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 \(\mathrm{PFA}_{\aleph _3}\).     

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Analysis of a defect correction method for geometric integrators

We discuss a new variant of Iterated Defect Correction (IDeC), which increases the range of applicability of the method. Splitting methods are utilized in conjunction with special integration methods for Hamiltonian systems, or other initial value problems for ordinary differential equations with a particular structure, to solve the neighboring problems occurring in the course of the IDeC iteration. We demonstrate that this acceleration technique serves to rapidly increase the convergence order of the resulting numerical approximations, up to the theoretical limit given by the order of certain superconvergent collocation methods. This project was supported by the Special Research Program SFB F011 ‘AURORA’ of the Austrian Science Fund FWF.     

Projective Well-orderings of the Reals

If there is no inner model with ω many strong cardinals, then there is a set forcing extension of the universe with a projective well-ordering of the reals.     

What is the Magnetic Moment of the Electron?

 Because of infrared effects the charged sectors of QED contain no eigenstates of the mass operator. The electron is therefore not definable as a Wigner particle. There exists no sharp, unambiguous, definition of the notion of a 1-electron state. The assignment of a fixed value of the magnetic moment – or similar quantities – to the electron is therefore at first problematic. It is not clear a priori that such a notion is meaningful. Conventionally this problem is solved by first calculating the desired quantity in an IR-regularized theory and then removing the regularization. If this method yields a finite value, that is considered sufficient proof of its soundness. This is clearly less than satisfactory. Here we propose a more convincing way of defining the intrinsic magnetic moment of the electron, which does not use any regularizations and is not based on an interaction with external fields. A pseudostatic 1-electron state is defined in a phenomenological way. Its magnetic moment, as defined here, does not depend on the unavoidable ambiguities inherent in this definition. The method leads to the same analytic expression as the conventional approach, thus preserving the excellent agreement between theory and experiment. Received: 9 December 2002 / Accepted: 11 March 2003 Published online: 5 May 2003 Communicated by H. Araki, D. Buchholz and K. Fredenhagen     

The convergence of shooting methods for singular boundary value problems

We investigate the convergence properties of single and multiple shooting when applied to singular boundary value problems. Particular attention is paid to the well-posedness of the process. It is shown that boundary value problems can be solved efficiently when a high order integrator for the associated singular initial value problems is available. Moreover, convergence results for a perturbed Newton iteration are discussed.

     

Analysis of Sulfate Patterns in Glycosaminoglycan Oligosaccharides by MS<Superscript>n</Superscript> Coupled to Infrared Ion Spectroscopy: the Case of GalNAc4S and GalNAc6S

We report distinctive spectroscopic fingerprints of the monosaccharide standards GalNAc4S and GalNAc6S by coupling mass spectrometry and ion spectroscopy in the 3-μm range. The disaccharide standards CSA and CSC are used to demonstrate the applicability of a novel approach for the analysis of sulfate position in GalNAc-containing glycosaminoglycans. This approach was then used for the analysis of a sample containing CSA and CSC disaccharides. Finally, we discuss the generalization of the coupling of mass spectrometry with ion spectroscopy for the structural analysis of glycosaminoglycans on a tetrasaccharide from dermatan sulfate source.

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