Ramsey partitions of integers and pair divisions

Ifk ₁ andk ₂ are positive integers, the partitionP = (₁,₂,..., _n ) ofk ₁+k ₂ is said to be a Ramsey partition for the pairk ₁,k ₂ if for any sublistL ofP, either there is a sublist ofL which sums tok ₁ or a sublist ofP –L which sums tok ₂. Properties of Ramsey partitions are discussed. In particular it is shown that there is a unique Ramsey partition fork ₁,k ₂ having the smallest numbern of terms, and in this casen is one more than the sum of the quotients in the Euclidean algorithm fork ₁ andk ₂.An application of Ramsey partitions to the following fair division problem is also discussed: Suppose two persons are to divide a cake fairly in the ratiok ₁k ₂. This can be done trivially usingk ₁+k ₂-1 cuts. However, every Ramsey partition ofk ₁+k ₂ also yields a fair division algorithm. This method yields fewer cuts except whenk ₁=1 andk ₂=1, 2 or 4.     

On the topology of compact smooth three-dimensional Levi-flat hypersurfaces

We study topological conditions that must be satisfied by a compactC ^∞ Levi-flat hypersurface in a two-dimensional complex manifold, as well as related questions about the holonomy of Levi-flat hypersurfaces. As a consequence of our work, we show that no two-dimensional complex manifold admits a subdomain Ω with compact nonemptyC ^∞ boundary such that Ω ? ?².     

Base-assisted intramolecular 6-acetoxypyranone-acetylene [5+2] cycloaddition. Synthesis and reactivity of novel oxa-tricyclo[5.3.1.0]undecenones

The synthesis of 3,11-dioxatricyclo[5.3.1.0^1,5]undeca-5,9-dien-8-ones is reported from suitable 5-substituted furfuryl alcohols bearing an acetylenic side-chain. Successive peracid-mediated oxidative rearrangement of furan carbinols and base-assisted intramolecular 1,3-dipolar cycloaddition afforded oxygen-bridged tricyclo-undecane derivatives. Stereoselective transformations of cycloadducts are also discussed.     

Influence des interactions attractives non liees sur les stabilites relatives d'ethers et d'acetates d'enol renfermant un groupe styryle

The measure of relative stabilities of β-alcoxystyrene isomers I: C₆H₅-CHCH-OR shows that the trans compound is the most stable when RCH₃ and C₂H₅ and the cis compound is the most stable when Ri-C₃H₇ and t-C₄H₉. The orientation of the OR group can be determined by RMN ¹³C. The stabilities of these molecules are discussed in terms of non bounded attractive interactions. This interpretation is confirmed by the measure of relative stabilities of α-methyl β-alcoxy (and acetoxy)-styrene isomers II: C₆H₅-C(CH3)CH-OR. (RCH₃, C₂H₅ et COCH₃).     

Penicillin biosynthesis: Retention of configuration at c-3 of valine during its incorporation into the arnstein tripeptide

[3-³H]-valine was efficiently synthesised from sodium α-ketoisovalerate. With a β-lactam negative mutant of C. acremonium, l-[1-¹⁴C-3-³H]-valine and dl-[1-¹⁴C-3-³H]-valine were independently incorporated into the Arnstein tripeptide dimer, i.e. Bis-δ-(l-α-aminodipyl)-l-cystinyl-bis-d-valine, with full retention of trieium at C-3 of the d-valine residue. This result strongly suggested retention of configuration at C-3 of valine when the tripeptide was biosynthesised, and further limited the number of possible mechanisms for the biosynthesis of penicillins.     

Ribosomal synthesis of unnatural peptides

Combinatorial libraries of non-biological polymers and drug-like peptides could in principle be synthesized from unnatural amino acids by exploiting the broad substrate specificity of the ribosome. The ribosomal synthesis of such libraries would allow rare functional molecules to be identified using technologies developed for the in vitro selection of peptides and proteins. Here, we use a reconstituted E. coli translation system to simultaneously re-assign 35 of the 61 sense codons to 12 unnatural amino acid analogues. This reprogrammed genetic code was used to direct the synthesis of a single peptide containing 10 different unnatural amino acids. This system is compatible with mRNA-display, enabling the synthesis of unnatural peptide libraries of 10(14) unique members for the in vitro selection of functional unnatural molecules. We also show that the chemical space sampled by these libraries can be expanded using mutant aminoacyl-tRNA synthetases for the incorporation of additional unnatural amino acids or by the specific posttranslational chemical derivitization of reactive groups with small molecules. This system represents a first step toward a platform for the synthesis by enzymatic tRNA aminoacylation and ribosomal translation of cyclic peptides comprised of unnatural amino acids that are similar to the nonribosomal peptides.     

Reactions of metal ions with dithizone in molten naphthalene

A comparative study has been made of the reactions of metal ions with dithizone in the two media, chloroform and molten naphthalene. The absorption spectra of several metal dithizonates, prepared in molten naphthalene, were recorded and were found to be the same as those of the dithizonates obtained by extraction of metal ions from aqueous solution with chloroform dithizone.     

Synthesis of marine sponge alkaloid hachijodine B and a comment on the structure of ikimine B and on the absolute configuration of niphatesine D

The syntheses of the proposed structures of hachijodine B 1, ikimine B 2 and niphatesine D 3 from S-citronellol are described. Our results suggest that the gross structures of hachijodine B and niphatesine D are correct, but that ikimine B was incorrectly assigned. We have also established that the previous absolute stereochemical assignment for niphatesine D is unreliable.     

SIMULTANEOUS MEASUREMENT OF ACTION SPECTRA FOR PHOTOREACTIONS I AND II OF PHOTOSYNTHESIS

Abstract— We have devised a method of obtaining simultaneous action spectra for photoreactions I and II by analysis of direct and indirect effects involved in enhancement. The method requires previous determination of the neutral wavelength which gives maximum quantum yield by virtue of equal fractions of open reaction centers ( p and q ) for each photoreaction. A sufficient intensity of the neutral wavelength is used as a constant background. Upon addition of a weak modulated measuring light of intensity I_m and wavelength λ _m two amperometric signals are obtained for rate of oxygen evolution. A modulated signal (AC¯) isolates the direct effect of I_m and gives action of photoreaction II as AC/ I_m . An increment in total rate (ΔDC) also includes an indirect effect of I_m in perturbing reaction center conditions ( p and q ). From analysis of interaction of the two photoreactions, action for photoreaction I can be estimated as (2 ΔDC-AC)/ I_m . The method is applicable to whole cells, properly scales the two action spectra to each other, and removes contribution of the State 1-State 2 phenomena. Action spectra were obtained for Chlorella.