Konstytucja polityczna Republiki Portugalii z 11 kwietnia 1933 r.

R/Z Index Theory

We define topological and analytic indices in R/Z K-theory and show that they are equal.

11 - IJCBMS- Dr. Praveen R

Synthesis of anacardic acids, 6-[8(Z),11(Z)-pentadecadienyl]salicylic acid and 6-[8(Z),11(Z),14-pentadecatrienyl]salicylic acid.

11-Chloro-3-methoxy-2-undecenal was synthesized from 8-bromooctanol, and an annelation reaction with this aldehyde and ethyl acetoacetate proceeded to give the ethyl 6-(8-chlorooctyl)salicylate. Ethyl 6-(8-chlorooctyl)salicylate was converted to ethyl 6-(7-formylheptyl)-2-methoxybenzoate through the iodide after protection of the phenolic hydroxyl group. Finally, the Wittig reaction with the al...

An Irrational Construction of R from Z

This paper reports on a construction of the real numbers in the ProofPower implementation of the HOL logic. Since the original construction was implemented, some major improvements to the ideas have been discovered. The improvements involve some entertaining mathematics: it turns out that the Dedekind cuts provide many routes one can travel to get from the ring of integers, Z, to the field of r...

DISTORTION ELEMENTS IN Diff ∞ (R/Z)

We consider the group of smooth diffeomorphisms of the circle. We show that any recurrent f (in the sense that {f}n∈Z is not discrete) is in fact a distortion element (in the sense that its iterates can be written as short compositions involving finitely many smooth diffeomorphisms). Thus rotations are distortion elements.