics and Space Administration under Research Grant NSG 110-61 and by the Office of Naval Research under Grant No. N00014-67-A-0404-0010.     

Projective deformation and biholomorphic equivalence of real hypersurfaces

The concepts of first order projective deformation, biholomorphic equivalence, and equivalence of induced Cauchy-Riemann structure are all equivalent for real analytic hypersurfaces in complex projective space. Studying the first concept leads to a realization of the Cauchy-Riemann structure bundle as a submanifold of the projective group. The Chern-Moser connection on this bundle can then be given in terms of the Maurer-Cartan form of the projective group, and equations analogous to the Gauss equations of Euclidean geometry give the Chern-Moser invariants. Research supported by the National Science Foundation under Grant Number MCS-8103370.     

Convergence of Newton‘’s Method and Uniqueness of the Solution of Equations in Banach SpacesⅡ

Some results on convergence of Newton‘s method in Banach spaces are established under the assumption that the derivative of the opderators satisfies the radius or center Lipschitz condition with a weak L average.     

Linear representation of a class of projective planes in a four dimensional projective space

Summary The theory of linear representations of projective planes developed by Bruck and one of the authors (Bose) in two earlier papers [J. Algebra1 (1964), pp. 85–102 and4 (1966), pp. 117–172] can be further extended by generalizing the concept of incidence adopted there. A linear representation is obtained for a class of non-Desarguesian projective planes illustrating this concept of generalized incidence. It is shown that in the finite case, the planes represented by the new construction are derived planes in the sense defined by Ostrom [Trans. Amer. Math Soc.111 (1964), pp. 1–18] and Albert [Boletin Soc. Mat. Mex,11 (1966), pp, 1–13] of the dual of translation planes which can be represented in a 4-space by the Bose-Bruck construction. An analogous interpretation is possible for the infinite case. This research was sponsored by the National Science Foundation under Grant No. GP-8624, and the U.S. Air Force Office of Scientific Research under Grant No. AFOSR-68-1406. This research was conducted while the author was visiting professor at the University of North Carolina at Chapel Hill. His research was also partially supported by C.N.R. Entrata in Redazione il 28 maggio 1970.     

On structure of small contractions of odd dimensional projective varieties

Let X be a smooth projective variety of dimension 2k-1 (k≥3) over the complex number field. Assume that fR: X→Y is a small contraction such that every irreducible component Ei of the exceptional locus of fR is a smooth subvariety of dimension k. It is shown that each Ei is isomorphic to the k-dimensional projective space Pk, the k-dimensional hyperquadric surface Qk in Pk 1, or a linear Pk-1-bundle over a smooth curve.     

On homologies of finite projective planes

Study of finite projective planes which have a polarity preserved by many homologies. This research was supported in part by NSF Grant GP-33223.     

The synthesis of optimal controls for linear,time-optimal problems with retarded controls

Optimization problems involving linear systems with retardations in the controls are studied in a systematic way. Some physical motivation for the problems is discussed. The topics covered are: controllability, existence and uniqueness of the optimal control, sufficient conditions, techniques of synthesis, and dynamic programming. A number of solved examples are presented.This research was supported in part by the National Aeronautics and Space Administration under Grant No. NGL-40-002-015, in part by the Air Force Office of Scientific Research under Grant No. AF-AFOSR-693-67B, and in part by the National Science Foundation under Grant No. GP-15132.     

Homogeneous linear differential equations with only periodic solutions

We determine all third order homogeneous linear differential equations with periodic coefficients and only periodic solutions. The method extends tonth order equations. As an application, we show that the Laguerre-Forsyth canonical form cannot be used for global investigations i projective differential geometry. Research supported by NSF Grant GP-8176. To the Memory of Eri Jabotinsky     

Filtrations, mixed complexes, and cyclic homology in mixed characteristic

We describe an approach to computing cyclic homology suited to ground rings of characteristicp or mixed characteristic. We illustrate the method by computing HC for coordinate rings of affine plane curves and for maximal orders of semisimple algebras.Partially supported by NSF Grant No. DMS-8807203 and NSA Grant No. MDA 904-92-H-3026.     

Harmonic maps between complex projective spaces

Using the DoCarmo-Wallach theory, we classify (homogeneous) polynomial harmonic maps of complex projective spaces into spheres and complex projective spaces in terms of finite dimensional moduli spaces. We make use of representation theory of the (special) unitary group to give, for a spherical range, the exact dimension and, for complex projective spaces, a lower bound of the dimension of the moduli spaces.Supported in part by NSF Grant DMS 8603172.Research done in part while the third named author was on leave at University of California, Berkeley.     

On Anstreicher's combined phase I—phase II projective algorithm for linear programming

Anstreicher has proposed a variant of Karmarkar's projective algorithm that handles standard-form linear programming problems nicely. We suggest modifications to his method that we suspect will lead to better search directions and a more useful algorithm. Much of the analysis depends on a two-constraint linear programming problem that is a relaxation of the scaled original problem.Research supported in part by NSF Grant ECS-8602534 and ONR Contract N00014-87-K-0212.