Easton’s theorem in the presence of Woodin cardinals

Under the assumption that δ is a Woodin cardinal and GCH holds, I show that if F is any class function from the regular cardinals to the cardinals such that (1) ${\kappa < {\rm cf}(F(\kappa))}$ , (2) ${\kappa < \lambda}$ implies ${F(\kappa) \leq F(\lambda)}$ , and (3) δ is closed under F, then there is a cofinality-preserving forcing extension in which 2 ^γ = F(γ) for each regular cardinal γ < δ, and in which δ remains Woodin. Unlike the analogous results for supercompact cardinals [Menas in Trans Am Math Soc 223:61–91, (1976)] and strong cardinals [Friedman and Honzik in Ann Pure Appl Logic 154(3):191–208, (2008)], there is no requirement that the function F be locally definable. I deduce a global version of the above result: Assuming GCH, if F is a function satisfying (1) and (2) above, and C is a class of Woodin cardinals, each of which is closed under F, then there is a cofinality-preserving forcing extension in which 2 ^γ = F(γ) for all regular cardinals γ and each cardinal in C remains Woodin.     

Generic Vopěnka’s Principle,remarkable cardinals,and the weak Proper Forcing Axiom

We introduce and study the first-order Generic Vopěnka’s Principle, which states that for every definable proper class of structures \(\mathcal {C}\) of the same type, there exist \(B\ne A\) in \(\mathcal {C}\) such that B elementarily embeds into A in some set-forcing extension. We show that, for \(n\ge 1\), the Generic Vopěnka’s Principle fragment for \(\Pi _n\)-definable classes is equiconsistent with a proper class of n-remarkable cardinals. The n-remarkable cardinals hierarchy for \(n\in \omega \), which we introduce here, is a natural generic analogue for the \(C^{(n)}\)-extendible cardinals that Bagaria used to calibrate the strength of the first-order Vopěnka’s Principle in Bagaria (Arch Math Logic 51(3–4):213–240, 2012). Expanding on the theme of studying set theoretic properties which assert the existence of elementary embeddings in some set-forcing extension, we introduce and study the weak Proper Forcing Axiom, \(\mathrm{wPFA}\). The axiom \(\mathrm{wPFA}\) states that for every transitive model \(\mathcal M\) in the language of set theory with some \(\omega _1\)-many additional relations, if it is forced by a proper forcing \(\mathbb P\) that \(\mathcal M\) satisfies some \(\Sigma _1\)-property, then V has a transitive model \(\bar{\mathcal M}\), satisfying the same \(\Sigma _1\)-property, and in some set-forcing extension there is an elementary embedding from \(\bar{\mathcal M}\) into \(\mathcal M\). This is a weakening of a formulation of \(\mathrm{PFA}\) due to Claverie and Schindler (J Symb Logic 77(2):475–498, 2012), which asserts that the embedding from \(\bar{\mathcal M}\) to \(\mathcal M\) exists in V. We show that \(\mathrm{wPFA}\) is equiconsistent with a remarkable cardinal. Furthermore, the axiom \(\mathrm{wPFA}\) implies \(\mathrm{PFA}_{\aleph _2}\), the Proper Forcing Axiom for antichains of size at most \(\omega _2\), but it is consistent with \(\square _\kappa \) for all \(\kappa \ge \omega _2\), and therefore does not imply \(\mathrm{PFA}_{\aleph _3}\).     

On successors of Jónsson cardinals

We show that, like singular cardinals, and weakly compact cardinals, Jensen's core model K for measures of order zero [4] calculates correctly the successors of Jónsson cardinals, assuming does not exist. Namely, if is a Jónsson cardinal then , provided that there is no non-trivial elementary embedding . There are a number of related results in ZFC concerning in V and inner models, for a Jónsson or singular cardinal. Received: 8 December 1998     

Cichoń’s diagram for uncountable cardinals

We develop a version of Cichoń’s diagram for cardinal invariants on the generalized Cantor space 2 ^κ or the generalized Baire space κ ^κ , where κ is an uncountable regular cardinal. For strongly inaccessible κ, many of the ZFC-results about the order relationship of the cardinal invariants which hold for ω generalize; for example, we obtain a natural generalization of the Bartoszyński–Raisonnier–Stern Theorem. We also prove a number of independence results, both with < κ-support iterations and κ-support iterations and products, showing that we consistently have strict inequality between some of the cardinal invariants.     

Going-down functors, the Künneth formula, and the Baum-Connes conjecture

We study the connection between the Baum-Connes conjecture for a locally compact group G with coeefficient A and the Künneth formula for the K-theory of tensor products by the corresponding crossed product . The main tool for this is obtained by an application of a general reduction procedure which allows us to analyze certain functors connected to the topological K-theory of a group in terms of their restrictions to compact subgroups. We also discuss several other interesting applications of this method, including a general extension result for the Baum-Connes conjecture.     

Ergodic properties of the Erdös measure, the entropy of the goldenshift, and related problems

We define a two-sided analog of the Erdös measure on the space of two-sided expansions with respect to the powers of the golden ratio, or, equivalently, the Erdös measure on the 2-torus. We construct the transformation (goldenshift) preserving both Erdös and Lebesgue measures on that is the induced automorphism with respect to the ordinary shift (or the corresponding Fibonacci toral automorphism) and proves to be Bernoulli with respect to both measures in question. This provides a direct way to obtain formulas for the entropy dimension of the Erdös measure on the interval, its entropy in the sense of Garsia-Alexander-Zagier and some other results. Besides, we study central measures on the Fibonacci graph, the dynamics of expansions and related questions.With 11 PiguresTo the memory of Paul ErdösSupported in part by the INTAS grant 93-0570. The first author was supported by the French foundation PRO MATHEMATICA. The first author expresses his gratitude to l'Institut de Mathématiques de Luminy for support during his stay in Marseille in 1996-97. The second author is grateful to the University of Stony Brook for support during his visit in February–March 1996 and to the Institute for Advanced studies of Hebrew University for support during his being there in 1997     

Students,mathematics, and technology: assessing the present – challenging the future

This paper is concerned with identifying and assessing the impacts of various technologically enriched approaches to mathematics learning. Its purpose is to address the usefulness of emerging knowledge to enhance practice, and to contribute to theorizing about technology-based learning. Hence the main drivers are intentions to identify and elaborate on obstacles, errors, and unresolved problems on the one hand, and positives and insights on the other. The paper tells a cautionary tale about expectations of technology-enhanced learning, while simultaneously uncovering a rich base from which to theorize and test new appreciations of what is involved when students, technology, and mathematics connect in learning settings. A summary assessment is that this area still very much represents work in progress, but there is now a heightened realization, at least among those not transfixed by technological blindness, that the search for ultimate answers will require much greater exploration of machine–mathematics–learner relationships.     

Whose deletion does not affect your payoff? The difference between the Shapley value, the egalitarian value, the solidarity value, and the Banzhaf value

This study provides a unified axiomatic characterization method of one-point solutions for cooperative games with transferable utilities. Any one-point solution that satisfies efficiency, the balanced cycle contributions property (BCC), and the axioms related to invariance under a player deletion is characterized as a corollary of our general result. BCC is a weaker requirement than the well-known balanced contributions property. Any one-point solution that is both symmetric and linear satisfies BCC. The invariance axioms necessitate that the deletion of a specific player from games does not affect the other players’ payoffs, and this deletion is different with respect to solutions. As corollaries of the above characterization result, we are able to characterize the well-known one-point solutions, the Shapley, egalitarian, and solidarity values, in a unified manner. We also studied characterizations of an inefficient one-point solution, the Banzhaf value that is a well-known alternative to the Shapley value.     

The Riemann Hypothesis, the Generalized Riemann Hypothesis, and the Cesáro Operator

A result due to Nyman establishes the equivalence of the Riemann hypothesis with the density of a set of functions in L ²[0, 1]. Here a large class of analytic functions is considered, which includes the Riemann zeta function and the Dirichlet L-functions as well as functions not given by a Dirichlet series. For each such function there is an associated integral operator T on L ²[0, 1] such that has no zeros in Re(s) > 1/2 iff the operator T has dense range iff a specified set of functions is dense in L ²[0, 1].      

On the uniform estimate in the Calabi-Yau theorem,Ⅱ

We show that a pluripotential proof of the uniform estimate in the Calabi-Yau theorem works also in the Hermitian case.     

Lehmer pairs of zeros,the de Bruijn-Newman constant Λ, and the Riemann Hypothesis

We give here a rigorous formulation for a pair of consecutive simple positive zeros of the functionH ₀ (which is closely related to the Riemann -function) to be a Lehmer pair of zeros ofH ₀. With this formulation, we establish that each such pair of zeros gives a lower bound for the de Bruijn-Newman constant (where the Riemann Hypothesis is equivalent to the assertion that 0). We also numerically obtain the following new lower bound for :

On the chromatic uniqueness of the graphW(n,n − 2, k)

This paper shows that the graphW(n, n – 2, k) is chromatically unique for any even integern 6 and any integerk 1.     

General Topology — the Monadic Case, Examples, Applications

The paper deals with monadic as well as monadic-free topological notions. For defining these monadic-free notions the notion of basic triple is introduced. A lot of monadic-free topological notions are presented, for instance that of -convergence structure, -hull operator and -uniform structure. By means of a generalized metric, e.g. a probabilistic metric, and the general notion of -zero approach introduced in this paper, a -uniform structure is generated. In case of a fuzzy metric the related -uniform structure defines in a canonic way a fuzzy topology which is used for developing a fuzzy analysis and fuzzy calculus.     

Foliations, Groupoids, and the Baum―Connes Conjecture

The BaumConnes conjecture establishes, for foliated manifolds, an analog of the well-known isomorphism between the topological K-theory of a locally compact space M and the analytic K-theory of the C ^*-algebra of continuous functions on M vanishing at infinity. In this work, we describe the principal notions involved in the statement of the conjecture and indicate its contemporary status. Bibliography: 11 titles.     

What part of the concept of acceleration is difficult to understand: the mathematics, the physics, or both?

In this paper, details of student difficulties in understanding the concept of acceleration and the mathematical and physical/intuitive sources of these are delineated by utilizing the teaching experiment methodology. As a result of the study, two anchoring analogies are proposed that can be used as a diagnostic tool for students’ alternative conceptions. These can be used in teaching to highlight the peculiarity of acceleration concept. This study portrays how seeing acceleration as ‘rate of change’ of a quantity (velocity) and recognizing the consequences of such a definition are hindered in certain ways which in turn negatively affect learning the concept of force. This is also an example that illustrates that a rather “simple” mathematical concept (i.e., rate of change) for the expert can become a complex phenomenon when embedded in a physical concept (i.e., acceleration) which is consistently found to be as a misconception among learners at various levels that is widely occurring and very resistant to change.     

Share the gain, share the pain? Almost transferable utility, changes in production possibilities, and bargaining solutions

We consider an n-person economy in which efficiency is independent of distribution but the cardinal properties of the agents’ utility functions may preclude transferable utility (a property we call “Almost TU”). Holding the disagreement point fixed, we show that Almost TU is a necessary and sufficient condition for all agents to either benefit jointly or suffer jointly with any change in production possibilities under well-behaved generalized utilitarian bargaining solutions (of which the Nash bargaining and the utilitarian solutions are special cases). We apply the result to policy analysis and to incentive compatibility.