To the memory of Marcus Ratzke Atom economy (AE) or atom utilization was one of the first defining terms in the sustainable chemistry movement. In contrast to the often-cited twelve (qualitative) principles of green chemistry, AE represents a metric for quantification purposes. The theoretical efficiency of a reaction expressed by its stoichiometric equation can be determined by AE Product [g/mol]/(Substrate 1 Substrate 2 ...) [g/mol] and compared with synthetic alternatives. Of course, the atom economy will be of limited use, if starting materials differ much in complexity, i.e., in the degree of refinement. In these cases, their syntheses have to be taken in consideration, too. But, the further the retrospect goes and the more preceding synthesis steps ramify, the more complex the calculation gets. To overcome this limitation, we introduce a stepwise approach that is enabled by a simple modification of the above formula (P product; S substrate; Syn. synthesis): AE P [g/mol]/((S1/AE(Syn. of S1)) (S2/AE(Syn. of S2)) ...) [g/mol]. To illustrate this equation, which is derived mathematically, the convergent multistep synthesis of the natural product trans-chrysanthemic acid is subjected to a stepwise method of calculation. The equation can be understood as a general expression for related ratios, i.e., there are corresponding modified equations for yield, selectivity, etc. In terms of the yield, it is no longer necessary to decide between the chains of the convergent synthesis, when possibly forced to ignore significant parts of the sequence. For demonstration purposes, the yield of the convergent synthesis of the natural product peridinin has been determined with a correspondingly modified equation.
Given an integral lattice L and a hyperbolic decomposition of some quotient L/pL, there is a simple technique for obtaining other lattices of the same dimension and discriminant as L⊥ … ⊥L. When applied to the D4 and E8 root lattices, for example, this yields a new sphere packing in ℝ32, which is denser than those known up to now, and an extremal type II lattice in ℝ64.
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