Globally generated vector bundles of rank 2 on a smooth quadric threefold

We investigate the existence of globally generated vector bundles of rank 2 with _c1≤3$c_1\le 3$ on a smooth quadric threefold and determine their Chern classes. As an automatic consequence, every rank 2 globally generated vector bundle on Q$Q$ with _c1=3$c_1=3$ is an odd instanton up to twist.     

Generating pairs and group actions

An action of a finite group on a closed 2-manifold is called almost free if it has a single orbit of points with nontrivial stabilizers. It is called large when the order of the group is greater than or equal to the genus of the surface. We prove that the orientation-preserving large almost free actions of G on closed orientable surfaces correspond to the Nielsen equivalence classes of generating pairs of G  . We classify the almost free actions on the surfaces of genera 3 and 4, find the large almost free actions of the alternating group _A5$A_5$, and give various other examples.     

Orthogonal Pfister involutions in characteristic two

We show that over a field of characteristic 2 a central simple algebra with orthogonal involution that decomposes into a product of quaternion algebras with involution is either anisotropic or metabolic. We use this to define an invariant of such orthogonal involutions that completely determines the isotropy behaviour of the involution. We also give an example of a non-totally decomposable algebra with orthogonal involution that becomes totally decomposable over every splitting field of the algebra.     

Logical omniscience as infeasibility

Logical theories for representing knowledge are often plagued by the so-called Logical Omniscience Problem. The problem stems from the clash between the desire to model rational agents, which should be capable of simple logical inferences, and the fact that any logical inference, however complex, almost inevitably consists of inference steps that are simple enough. This contradiction points to the fruitlessness of trying to solve the Logical Omniscience Problem qualitatively if the rationality of agents is to be maintained. We provide a quantitative solution to the problem compatible with the two important facets of the reasoning agent: rationality and resource boundedness. More precisely, we provide a test for the logical omniscience problem in a given formal theory of knowledge. The quantitative measures we use are inspired by the complexity theory. We illustrate our framework with a number of examples ranging from the traditional implicit representation of knowledge in modal logic to the language of justification logic, which is capable of spelling out the internal inference process. We use these examples to divide representations of knowledge into logically omniscient and not logically omniscient, thus trying to determine how much information about the reasoning process needs to be present in a theory to avoid logical omniscience.     

Reverse mathematics and initial intervals

In this paper we study the reverse mathematics of two theorems by Bonnet about partial orders. These results concern the structure and cardinality of the collection of initial intervals. The first theorem states that a partial order has no infinite antichains if and only if its initial intervals are finite unions of ideals. The second one asserts that a countable partial order is scattered and does not contain infinite antichains if and only if it has countably many initial intervals. We show that the left to right directions of these theorems are equivalent to _ACA0${\mathsf{ACA}}_0$ and _ATR0${\mathsf{ATR}}_0$, respectively. On the other hand, the opposite directions are both provable in _WKL0${\mathsf{WKL}}_0$, but not in _RCA0${\mathsf{RCA}}_0$. We also prove the equivalence with _ACA0${\mathsf{ACA}}_0$ of the following result of Erdös and Tarski: a partial order with no infinite strong antichains has no arbitrarily large finite strong antichains.     

Non-annulation effective et positivité locale des fibrés en droites amples adjoints

We show how to use effective non-vanishing to prove that Seshadri constants of some ample divisors are bigger than 1 on smooth threefolds whose anticanonical bundle is nef or on Fano varieties of small coindice. We prove the effective non-vanishing conjecture of Ionescu–Kawamata in dimension 3 in the case of line bundles of “high” volume.     

Continuation of multiple power series in a sectorial domain

For multiple power series centered at the origin we consider the problem of its analytic continuability into a sectorial domain. The condition for continuability is formulated in terms of a holomorphic function that interpolates the series coefficients. For series in one variable this problem has been studied in the works of E. Lindelöf, N. Arakelian, and others.     

An iterative substructuring algorithm for equilibrium equations

Summary The topic of iterative substructuring methods, and more generally domain decomposition methods, has been extensively studied over the past few years, and the topic is well advanced with respect to first and second order elliptic problems. However, relatively little work has been done on more general constrained least squares problems (or equivalent formulations) involving equilibrium equations such as those arising, for example, in realistic structural analysis applications. The potential is good for effective use of iterative algorithms on these problems, but such methods are still far from being competitive with direct methods in industrial codes. The purpose of this paper is to investigate an order reducing, preconditioned conjugate gradient method proposed by Barlow, Nichols and Plemmons for solving problems of this type. The relationships between this method and nullspace methods, such as the force method for structures and the dual variable method for fluids, are examined. Convergence properties are discussed in relation to recent optimality results for Varga's theory ofp-cyclic SOR. We suggest a mixed approach for solving equilibrium equations, consisting of both direct reduction in the substructures and the conjugate gradient iterative algorithm to complete the computations.Dedicated to R. S. Varga on the occasion of his 60th birthdayResearch completed while pursuing graduate studies sponsored by the Department of Mathematical Sciences, US Air Force Academy, CO, and funded by the Air Force Institute of Technology, WPAFB, OHResearch supported by the Air Force under grant no. AFOSR-88-0285 and by the National Science Foundation under grant no. DMS-89-02121     

Interval method for bounding level sets: Revisited and tested with global optimization problems

An interval method for bounding level sets, modified to increase its efficiency and to get sharper bounding boxes, is presented. The new algorithm was tested with standard global optimization test problems. The test results show that, while the modified method gives a more valuable, guaranteed reliability result set, it is competitive with non-interval methods in terms of CPU time and number of function evaluations.This work was supported by Grant OTKA 1074/1987, and in part by DAAD Fellowship No. 314/108/004/8 during the author's stay at Düsseldorf University.     

Preconditionings and splittings for rectangular systems

Summary Recently Eiermann, Marek, and Niethammer have shown how to applysemiiterative methods to a fixed point systemx=Tx+c which isinconsistent or in whichthe powers of the fixed point operator T have no limit, to obtain iterative methods which converge to some approximate solution to the fixed point system. In view of their results we consider here stipulations on apreconditioning QAx=Qb of the systemAx=b and, separately, on asplitting A=M–N which lead to fixed point systems such that, with the aid of a semiiterative method, the iterative scheme converges to a weighted Moore-Penrose solution to the systemAx=b. We show in several ways that to obtain a meaningful limit point from a semiiterative method requires less restrictions on the splittings or the reconditionings than those which have been required in the classical Picard iterative method (see, e.g., the works of Berman and Plemmons, Berman and Neumann, and Tanabe).We pay special attention to the case when the weighted Moore-Penrose solution which is sought is the minimal norm least squares solution toAx=b.Research supported by the Deutsche ForschungsgemeinschaftPartially supported by AFOSR research grant 88-0047