Axiomatizing relativistic dynamics without conservation postulates

A part of relativistic dynamics (or mechanics) is axiomatized by simple and purely geometrical axioms formulated within first-order logic. A geometrical proof of the formula connecting relativistic and rest masses of bodies is presented, leading up to a geometric explanation of Einstein's famous $E=mc^2$. The connection of our geometrical axioms and the usual axioms on the conservation of mass, momentum and four-momentum is also investigated.Comment: 21 pages, 7 figure

A logic road from special relativity to general relativity

We present a streamlined axiom system of special relativity in first-order logic. From this axiom system we "derive" an axiom system of general relativity in two natural steps. We will also see how the axioms of special relativity transform into those of general relativity. This way we hope to make general relativity more accessible for the non-specialist

Twin Paradox and the logical foundation of relativity theory

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 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.Comment: 24 pages, 6 figure

Cross-Dehydrogenative Couplings between Indoles and β-Keto Esters : Ligand-Assisted Ligand Tautomerization and Dehydrogenation via a Proton-Assisted Electron Transfer to Pd(II)

Cross-dehydrogenative coupling reactions between -ketoesters and electron-rich arenes, such as indoles, proceed with high regiochemical fidelity with a range of -ketoesters and indoles. The mechanism of the reaction between a prototypical -ketoester, ethyl 2-oxocyclopentanonecarboxylate and N-methylindole, has been studied experimentally by monitoring the temporal course of the reaction by 1H NMR, kinetic isotope effect studies, and control experiments. DFT calculations have been carried out using a dispersion-corrected range-separated hybrid functional (B97X-D) to explore the basic elementary steps of the catalytic cycle. The experimental results indicate that the reaction proceeds via two catalytic cycles. Cycle A, the dehydrogenation cycle, produces an enone intermediate. The dehydrogenation is assisted by N-methylindole, which acts as a ligand for Pd(II). The compu-tational studies agree with this conclusion, and identify the turnover-limiting step of the dehydrogenation step, which involves a change in the coordination mode of the -keto ester ligand from an O,O’-chelate to an C-bound Pd enolate. This ligand tautom-erization event is assisted by the -bound indole ligand. Subsequent scission of the ’-C–H bond takes place via a proton-assisted electron transfer mechanism, where Pd(II) acts as an electron sink and the trifluoroacetate ligand acts as a proton acceptor, to pro-duce the Pd(0) complex of the enone intermediate. The coupling is completed in cycle B, where the enone is coupled with indole. Pd(TFA)2 and TFA-catalyzed pathways were examined experimentally and computationally for this cycle, and both were found to be viable routes for the coupling step