Mirror symmetry for a toric variety involves Laurent polynomials whose symplectic topology is related to the algebraic geometry of the toric variety. We show that there is a monodromy action on the monomially admissible Fukaya-Seidel categories of these Laurent polynomials as the arguments of their coefficients vary that corresponds under homological mirror symmetry to tensoring by a line bundle naturally associated to the monomials whose coefficients are rotated. In the process, we introduce the monomially admissible Fukaya-Seidel category as a new interpretation of the Fukaya-Seidel category of a Laurent polynomial on (C * ) n , which has other potential applications, and give evidence of homological mirror symmetry for non-compact toric varieties.
We use the Hofer norm to show that all Hamiltonian diffeomorphisms with compact support in R 2n that displace an open connected set with a nonzero Hofer-Zehnder capacity move a point farther than a capacitydependent constant. In R 2 , this result is extended to all compactly supported area-preserving homeomorphisms. Next, using the spectral norm, we show the result holds for Hamiltonian diffeomorphisms on closed surfaces. We then show that all area-preserving homeomorphisms of S 2 and RP 2 that displace the closure of an open connected set of fixed area move a point farther than an area-dependent constant.2010 Mathematics Subject Classification. Primary: 57R17. Key words and phrases. Spectral Norm, Displaced Disks, Hofer Norm. We would like to thank Dmitri Burago for informing us of the problem of displacing sets on the sphere and to again thank Dmitri Burago, Sergei Ivanov, and Leonid Poltervich for the idea to use the spectral norm to solve this problem. We would also like to thank Misha Guysinsky for his guidance and for many useful discussions and comments. We thank Sobhan Seyfaddini for suggesting that his results in [10] would allow us to prove the result on the cotangent bundle.
The experimental procedures of time-resolved laser photolysis studies of pyrene excimer formation in solution have been scrutinized and the appropriate modifications have been implemented. Contrary to the experimental methods applied in all related previous work, the selection of a suitable excitation wavelength (such that the corresponding pyrene absorbance is less than 0.5 absorbance units) utilized in our study results in simple homogeneous kinetics. Consequently, the rate parameters obtained and the mechanism proposed differ significantly from those published previously. The rate constant values of the unimolecular decay of the pyrene monomer, the unimolecular decay of the pyrene excimer, and the excimer formation in decane solution (η = 0.860 mPa s) at 25 °C are (2.38 ± 0.01) × 10(6) s(-1), (2.78 ± 0.02) × 10(7) s(-1), and (3.11 ± 0.06) × 10(9) M(-1) s(-1), respectively. The dissociation of the excimer to form a singlet excited state pyrene and a ground state pyrene was shown to be negligible. The energies of activation corresponding to the monomer and excimer unimolecular decays were found to be 2.51 ± 0.07 and 25.7 ± 0.7 kJ mol(-1), respectively. Also, our temperature resolved laser photolysis data revealed that the excimer formation has a negative energy of activation equal to -11.2 ± 0.5 kJ mol(-1). This unique phenomenon may be attributed to steric effects in the collision of the reactants. The current findings are important for the correct data analysis and interpretation in many applications of the pyrene excimer.
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