Universal forcing notions and ideals

Our main result states that a finite iteration of Universal Meager forcing notions adds generic filters for many forcing notions determined by universality parameters. We also give some results concerning cardinal characteristics of the σ-ideals determined by those universality parameters. Both authors acknowledge support from the United States-Israel Binational Science Foundation (Grant no. 2002323). Also, we would like to thank the referee for valuable comments and suggestions concerning the exposition of the paper. This is publication 845 of the second author     

Perfect sets of random reals

We show that the existence of a perfect set of random reals over a modelM ofZFC does not imply the existence of a dominating real overM, thus answering a well-known open question (see [BJ 1] and [JS 2]). We also prove that (the product of two copies of the random algebra) neither adds a dominating real nor adds a perfect set of random reals (this answers a question that A. Miller asked during the logic year at MSRI). The first author would like to thank the MINERVA-foundation for supporting him. The second author would like to thank the Basic Research Foundation (the Israel Academy of Sciences and Humanities) for supporting him.     

Baire property and axiom of choice

We show that without using inaccessible cardinals it is possible to get models of “ZF+all sets of reals have the Baire property +DC(ω₁)” and “ZFC+all projective sets have the Baire property+the union of less than ω₂ many meager sets is meager”, answering two well-known open questions of Woodin and Judah, respectively. The authors would like to thank the Israel Academy of Sciences BSF for partial support. The second author would like to thank the Landau Center for Mathematical Analysis, supported by the Minerva Foundation (Germany).     

Global maxima of real-valued functions

In this paper, sufficient conditions for a local maximum to be global are discussed. The result is that a real continuously differentiable functionf, defined on a subset ofR ⁿ, under fairly weak conditions, is unimodal iff has a strict local maximum at any stationary point.The author would like to thank the referee for detailed and constructive comments.     

Martin’s axiom does not imply that every two ℕ1-dense sets of reals are isomorphic

Assuming the consistency of ZFC we prove the claim in the title by showing the consistency with ZFC of: There exists a set of realsA such that every function fromA toA is order preserving on an uncountable set. We prove related results among which is the consistency with ZFC of: Every function from the reals into the reals is monotonic on an uncountable set. I would like to thank the United States-Israel Binational Science Foundation for supporting this research by a grant.     

Imaginaries in Hilbert spaces

We characterise imaginaries (up to interdefinability) in Hilbert spaces using a Galois theory for compact unitary groups.The authors would like to thank Frank Wagner and the CIRM for their hospitality during the Simpleton 2002 meeting during which the discussions that led to this paper took place.The first author would like to thank Ilan Hirshberg for a few important hintsAt the time of the writing of this paper, the first author was a graduate student with the Équipe de logique mathématique of Université Paris VIIMathematics Subject Classification (2000): 03C45, 03C95     

Commutator conditions and splitting automorphisms for stable groups

Summary We show that a stable groupG satisfying certain commutator conditions is nilpotent. Furthermore, a soluble stable group with generically splitting automorphism of prime order is nilpotent-by-finite. In particular, a soluble stable group with a generic element of prime order is nilpotent-by-finite.The author would like to thank Ali Nesin for pointing out an error in an earlier version of this paper and various valuable comments     

Natural dualities for varieties of BL-algebras

BL-algebras are the Lindenbaum algebras for Hájek's Basic Logic, just as Boolean algebras correspond to the classical propositional calculus. The finite totally ordered BL-algebras are ordinal sums of MV-chains. We develop a natural duality, in the sense of Davey and Werner, for each subvariety generated by a finite BL-chain, and we use it to describe the injective and the weak injective members of these classes. The preliminary research for this paper was carried out while the second author was visiting Salerno University. The second author would like to thank the first author and Salerno University for their hospitality. The second author acknowledges partial supports from Salerno University and from the belgian Fonds National de la Recherche Scientifique.     

Understanding preservation theorems,II

We present an exposition of much of Sections VI.3 and XVIII.3 from Shelah's book Proper and Improper Forcing. This covers numerous preservation theorems for countable support iterations of proper forcing, including preservation of the property “no new random reals over V ”, the property “reals of the ground model form a non‐meager set”, the property “every dense open set contains a dense open set of the ground model”, and preservation theorems related to the weak bounding property, the weak ^ωω ‐bounding property, and the property “the set of reals of the ground model has positive outer measure” (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Construction of constant mean curvature <Emphasis Type="Italic">n</Emphasis>-noids from holomorphic potentials

Using a Weierstrass type representation of constant mean curvature surfaces, we give a general method for constructing constant mean curvature n-noids (of genus 0) from holomorphic potentials, where n ≥ 3. The ends of these surfaces are embedded and asymptotically approach Delaunay surfaces, while the surfaces are in general not even almost embedded. In particular, a 3-parameter family of constant mean curvature trinoids is constructed. Part of this work was done, while the first named author held a Lehrstuhlvertretung at the University of Augsburg. He would like to thank the University of Augsburg for its hospitality. He would also like to acknowledge partial support by DFG-grant DO 776.     

An example of bad convex function

An example of a convex function having the gradient at each point (x, 0, ..., 0),x>0, which does not converge, whenx tends to zero, is given.The author would like to thank Professor Giannessi for his help, which led to an improved version of this paper.     

On the simplicity of the automorphism group ofP(ω)/fin

Summary We prove that the automorphism group ofP()/fin remains simple if ₂ Cohen reals are added to a model of ZFC+CH.This paper is based on a part of author's Dissertation [4] written under the supervision of Professor Sabine Koppelberg whom the author would like to thank for her encouragement during the preparation of the thesis. Thanks are also due to the referee for some valuable suggestions on improvement of the formulation     

Measured creatures

We prove that two basic questions on outer measure are undecidable. First we show that consistently every sup-measurable functionf: ℝ² → ℝ is measurable. The interest in sup-measurable functions comes from differential equations and the question for which functionsf: ℝ² → ℝ the Cauchy problemy′=f(x,y), y(x₀)=y₀ has a unique almost-everywhere solution in the classAC _t(ℝ) of locally absolutely continuous functions on ℝ. Next we prove that consistently every functionf: ℝ → ℝ is continuous on some set of positive outer Lebesgue measure. This says that in a strong sense the family of continuous functions (from the reals to the reals) is dense in the space of arbitrary such functions. For the proofs we discover and investigate a new family of nicely definable forcing notions (so indirectly we deal with nice ideals of subsets of the reals—the two classical ones being the ideal of null sets and the ideal of meagre ones). Concerning the method, i.e., the development of a family of forcing notions, the point is that whereas there are many such objects close to the Cohen forcing (corresponding to the ideal of meagre sets), little has been known on the existence of relatives of the random real forcing (corresponding to the ideal of null sets), and we look exactly at such forcing notions. The first author thanks The Hebrew University of Jerusalem for support during his visits to Jerusalem and the KBN (Polish Committee of Scientific Research) for partial support through grant 2P03A03114. The research of the second author was partially supported by the Israel Science Foundation. Publication 736.     

The monoid of strong endomorphisms of a graph

We explicitly find all the idempotents in eachL(R)-class and all the inverses of each element of the strong endomorphism monoid of a graph. The number of these idempotents and inverses is also obtained. The author is deeply indebted to Professor Dr. T. E. Hall for his stimulating questions about this theme and much improvement made to an earlier version of this paper. The author would like to thank Professor Dr. U. Knauer and Dr. E. Wilkeit for helpful comments.     

Lipschitz continuity ofV-subdifferentials of convex operators

It is proved that theV-subdifferential of a convex operator is locally Lipschitzian on the set of points at which it is continuous and subdifferentiable.Proposition 2.2 was originally stated for Holder continuity ofV-subdifferentials. The author would like to thank J. P. Penot for a very helpful suggestion which led to the present form of this proposition.     

Green's relations on the strong endomorphism monoid of a graph

Combinatorial characteristics of Green's relations on the monoid of strong endomorphisms of graphs are explored. As an application, a short proof of the fact thats End(G) is regular is given. The author would like to thank Professor Dr. U. Knauer and Dr. E. Wilkeit for valuable advice and Professor T. E. Hall for helpful suggestions and corrections made to an earlier version of this paper.     

A generalization of the rees theorem to a class of regular semigroups

Let S be a regular semigroup for which Green's relations J and D coincide, and which is max-principal in the sense that every element of S is contained in maximal principal right, left and two-sided ideals of S. A construction is given of a max-principal regular semigroup W with J=D, which is also principally separated in the sense that distinct maximal principal right (or left) ideals of S are disjoint, and an epimorphism ψ: W→S that preserves maximality of principal left, right, and two sided ideals, and is in a sense locally one-to-one. If S is completely simple, this construction reduces to the Rees matrix representation of S. The main result of this paper has its origin in an incorrect result contained in the author's doctoral dissertation which was written at the University of California (Berkeley) under Professor John Rhodes. This theorem was first established for finite regular semigroups in [1] (Corollary 2.3), and the present generalization of this result to infinite semigroups was suggested by Professor A. H. Clifford, who the author would like to thank for this as well as his generous encouragement and many helpful editorial suggestions. The author would also like to thank Professor Rhodes for his encouragement.     

A two‐grid SA‐AMG convergence bound that improves when increasing the polynomial degree

In this paper, we consider the convergence rate of a smoothed aggregation algebraic multigrid method, which uses a simple polynomial (1 ? t)^ν or an optimal Chebyshev‐like polynomial to construct the smoother and prolongation operator. The result is purely algebraic, whereas a required main weak approximation property of the tentative interpolation operator is verified for a spectral element agglomeration version of the method. More specifically, we prove that, for partial differential equations (PDEs), the two‐grid method converges uniformly without any regularity assumptions. Moreover, the convergence rate improves uniformly when the degree of the polynomials used for the smoother and the prolongation increases. Such a result, as is well‐known, would imply uniform convergence of the multilevel W‐cycle version of the algorithm. Numerical results, for both PDE and non‐PDE (graph Laplacian) problems are presented to illustrate the theoretical findings. Published 2016. This article is a U.S. Government work and is in the public domain in the USA.     

Common homoclinic points of commuting toral automorphisms

The points homoclinic to 0 under a hyperbolic toral automorphism form the intersection of the stable and unstable manifolds of 0. This is a subgroup isomorphic to the fundamental group of the torus. Suppose that two hyperbolic toral automorphisms commute so that they determine a ℤ²-action, which we assume is irreducible. We show, by an algebraic investigation of their eigenspaces, that they either have exactly the same homoclinic points or have no homoclinic point in common except 0 itself. We prove the corresponding result for a compact connected abelian group, and compare the two proofs. The second author would like to thank the Austrian Academy of Sciences and the Royal Society for partial support while this work was done.