Non-circular,non-well-founded set universes

We show that there are universes of sets which contain descending ?-sequences of length α for every ordinal α, though they do not contain any ?-cycle. It is also shown that there is no set universe containing a descending ?-sequence of length On. MSC: 03E30; 03E65.     

A note on a theorem of Kanovei

We give a short proof of a theorem of Kanovei on separating induction and collection schemes for _n formulas using families of subsets of countable models of arithmetic coded in elementary end extensions.Mathematics Subject Classification (2000): 03C62     

Dependent Choices and Anti‐Foundation

In Zermelo‐Fraenkel set theory without the Axiom of Foundation we study the schema version of the principle of dependent choices in connection with Aczel's antifoundation axiom (a part of it), Boffa's anti‐foundation axiom, and axiom of collection.     

Remarks on Levy's reflection axiom

Adding higher types to set theory differs from adding inaccessible cardinals, in that higher type arguments apply to all sets rather than just ordinary ones. Levy's reflection axiom is justified, by considering the principle that we can pretend that the universe is a set, together with methods of Gaifman [8]. We reprove some results of Gaifman, and some facts about Levy's reflection axiom, including the fact that adding higher types yields no new theorems about sets. Some remarks on standard models are made. An obvious strengthening of Levy's axiom to higher types is considered, which implies the existence of indescribable cardinals. Other remarks about larger cardinals are made; some questions of Gloede [9] are settled. Finally we argue that the evidence for V = L is strong, and that CH is certainly true. MSC: 03E30, 03E55.     

A Note on Relative Efficiency of Axiom Systems

We introduce a notion of relative efficiency for axiom systems. Given an axiom system A_β for a theory T consistent with S¹₂, we show that the problem of deciding whether an axiom system A_α for the same theory is more efficient than A_β is II₂-hard. Several possibilities of speed-up of proofs are examined in relation to pairs of axiom systems A_α, A_β, with A_α ? A_β, both in the case of A_α, A_β having the same language, and in the case of the language of A_α extending that of A_β: in the latter case, letting Pr_α, Pr_β denote the theories axiomatized by A_α, A_β, respectively, and assuming Pr_α to be a conservative extension of Pr_β, we show that if A_α — A_β contains no nonlogical axioms, then A_α can only be a linear speed-up of A_β; otherwise, given any recursive function g and any A_β, there exists a finite extension A_α of A_β such that A_α is a speed-up of A_β with respect to g. Mathematics Subject Classification: 03F20, 03F30.     

A note on effective ultrapowers: Uniform failure of bounded collection

By suitably adapting an argument of Hirschfeld (see [2, Chapter 9]), we show that there is a single Δ₁ formula that defeats “bounded collection” for any model of II₂ Arithmetic that is either a recursive ultrapower or an existentially complete model. Some related facts are noted. MSC: 03F30, 03C62.     

A syzygetic approach to the smoothability of zero-dimensional schemes

We consider the question of which zero-dimensional schemes deform to a collection of distinct points; equivalently, we ask which Artinian k-algebras deform to a product of fields. We introduce a syzygetic invariant which sheds light on this question for zero-dimensional schemes of regularity two. This invariant imposes obstructions for smoothability in general, and it completely answers the question of smoothability for certain zero-dimensional schemes of low degree. The tools of this paper also lead to other results about Hilbert schemes of points, including a characterization of nonsmoothable zero-dimensional schemes of minimal degree in every embedding dimension d?4.     

Expanding the Applicability of High-Order Traub-Type Iterative Procedures

We propose a collection of hybrid methods combining Newton’s method with frozen derivatives and a family of high-order iterative schemes. We present semilocal convergence results for this collection on a Banach space setting. Using a more precise majorizing sequence and under the same or weaker convergence conditions than the ones in earlier studies, we expand the applicability of these iterative procedures.     

Convergent difference schemes for nonlinear parabolic equations and mean curvature motion

Summary. Explicit finite difference schemes are given for a collection of parabolic equations which may have all of the following complex features: degeneracy, quasilinearity, full nonlinearity, and singularities. In particular, the equation of “motion by mean curvature” is included. The schemes are monotone and consistent, so that convergence is guaranteed by the general theory of approximation of viscosity solutions of fully nonlinear problems. In addition, an intriguing new type of nonlocal problem is analyzed which is related to the schemes, and another very different sort of approximation is presented as well. Received January 10, 1995     

On efficiency of nonmonotone Armijo-type line searches

Monotonicity and nonmonotonicity play a key role in studying the global convergence and the efficiency of iterative schemes employed in the field of nonlinear optimization, where globally convergent and computationally efficient schemes are explored. This paper addresses some features of descent schemes and the motivation behind nonmonotone strategies and investigates the efficiency of an Armijo-type line search equipped with some popular nonmonotone terms. More specifically, we propose two novel nonmonotone terms, combine them into Armijo’s rule and establish the global convergence of sequences generated by these schemes. Furthermore, we report extensive numerical results and comparisons indicating the performance of the nonmonotone Armijo-type line searches using the most popular search directions on the CUTEst test collection of unconstrained problems. We finally apply the considered nonmonotone schemes to a deblurring problem to recover a blurred/noisy image.     

A Survey of Applications of Markov Decision Processes

A collection of papers on the application of Markov decision processes is surveyed and classified according to the use of real life data, structural results and special computational schemes. Observations are made about various features of the applications.     

Optimal scaling of balls and polyhedra

The concern is with solving as linear or convex quadratic programs special cases of the optimal containment and meet problems. The optimal containment or meet problem is that of finding the smallest scale of a set for which some translation contains a set or meets each element in a collection of sets, respectively. These sets are unions or intersections of cells where a cell is either a closed polyhedral convex set or a closed solid ball.This work was supported in part by the Department of Energy Contract DE-AC03-76-SF00326, PA No. DE-AT-03-76ER72018; National Science Foundation Grants MCS79-03145 and SOC78-16811; and the Army Research Office—Durham, Contract DAAG-29-78-G-0026.     

A Gap Cohomology Group

Dan Talayco has recently defined the gap cohomology group of a tower in p(ω)/fin of height ω₁. This group is isomorphic to the collection of gaps in the tower modulo the equivalence relation given by two gaps being equivalent (cohomologous) if their levelwise symmetric difference is not a gap in the tower, the group operation being levelwise symmetric difference. Talayco showed that the size of this group is always at least 2^N0 and that it attains its greatest possible size, 2^N1, if ? holds and also in some generic extensions in which CH fails, for example on adding many Cohen or random reals. In this paper it is shown that there is always some tower whose gap cohomology group has size 2^N1. It is still open as to whether there are models in which there are towers whose gap cohomology group has size less than 2^ω1.     

On the structure of kripke models of heyting arithmetic

Since in Heyting Arithmetic (HA) all atomic formulas are decidable, a Kripke model for HA may be regarded classically as a collection of classical structures for the language of arithmetic, partially ordered by the submodel relation. The obvious question is then: are these classical structures models of Peano Arithmetic (PA)? And dually: if a collection of models of PA, partially ordered by the submodel relation, is regarded as a Kripke model, is it a model of HA? Some partial answers to these questions were obtained in [6], [3], [1] and [2]. Here we present some results in the same direction, announced in [7]. In particular, it is proved that the classical structures at the nodes of a Kripke model of HA must be models of IΔ₁ (PA^- with induction for provably Δ₁ formulas) and that the relation between these classical structures must be that of a Δ₁-elementary submodel. MSC: 03F30, 03F55.     

格上点式一致结构的刻划与点式度量化定理

本文通过对格上远域映射,特别是△-映射和＊-映射的性质的研究,给出了格上点式一致结构的若干简明刻划,证明了几个重要的格上点式度量化定理,并提出了与格上点式度量理论相协调的若干分离性公理.     

A note on Kneser-Haken finiteness

Kneser-Haken finiteness asserts that for each compact 3-manifold there is an integer such that any collection of c(M)$"> closed, essential, 2-sided surfaces in must contain parallel elements. We show here that if is closed, then twice the number of tetrahedra in a (pseudo)-triangulation of suffices for .

     

On End-Extensions of Models of ¬exp

Every model of IΔ₀ is the tally part of a model of the stringlanguage theory Th-FO (a main feature of which consists in having induction on notation restricted to certain AC⁰. sets). We show how to “smoothly” introduce in Th-FO the binary length function, whereby it is possible to make exponential assumptions in models of Th-FO. These considerations entail that every model of IΔ₀ + ¬exp is a proper initial segment of a model of Th-FO and that a modicum of bounded collection is true in these models. Mathematics Subject Classification: 03F30, 03C62, 68Q15.     

Sequential and parallel synchronous alternating iterative methods

The so-called parallel multisplitting nonstationary iterative Model A was introduced by Bru, Elsner, and Neumann [Linear Algebra and its Applications 103:175-192 (1988)] for solving a nonsingular linear system using a weak nonnegative multisplitting of the first type. In this paper new results are introduced when is a monotone matrix using a weak nonnegative multisplitting of the second type and when is a symmetric positive definite matrix using a -regular multisplitting. Also, nonstationary alternating iterative methods are studied. Finally, combining Model A and alternating iterative methods, two new models of parallel multisplitting nonstationary iterations are introduced. When matrix is monotone and the multisplittings are weak nonnegative of the first or of the second type, both models lead to convergent schemes. Also, when matrix is symmetric positive definite and the multisplittings are -regular, the schemes are also convergent.

     

3-dimensional sundials

R. Hartshorne and A. Hirschowitz proved that a generic collection of lines on ℙ ⁿ , n≥3, has bipolynomial Hilbert function. We extend this result to a specialization of the collection of generic lines, by considering a union of lines and 3-dimensional sundials (i.e., a union of schemes obtained by degenerating pairs of skew lines).