A note on the Σ1 spectrum of a theory

Let T be a suitable system of classical set theory. We will show, that the Σ₁ spectrum of T, i.e. the set of ordinals having good Σ₁ definition in T is an initial segment of the ordinals. Received: 11 October 1999 / Published online: 12 December 2001     

Time-resolved fluorescence of the bacteriophage T4 capsid protein gp23

The time-resolved fluorescence properties of the bacteriophage T4 capsid protein gp23 are investigated. The structural characteristics of this protein are largely unknown and can be probed by recording time-resolved and decay-associated fluorescence spectra and intensity decay curves using a 200 ps-gated intensified CCD-camera. Spectral and decay data are recorded simultaneously, which makes data acquisition fast compared to time-correlated single-photon counting. A red-shift of the emission maximum within the first nanosecond of decay is observed, which can be explained by the different decay-associated spectra of fluorescence lifetimes of the protein in combination with dipolar relaxation. In addition, iodide quenching experiments are performed, to study the degree of exposure of the various tryptophan residues. A model for the origin of the observed lifetimes of 0.032 +/- 0.003, 0.39 +/- 0.06, 2.1 +/- 0.1 and 6.8 +/- 0.8 ns is presented: the 32 ps lifetime can be assigned to the emission of a buried tryptophan residue, the 0.4 and 2.1 ns lifetimes to two partly buried residues, and the 6.8 ns lifetime to a single tryptophan outside the bulk of the folded gp23.     

The fully deprotonated anion of 1,3,5-benzene-triphosphonic acid: 1H, 31P,and 13C{1H} NMR and some comments on corresponding [AX]3 and AXX′2 spin systems and spectra

Abstract

The fully deprotonated anion of 1,3,5-benzene-triphosphonic acid 1,3,5-C₆H₃[PO₃^2?]₃ gives rise to deceptively simple ¹H and ³¹P NMR spectra due to a corresponding [AX]₃ spin system. The ¹³C{¹H} NMR spectrum reveals two ¹³C isotopomers which are identified via AXX′₂ systems. Results from analysis and iteration are described.     

An ordinal analysis for theories of self-referential truth

The first attempt at a systematic approach to axiomatic theories of truth was undertaken by Friedman and Sheard (Ann Pure Appl Log 33:1–21, 1987). There twelve principles consisting of axioms, axiom schemata and rules of inference, each embodying a reasonable property of truth were isolated for study. Working with a base theory of truth conservative over PA, Friedman and Sheard raised the following questions. Which subsets of the Optional Axioms are consistent over the base theory? What are the proof-theoretic strengths of the consistent theories? The first question was answered completely by Friedman and Sheard; all subsets of the Optional Axioms were classified as either consistent or inconsistent giving rise to nine maximal consistent theories of truth.They also determined the proof-theoretic strength of two subsets of the Optional Axioms. The aim of this paper is to continue the work begun by Friedman and Sheard. We will establish the proof-theoretic strength of all the remaining seven theories and relate their arithmetic part to well-known theories ranging from PA to the theory of ${\Sigma^1_1}The first attempt at a systematic approach to axiomatic theories of truth was undertaken by Friedman and Sheard (Ann Pure Appl Log 33:1–21, 1987). There twelve principles consisting of axioms, axiom schemata and rules of inference, each embodying a reasonable property of truth were isolated for study. Working with a base theory of truth conservative over PA, Friedman and Sheard raised the following questions. Which subsets of the Optional Axioms are consistent over the base theory? What are the proof-theoretic strengths of the consistent theories? The first question was answered completely by Friedman and Sheard; all subsets of the Optional Axioms were classified as either consistent or inconsistent giving rise to nine maximal consistent theories of truth.They also determined the proof-theoretic strength of two subsets of the Optional Axioms. The aim of this paper is to continue the work begun by Friedman and Sheard. We will establish the proof-theoretic strength of all the remaining seven theories and relate their arithmetic part to well-known theories ranging from PA to the theory of S¹₁{\Sigma^1_1} dependent choice.     

Synthesis of Inorganic Structural Isomers By Diffusion‐Constrained Self‐Assembly of Designed Precursors: A Novel Type of Isomerism

The structure of precursors is used to control the formation of six possible structural isomers that contain four structural units of PbSe and four structural units of NbSe₂: [(PbSe)_1.14]₄[NbSe₂]₄, [(PbSe)_1.14]₃[NbSe₂]₃[(PbSe)_1.14]₁[NbSe₂]₁, [(PbSe)_1.14]₃[NbSe₂]₂[(PbSe)_1.14]₁[NbSe₂]₂, [(PbSe)_1.14]₂[NbSe₂]₃[(PbSe)_1.14]₂[NbSe₂]₁, [(PbSe)_1.14]₂[NbSe₂]₂[(PbSe)_1.14]₁[NbSe₂]₁[(PbSe)_1.14]₁[NbSe₂]₁, [(PbSe)_1.14]₂[NbSe₂]₁[(PbSe)_1.14]₁[NbSe₂]₂[(PbSe)_1.14]₁[NbSe₂]₁. The electrical properties of these compounds vary with the nanoarchitecture. For each pair of constituents, over 20 000 new compounds, each with a specific nanoarchitecture, are possible with the number of structural units equal to 10 or less. This provides opportunities to systematically correlate structure with properties and hence optimize performance.     

Hexanuclear copper(II) cages built on a central {μ3-O···H···μ3-O} moiety, 1,3-bis(dimethylamino)-2-propanolato and capping R-phosphonates: crystal structures, magnetic behavior, and DFT studies

The syntheses, structural characterization, and magnetic behavior of two new hexanuclear copper(II) complexes derived from R-phosphonic acids and 1,3-bis(dimethylamino)-2-propanol (Hbdmap) with formulas [Cu(6)(μ-bdmap)(3)(μ(3)-Ph-PO(3))(2)(μ(3)-O···H···μ(3)-O)(ClO(4))(2)(H(2)O)]·5H(2)O (1) and [Cu(6)(μ-bdmap)(3)(μ(3)-t-Bu-PO(3))(2)(μ(3)-O···H···μ(3)-O)(μ(1,3)-dca)(dca)(H(2)O)]·6H(2)O (2) (Ph-H(2)PO(3) = phenylphosphonic acid, t-Bu-H(2)PO(3) = tert-butylphosphonic acid, dca = dicyanamide) are reported. Compounds 1 and 2 are hexanuclear 3.111 R-phosphonate(2-)/1,3-bis(dimethylamino)-2-propanolato(1-) cages including in the center the [μ(3)-O···H···μ(3)-O](3-) unit. The temperature dependence of the magnetic properties of 1 and 2 clearly indicates an overall strong antiferromagnetic coupling confirmed by DFT calculations.