Epireflections and supercompact cardinals

We prove that the existence of arbitrarily large supercompact cardinals implies that every absolute epireflective class of objects in a balanced accessible category is a small-orthogonality class. In other words, if L is a localization functor on a balanced accessible category such that the unit morphism X→LX is an epimorphism for all X and the class of L-local objects is defined by an absolute formula, then the existence of a sufficiently large supercompact cardinal implies that L is a localization with respect to some set of morphisms.     

On coding uncountable sets by reals

If A ? ω₁, then there exists a cardinal preserving generic extension ??[A ][x ] of ??[A ] by a real x such that 1) A ∈ ??[x ] and A is Δ₁^HC (x) in ??[x ]; 2) x is minimal over ??[A ], that is, if a set Y belongs to ??[x ], then either x ∈ ??[A, Y ] or Y ∈ ??[A ]. The forcing we use implicitly provides reshaping of the given set A (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Locally-generic Boolean algebras and cardinal sequences

Given a sequence of cardinals of length less than , with each cardinal in the sequence being either or , we construct a -poset (see Defnition 1 below) which, with a natural topology, becomes a locally-compact, Hausdorff, scattered space with cardinal sequence . The algebra of the clopen subsets of its one-point compactification yields, in turn, a superatomic Boolean algebra with as its cardinal sequence. The posets are locallygeneric, that is, they are constructed generically over countable sets. This gives them additional chain properties, specially under Under Martin's Axiom, the construction allows any cardinals in the sequence, provided it has length Finally, we modify a forcing argument of Baumgartner-Shelah [B-S], to build -posets for any given cardinal sequence of length with each cardinal in the sequence being either or . Received September 2, 1998; accepted in final form September 13, 2001.     

Correlation of global electrophilicity with the activation energy in single-step concerted reactions

The experimental energy of activation (Ea) of the single-step concerted oxidation process of aliphatic primary alcohols by quinolinium bromochromate (QBC) are correlated with the theoretically evaluated global electrophilicity values (w) [as proposed by Parr et al. (J. Am. Chem. Soc. 1999, 121, 1922)]. Conceptual justification in favor of correlating w of the substrate with Ea involved in a single-step concerted reaction is also discussed. The evaluated w values at HF/cc-pVTZ and MP2/6-31G(d,p) methods are found to be as expected (when we consider structural aspects), although there are some inconsistencies in other methods [e.g., HF/6-31G(d,p), B3LYP/cc-pVTZ, BLYP/dnp, PW91/dnp, PWC/dnp, VWN/dnp]. The reasons for the inconsistencies, even with a superior B3LYP/cc-pVTZ method, are discussed thoroughly. It is observed that the higher the value of w, the more the value of Ea involved in the process of oxidation of primary alcohols by QBC. The present study also reveals that the apparent success of insignificant (i.e., much smaller) local electrophilicity values (S+O(OH)), evaluated using Hirshfeld population analysis (HPA), in explaining observed trend of experimental Ea values turns out to be ambiguous when more significant (i.e., much larger) local nucleophilicity values (S-O(OH)) are also compared. This is evident from the corresponding correlation coefficient values.     

Cyclo(l‐leucyl‐α,β‐dehydro­phenyl­alanine): the first diketopiperazine containing an α,β‐dehydro­phenyl­alanine residue

The title compound (systematic name: 3‐benzyl­idene‐6‐iso­butyl­piperazine‐2,5‐dione), C₁₅H₁₈N₂O₂, an α,β‐dehydro­phenyl­alanine containing diketopiperazine, crystallizes in the space group P1 with two mol­ecules in the asymmetric unit arranged antiparallel to one another. The α,β‐dehydro­phenyl­alanine (ΔPhe) residue in this cyclic peptide retains its planarity but deviates from the standard conformations observed in its linear analogues. Each type of mol­ecule forms a linear chain with mol­ecules of the same type via pairwise N—H⋯O hydrogen bonds, while weaker C—H⋯O inter­actions link the chains together to form a three‐dimensional network.     

Superstrong and other large cardinals are never Laver indestructible

Superstrong cardinals are never Laver indestructible. Similarly, almost huge cardinals, huge cardinals, superhuge cardinals, rank-into-rank cardinals, extendible cardinals, 1-extendible cardinals, 0-extendible cardinals, weakly superstrong cardinals, uplifting cardinals, pseudo-uplifting cardinals, superstrongly unfoldable cardinals, Σ_n-reflecting cardinals, Σ_n-correct cardinals and Σ_n-extendible cardinals (all for n ≥  3) are never Laver indestructible. In fact, all these large cardinal properties are superdestructible: if κ exhibits any of them, with corresponding target θ, then in any forcing extension arising from nontrivial strategically <κ-closed forcing \({\mathbb{Q} \in V_\theta}\), the cardinal κ will exhibit none of the large cardinal properties with target θ or larger.     

Parameterized partition relations on the real numbers

We consider several kinds of partition relations on the set of real numbers and its powers, as well as their parameterizations with the set of all infinite sets of natural numbers, and show that they hold in some models of set theory. The proofs use generic absoluteness, that is, absoluteness under the required forcing extensions. We show that Solovay models are absolute under those forcing extensions, which yields, for instance, that in these models for every well ordered partition of there is a sequence of perfect sets whose product lies in one piece of the partition. Moreover, for every finite partition of there is and a sequence of perfect sets such that the product lies in one piece of the partition, where is the set of all infinite subsets of X. The proofs yield the same results for Borel partitions in ZFC, and for more complex partitions in any model satisfying a certain degree of generic absoluteness. This work was supported by the research projects MTM 2005-01025 of the Spanish Ministry of Science and Education and 2005SGR-00738 of the Generalitat de Catalunya. A substantial part of the work was carried out while the second-named author was ICREA Visiting Professor at the Centre de Recerca Matemàtica in Bellaterra (Barcelona), and also during the first-named author’s stays at the Instituto Venezolano de Investigaciones Científicas and the California Institute of Technology. The authors gratefully acknowledge the support provided by these institutions.