Mathematical bases of ancient Egyptian architecture and graphic art

In ancient Egyptian buildings, such as pyramids and pylons with inclined walls, the inclination was contrived to give a horizontal displacement measured in palms, half-palms or quarter-palms for a drop of one royal cubit of seven palms. The inclination of the edges between adjacent walls was not relevant to the design. Evidence is presented suggesting that in two-dimensional art the obliquity of slanting lines may have been determined by a similar system, but with a drop of six units instead of seven. After the introduction of the squared grid, obliquity was probably controlled by reference to grid intersections. There is no valid reason for supposing that the irrational numbers π and φ were implicated, except coincidentally, in the inclination of pyramids or in the obliquity of slanting lines.     

Two-photon absorption spectroscopy of iodine monochloride in the 5 eV region

Two-photon high resolution sequential spectroscopy has been used to excite iodine monochloride from X¹Σ⁺ ground state to the intermediate A³Π₁ state and thence to a final electronic state at 4.82 eV. Vibrational and rotational analyses of this state have been carried out for both isotopic species. For I³⁵Cl, T_e = 38916.0 cm^?1 ω_e = 168.99 cm^?1, ω_ex_e = 0.357 cm^?1 and B_e = 0.05685 cm^?1. The state probably has Ω = 1 in case (c) coupling approximation. It is also shown how to two-photon technique enables rotational line structure of the A ← X transition to be selectively excited for either isotopic species at a resolution of 500000, from an absorption mixture containing natural iodine monochloride plus its iodine dissociation product at equilibrium vapour pressure.     

Synthesis of 3'-deoxynucleosides with 2-oxabicyclo[3.1.0]hexane sugar moieties: addition of difluorocarbene to a 3',4'-unsaturated uridine derivative and 1,2-dihydrofurans derived from D- and L-xylose1

Syntheses of 3'-deoxy analogues of adenosine, cytidine, and uridine with a 2,2-difluorocyclopropane ring fused at C3'-C4' are described. Treatment of a 2',5'-protected-3',4'-unsaturated derivative of uridine with difluorocarbene [generated from (CF3)2Hg and NaI] gave a diastereomeric mixture of the 3',4'-difluoromethylene compounds (alpha-L-arabino/beta-D-ribo, approximately 5:4). The limited stereoselectivity for addition at the beta face results from competitive steric hindrance by an allylic 4-methoxybenzyloxy group at C2' on the alpha face and a homoallylic nucleobase at C1' on the beta face. Protected uracil derivatives were converted into their cytosine counterparts via 4-(1,2,4-triazol-1-yl) intermediates. Treatment of 1,2-dihydrofurans derived from D- and L-xylose with difluorocarbene resulted in stereospecific addition at the beta face (anti to the 1,2-O-isopropylidene group on the alpha face). Glycosylations with activated enantiomeric sugar derivatives with the fused difluorocyclopropane ring on the beta face gave protected adenine nucleosides, whereas attempted glycosylation with an alpha-fused derivative gave multiple products. Removal of base- and sugar-protecting groups gave new difluoromethylene-bridged nucleoside analogues.     

Higher Order Inference On A Treatment Effect Under Low Regularity Conditions

We describe a novel approach to nonparametric point and interval estimation of a treatment effect in the presence of many continuous confounders. We show that the problem can be reduced to that of point and interval estimation of the expected conditional covariance between treatment and response given the confounders. Our estimators are higher order U-statistics. The approach applies equally to the regular case where the expected conditional covariance is root-n estimable and to the irregular case where slower nonparametric rates prevail.     

Tiling Lattices with Sublattices,I

Call a coset C of a subgroup of Z^d{\bf Z}^{d} a Cartesian coset if C equals the Cartesian product of d arithmetic progressions. Generalizing Mirsky–Newman, we show that a non-trivial disjoint family of Cartesian cosets with union Z^d{\bf Z}^{d} always contains two cosets that differ only by translation. Where Mirsky–Newman’s proof (for d=1) uses complex analysis, we employ Fourier techniques. Relaxing the Cartesian requirement, for d>2 we provide examples where Z^d{\bf Z}^{d} occurs as the disjoint union of four cosets of distinct subgroups (with one not Cartesian). Whether one can do the same for d=2 remains open.     

Organometallic Polyelectrolytes: Synthesis,Characterization and Layer‐By‐Layer Deposition of Cationic Poly(ferrocenyl(3‐ammoniumpropyl)‐methylsilane)

The water soluble poly(ferrocenylsilane) polycation, poly(ferrocenyl(3‐ammoniumpropyl)methylsilane), was synthesized by transition metal‐catalyzed ring‐opening polymerization of the novel [1]ferrocenophane Fe(η‐C₅H₄)₂SiCH₃(CH₂)₃Cl and by subsequent side group modification. Amination of the chloropropyl moieties using potassium 1,1,3,3‐tetramethyldisilazide followed by acidic hydrolysis produced the polycation. The polycation was employed together with poly(sodium vinylsulfonate) in the electrostatic layer‐by‐layer self‐assembly process to form organometallic multilayers on quartz. The multilayer fabrication process was monitored using UV/Vis absorption spectroscopy and XPS.