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Twin Paradox and the Logical Foundation of Relativity Theory 总被引:3,自引:0,他引:3
We study the foundation of space-time theory in the framework of first-order logic (FOL). Since the foundation of mathematics has been successfully carried through (via set theory) in FOL, it is not entirely impossible to do the same for space-time theory (or relativity). First we recall a simple and streamlined FOL-axiomatization Specrel of special relativity from the literature. Specrel is complete with respect to questions about inertial motion. Then we ask ourselves whether we can prove the usual relativistic properties of accelerated motion (e.g., clocks in acceleration) in Specrel. As it turns out, this is practically equivalent to asking whether Specrel is strong enough to “handle” (or treat) accelerated observers. We show that there is a mathematical principle called induction (IND) coming from real analysis which needs to be added to Specrel in order to handle situations involving relativistic acceleration. We present an extended version AccRel of Specrel which is strong enough to handle accelerated motion, in particular, accelerated observers. Among others, we show that~the Twin Paradox becomes provable in AccRel, but it is not provable without IND. 相似文献
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The conversion and residence time were investigated during the bulk polymerization of styrene in a twin screw extruder. It was found that polymerization mainly occurred in the zone between 400 and 1000 mm along the screw axis in the extruder, corresponding to the residence time of the reactants ranging from 1 to 4 min in the extruder. Furthermore, the processing conditions (feed rate, screw rotation rate) and average molecular weight of the polymer have a great effect on the residence time. Based on dimensionless analysis, a model of the residence time has been built‐up, which has been confirmed by the results of realistic measurements. A kinetic model of the polymerization has also been established under the assumption that the screw extruder can be regarded as an ideal plug flow reactor. Copyright © 2004 John Wiley & Sons, Ltd. 相似文献
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The preparation and properties of cyanobiphenyl liquid crystal droplets encapsulated by the polymerizable lecithin 1,2-bis(10,12-tricosadiynoyl)-sn-glyero-3-phosphocholine (DC8,9PC) are described. Under a wide variety of preparation conditions the droplets obtain a diameter of approximately 10 mum. These droplets are stable for periods of over one year at room temperature. Furthermore, they are stable upon temperature cycling between the nematic and isotropic phases and between the smectic A to nematic to isotropic phase transitions. 相似文献
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Falko Böttger‐Hiller Patrick Kempe Dr. Gerhard Cox Dr. Alexander Panchenko Dr. Nicole Janssen Dr. Albrecht Petzold Prof. Dr. Thomas Thurn‐Albrecht Lars Borchardt Dr. Marcus Rose Prof. Dr. Stefan Kaskel Colin Georgi Prof. Dr. Heinrich Lang Prof. Dr. Stefan Spange 《Angewandte Chemie (International ed. in English)》2013,52(23):6088-6091
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Chinmay G. Nardele S. K. Asha 《Journal of polymer science. Part A, Polymer chemistry》2012,50(14):2770-2785
We report systematic studies on a homologous series of twin liquid crystalline (LC) molecules based on phenyl and naphthyl azobenzene ( PnP and NpnNp ) as well as segmented copolyesters based on them. The twin series had the structure azobenzene–oligooxyethylene–azobenzene, where the ethyleneoxy length was varied from 2 to 6 units. The LC properties of the twin series depended on the chemical structure of the azochromophore and also the length of the central oligooxyethylene segment. The PnP series exhibited smectic LC properties for n > three oligooxyethylene units. Conversely, NpnNp series exhibited spherulitic phases only for the shortest member – Np2Np . One non‐LC short spacer twin ( P2P ) and one LC long spacer twin ( P6P ) were incorporated as part of a main chain polyester composed of fully aliphatic segments of sebacate and di or tetraethylene glycol (DEG/TEG) units by melt polycondensation. Non‐LC P2P formed LC polymers even at low (5 mol %) incorporation in DEG‐based copolymers, whereas the LC‐ P6P could do so only at 30 mol % incorporation. The LC properties of the twin molecules as well as copolymers were studied using differential scanning calorimetry, polarized light microscopy (PLM) along with variable temperature wide angle X‐ray diffraction. © 2012 Wiley Periodicals, Inc. J Polym Sci Part A: Polym Chem, 2012 相似文献
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《高分子科学杂志,A辑:纯化学与应用化学》2013,50(12):1427-1443
Three different series of twin nonlinear optic (NLO) molecules were studied, in which the two NLO chromophores are linked by a central flexible polymethylene spacer. The first series, which had two azobenzene chromophores (Azo-twins), was designed to also exhibit liquid crystallinity. Most of the members of this series exhibited a nematic mesophase. The second series had two 4-nitrophenol units as chromphores (PNP-twins), while the third one was based on 4-alkylsulfonyl-4'-alkoxy azobenzene chromophores (Sulfazo-twins). These twin NLO systems exhibited interesting odd-even oscillations in their second harmonic generation (SHG) efficiencies in the powder form. When the spacer had an odd number of methylene groups, they exhibited significantly higher powder SHG efficiency than their even counterparts, with the even ones most often exhibiting no detectable SH signal. Preliminary single crystal X-ray diffraction studies performed on the PNP-twin series showed that while the even members possess a molecular center of symmetry and pack centro-symmetrically, the odd ones do not, leading to the observed alternation. The orientational-disordering dynamics of two of the twin series – the PNP and the Sulfazo-twin series, doped in a poly(methyl methacrylate) matrix, was also studied by monitoring the SH-signal decay in electric field poled samples. Interestingly, the maximum attainable SH signal, χ(2), in the poled samples also showed an odd-even oscillation with the odd ones again exhibiting a higher value of χ(2). The temporal stability of the SHG intensity at 70°C, after the removal of the applied corona, was also studied and the relaxation of the chromophores was found to follow a biexponential decay. The slower relaxation component exhibits a spacer length dependence, which suggests the interplay of two factors in governing the temporal stability in such polymer doped twin systems, one is the conformational discomfort experienced by the spacer in adopting a U-shaped geometry, and the other the electrostatic repulsion when two aligned dipoles lie very close to each other. 相似文献
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Dr. Somnath Dey Dr. Andreas Schönleber Prof. Dr. Sander van Smaalen Dr. Wolfgang Morgenroth Prof. Dr. Finn Krebs Larsen 《Chemistry (Weinheim an der Bergstrasse, Germany)》2022,28(13):e202104151
Phase transitions in molecular crystals are often determined by intermolecular interactions. The cage complex of [Co(C12H30N8)]3+ ⋅ 3 NO3− is reported to undergo a disorder-order phase transition at Tc1 ≈133 K upon cooling. Temperature-dependent neutron and synchrotron diffraction experiments revealed satellite reflections in addition to main reflections in the diffraction patterns below Tc1. The modulation wave vector varies as function of temperature and locks in at Tc3≈98 K. Here, we demonstrate that the crystal symmetry lowers from hexagonal to monoclinic in the incommensurately modulated phases in Tc1<T<Tc3. Distinctive levels of competitions: trade-off between longer N−H⋅⋅⋅O and shorter C−H⋅⋅⋅O hydrogen bonds; steric constraints to dense C−H⋅⋅⋅O bonds give rise to pronounced modulation of the basic structure. Severely frustrated crystal packing in the incommensurate phase is precursor to optimal balance of intermolecular interactions in the lock-in phase. 相似文献