Covering n-Segment Unit Arcs Is Not Sufficient

In 1974 Gerriets and Poole conjectured for n = 3 that a convex set in the plane which contains a congruent copy of every n-segment polygonal arc of unit length must be a cover for the family of all unit arcs. We disprove this general conjecture by describing for each positive integer n a convex region R_n that contains a ongruent copy of every n-segment unit arc but not a congruent copy of every unit arc.     

Synthesis of Isagarin,a Tetracyclic Naphthoquinone via a Palladium‐Catalyzed Cyclization

Total synthesis of isagarin (2) has been accomplished in five steps with 24.1% overall yield from 2‐bromonaphthoquinone (6). The key step is a palladium‐catalyzed cyclization to form a bicyclic ketal ring.     

Real algebraic curves of constant width

Rabinowitz constructed a parametric curve of constant width and expressed it as a plane algebraic curve; however, the algebraic curve also contains isolated points separate from the original curve. We show how to modify his example in order to produce a curve with no isolated points. We then conjecture a method for producing a family of such curves and prove the conjecture in several cases.     

The broadest three-segment unit arc

We describe the broadest three-segment unit arc in the plane, and we conclude with some conjectures about the broadest n-segment unit arc for n > 3. Communicated by Imre Bárány     

A new flavonoid from the leaves of Atalantia monophylla (L.) DC

A new flavonoid, atalantraflavone (1) as well as eight known compounds including atalantoflavone (2), racemoflavone (3), 5,4′-dihydroxy-(3″,4″-dihydro-3″,4″-dihydroxy)-2″,2″-dimethylpyrano-(5″,6″:7,8)-flavone (4), lupalbigenin (5), anabellamide (6), citrusinine I (7), p-hydroxybenzaldehyde (8), and frideline (9), were isolated from the leaves of Atalantia monophylla (L.) DC. Focusing on Alzheimer’s disease, acetylcholine esterase (AChE) inhibition and antioxidant activity were evaluated using the modified Ellman’s method and the ABTS scavenging assay, respectively. It was found that isoflavonoid 5, lupalbigenin, showed 79% inhibition to AChE and was 1.4-fold stronger than the tacrine standard. In addition, acridone 7, citrusinine I, displayed 90.68% antioxidant activity.     

Functional group-mediated biotransformation by Curvularia lunata NRRL 2178: synthesis of 3-dehydro-2-deoxy-ecdysteroids from the 3-hydroxy-2-mesyloxy analogues

Microbial transformation of ecdysteroids with 3-hydroxy-2-mesyloxy functional group by the fungus Curvularia lunata NRRL 2178 furnished 3-dehydro-2-deoxy analogues. The metabolites included 3-dehydro-2-deoxy analogues of 20-hydroxyecdysone, pterosterone, ponasterone A, 20-hydroxyecdysone 20,22-acetonide, shidasterone, and poststerone. The mild biotransformation conditions prevented the metabolites from C-5 epimerization.     

Wetzel’s sector covers unit arcs

Periodica Mathematica Hungarica - We settle J. Wetzel’s 1970’s conjecture and show that a $$30^{\circ }$$ circular sector of unit radius can accommodate every planar arc of unit length....