Additivity and exponentiality are alien to each other

Let (S, +) be a (semi)group and let (R,+, ·) be an integral domain. We study the solutions of a Pexider type functional equation $$f(x+y) + g(x+y) = f(x) + f(y) + g(x)g(y)$$ for functions f and g mapping S into R. Our chief concern is to examine whether or not this functional equation is equivalent to the system of two Cauchy equations $$\left\{\begin{array}{@{}ll} f(x+y) = f(x) + f(y)\\ g(x+y) = g(x)g(y)\end{array}\right.$$ for every ${x,y \in S}$ .     

Solution of a functional equation arising from utility that is both separable and additive

The problem of determining all utility measures over binary gambles that are both separable and additive leads to the functional equation

The following conditions are more or less natural to the problem: strictly increasing, strictly decreasing; both map their domains onto intervals ( onto a , onto ); thus both are continuous, , , , , . We determine, however, the general solution without any of these conditions (except , , both into). If we exclude two trivial solutions, then we get as general solution (, ; for ), which satisfies all the above conditions. The paper concludes with a remark on the case where the equation is satisfied only almost everywhere.

     

Polymer-bound flavins. I. Influence of polarity of the polymer chain on properties of bound flavin

The preparation of flavin-containing, water-soluble terpolymers with varying contents of quaternary ammonium and neutral monomer units is described. The composition of the polymer, that is, the polarity of the polymer chain, affects the UV/VIS absorbance spectrum of the bound flavin moiety by influencing the flavin-solvent interaction. Aerobic oxidation of 1-benzyl-1,4-dihydronicotinamide (BNAH) is accelerated as much as 17-fold by polymer-bound flavins, compared with free flavin. Depending on the ratio of polar/nonpolar groups in the flavin polymer and the 2-propanol content of the aqueous reaction medium (pseudo) first-order or Michaelis-Menten type kinetics were observed as a function of substrate concentration. The influence of the polymer composition on flavin/BNAH complex formation (K_m^?1) and rate-limiting hydride transfer step (k₂) is discussed. Accumulation of the substrate in the polymer domain due to apolar interactions is one of the major factors that determine the observed rate enhancements.     

A Concise Total Synthesis of (+)‐Tetrabenazine and (+)‐α‐Dihydrotetrabenazine

Highly concise asymmetric total syntheses of (+)‐tetrabenazine ( 1 ), a drug for the treatment of chorea associated with Huntington’s disease, and of (+)‐α‐dihydrotetrabenazine ( 2 ), an active metabolite of 1 , have been accomplished. Our synthetic route features a trans‐selective enol etherification, followed by an unprecedented cation‐dependent aza‐Claisen rearrangement to establish the carbon framework and two stereogenic centers of tetrabenazine. The syntheses consist of seven steps (34 % overall yield) for (+)‐ 2 and eight steps (22 % overall yield) for (+)‐ 1 .     

Millijoule laser pulse delivery for spark ignition through kagome hollow-core fiber

We report on power handling oriented design of kagome lattice hollow-core fiber and demonstrate through it for the first time nanosecond laser pulses induced spark ignition in a friendly manner. Two different core designs and transmission bands are investigated and evaluated. The energy threshold damage was measured to be in excess of the 10 mJ level and the output power density is approaching the TW/cm2 after focusing; demonstrating the outstanding ability of such fiber for high power delivery.