Reverse mathematics and initial intervals

In this paper we study the reverse mathematics of two theorems by Bonnet about partial orders. These results concern the structure and cardinality of the collection of initial intervals. The first theorem states that a partial order has no infinite antichains if and only if its initial intervals are finite unions of ideals. The second one asserts that a countable partial order is scattered and does not contain infinite antichains if and only if it has countably many initial intervals. We show that the left to right directions of these theorems are equivalent to _ACA0${\mathsf{ACA}}_0$ and _ATR0${\mathsf{ATR}}_0$, respectively. On the other hand, the opposite directions are both provable in _WKL0${\mathsf{WKL}}_0$, but not in _RCA0${\mathsf{RCA}}_0$. We also prove the equivalence with _ACA0${\mathsf{ACA}}_0$ of the following result of Erdös and Tarski: a partial order with no infinite strong antichains has no arbitrarily large finite strong antichains.     

A note on Griffiths infinitesimal invariant for curves

Given a generic curve of genus $g\ge 4$ and a smooth point $L\in W_{g-1}^{1}(C)$ , whose linear system is base-point free, we consider the Abel–Jacobi normal function associated with $L^{\otimes 2}\otimes \omega _{C}^{-1}$ , when $(C,L)$ varies in moduli. We prove that its infinitesimal invariant reconstructs the couple $(C,L)$ . When $g=4$ , we obtain the generic Torelli theorem proved by Griffiths.     

Inner solvers for interior point methods for large scale nonlinear programming

This paper deals with the solution of nonlinear programming problems arising from elliptic control problems by an interior point scheme. At each step of the scheme, we have to solve a large scale symmetric and indefinite system; inner iterative solvers, with an adaptive stopping rule, can be used in order to avoid unnecessary inner iterations, especially when the current outer iterate is far from the solution. In this work, we analyse the method of multipliers and the preconditioned conjugate gradient method as inner solvers for interior point schemes. We discuss the convergence of the whole approach, the implementation details and report the results of numerical experimentation on a set of large scale test problems arising from the discretization of elliptic control problems. A comparison with other interior point codes is also reported. This research was supported by the Italian Ministry for Education, University and Research (MIUR) projects: FIRB Project: “Parallel Nonlinear Numerical Optimization PN ² O” (grant n. RBAU01JYPN, ) and COFIN/PRIN04 Project “Numerical Methods and Mathematical Software for Applications” (grant n. 2004012559, ).     

On a quadratic functional occurring in a bivariate scattered data interpolation problem

Sunto L’applicazione di noti metodi che utilizzano funzioni di tipo blending per la costruzione di funzioni bivariate C¹ per l’interpolazione di dati, richiede la conoscenza delle derivate parziali del primo ordine ai vertici di una triangolazione sottostante. In questo lavoro consideriamo il metodo proposto da Nielson, che consiste nel calcolare stime delle derivate parziali del primo ordine minimizzando un opportuno funzionale quadratico, caratterizzato da parametri di tensione non negativi. Scopo del lavoro è l’analisi di alcune proprietà particolari di questo funzionale per la costruzione di algoritmi efficienti e robusti per la determinazione delle stime suddette delle derivate quando si ha a che fare con insiemi di dati di grandi dimensioni. Abstract The application of widely known blending methods for constructingC ¹ bivariate functions interpolating scattered data requires the knowledge of the partial derivatives of first order at the vertices of an underlying triangulation. In this paper we consider the method proposed by Nielson that consists in computing estimates of the first order partial derivatives by minimizing an appropriate quadratic functional, characterized by nonnegative tension parameters. The aim of the paper is to analyse some peculiar properties of this functional in order to construct robust and efficient algorithms for determining the above estimates of the derivatives when we are concerned with extremely large data sets.      

Character degree graphs that are complete graphs

Let be a finite group, and write for the set of degrees of irreducible characters of . We define to be the graph whose vertex set is , and there is an edge between and if . We prove that if is a complete graph, then is a solvable group.

     

Mean value formulas for classical solutions to uniformly parabolic equations in the divergence form with non-smooth coefficients

We prove surface and volume mean value formulas for classical solutions to uniformly parabolic equations in the divergence form with low regularity of the coefficients. We then use them to prove the parabolic strong maximum principle and the parabolic Harnack inequality. We emphasize that our results only rely on the classical theory, and our arguments follow the lines used in the original theory of harmonic functions. We provide two proofs relying on two different formulations of the divergence theorem, one stated for sets with almost C¹-boundary, the other stated for sets with finite perimeter.     

Similar Structural Dynamics for the Degradation of CH3NH3PbI3 in Air and in Vacuum

We investigate the degradation path of MAPbI₃ (MA=methylammonium) films over flat TiO₂ substrates at room temperature by means of X‐ray diffraction, spectroscopic ellipsometry, X‐ray photoelectron spectroscopy, and high‐resolution transmission electron microscopy. The degradation dynamics is found to be similar in air and under vacuum conditions, which leads to the conclusion that the occurrence of intrinsic thermodynamic mechanisms is not necessarily linked to humidity. The process has an early stage, which drives the starting tetragonal lattice in the direction of a cubic atomic arrangement. This early stage is followed by a phase change towards PbI₂. We describe how this degradation product is structurally coupled with the original MAPbI₃ lattice through the orientation of its constituent PbI₆ octahedra. Our results suggest a slight octahedral rearrangement after volatilization of HI+CH₃NH₂ or MAI, with a relatively low energy cost. Our experiments also clarify why reducing the interfaces and internal defects in the perovskite lattice enhances the stability of the material.     

Mechanism insights into an unexpected isomerisation reaction observed during conversion of N-benzyl piperazinones to corresponding N-benzyl thio-piperazinones

An interesting and surprising rearrangement was observed during the reaction of 4-N-benzyl piperazinone derivatives with Lawesson’s reagent as a thionating agent. Investigation into the possible mechanism responsible for these results is reported herein.     

Fast analysis of amino acids in wine by capillary electrophoresis with laser-induced fluorescence detection

A fast analytical method has been developed for the determination of nine amino acids, together with serotonin, in wine samples of different origin and vintage. The method is based on capillary electrophoresis coupled to laser-induced fluorescence detection. Separation was obtained by using a fused-silica capillary (75?μm id, 74.0?cm total length, 60.0?cm length to detector) and a background electrolyte composed of carbonate buffer (20?mM, pH 9.2), applying a 20?kV voltage. Direct hydrodynamic injection of wine samples was made after an original microwave-assisted derivatisation step with 5-(4,6-dichlorotriazinyl)aminofluorescein. Fluorescence was induced by an Ar-Ion laser, exciting at 488?nm. Good linearity (r(2) >0.9990) was obtained for all considered analytes and sensitivity was also good, with limits of detection in the 7-50?ng/mL range. The method was successfully applied for the analysis of commercial Italian wines and thus seems to be suitable for the determination of the relevant amino acids and serotonin, providing good results in terms of accuracy and precision, together with the advantage of a very fast, microwave-assisted derivatisation procedure. Future applications of the method are planned to check for wine adulterations and commercial frauds.     

Water-soluble peptide-coated nanoparticles: control of the helix structure and enhanced differential binding to immune cells

The stabilizing action of C(α)-tetrasubstituted α-amino acids inserted into a sequence of short peptides allowed for the first time the preparation of water-soluble nanoparticles of different materials coated with a helix-structured undecapeptide. This peptide coating strongly favors nanoparticle uptake by human immune system cells.