Regressions and monotone chains II: The poset of integer intervals

A regressive function (also called a regression or contractive mapping) on a partial order P is a function mapping P to itself such that (x)x. A monotone k-chain for is a k-chain on which is order-preserving; i.e., a chain x ₁<...ksuch that (x ₁)...(x_k). Let P _nbe the poset of integer intervals {i, i+1, ..., m} contained in {1, 2, ..., n}, ordered by inclusion. Let f(k) be the least value of n such that every regression on P _nhas a monotone k+1-chain, let t(x,j) be defined by t(x, 0)=1 and t(x,j)=x ^t(x,j–1). Then f(k) exists for all k (originally proved by D. White), and t(2,k) < f(K) <t( + _k, k) , where _k 0 as k. Alternatively, the largest k such that every regression on P _nis guaranteed to have a monotone k-chain lies between lg^*(n) and lg^*(n)–2, inclusive, where lg^*(n) is the number of appliations of logarithm base 2 required to reduce n to a negative number. Analogous results hold for choice functions, which are regressions in which every element is mapped to a minimal element.     

Revised Robertson's test theory of special relativity: Space-time structure and dynamics

The experimental testing of the Lorentz transformations is based on a family of sets of coordinate transformations that do not comply in general with the principle of equivalence of the inertial frames. The Lorentz and Galilean sets of transformations are the only member sets of the family that satisfy this principle. In the neighborhood of regular points of space-time, all members in the family are assumed to comply with local homogeneity of space-time and isotropy of space in at least one free-falling elevator, to be denoted as Robertson'sab initio rest frame [H. P. Robertson,Rev. Mod. Phys. 21, 378 (1949)].Without any further assumptions, it is shown that Robertson's rest frame becomes a preferred frame for all member sets of the Robertson family except for, again, Galilean and Einstein's relativities. If one now assumes the validity of Maxwell-Lorentz electrodynamics in the preferred frame, a different electrodynamics spontaneously emerges for each set of transformations. The flat space-time of relativity retains its relevance, which permits an obvious generalization, in a Robertson context, of Dirac's theory of the electron and Einstein's gravitation. The family of theories thus obtained constitutes a covering theory of relativistic physics.A technique is developed to move back and forth between Einstein's relativity and the different members of the family of theories. It permits great simplifications in the analysis of relativistic experiments with relevant Robertson's subfamilies. It is shown how to adapt the Clifford algebra version of standard physics for use with the covering theory and, in particular, with the covering Dirac theory.Part of this work was done at the Department of Physics, Utah State University, Logan, Utah 84322.     

The influence of metal cluster size on adsorption energies: CO adsorbed on Au clusters supported on TiO2

Infrared reflection absorption spectroscopy (IRAS) has been used to study CO adsorption on Au clusters ranging in size from 1.8 to 3.1 nm, supported on TiO(2). The adsorbed CO vibrational frequency blue-shifts slightly (approximately 4 cm(-)(1)) compared to that adsorbed on bulk Au, whereas the heats of adsorption (-DeltaH(ads)) increase sharply with decreasing cluster size, from 12.5 to 18.3 kcal/mol.     

A thiazole-containing tripodal ligand: synthesis, characterization, and interactions with metal ions and matrix metalloproteinases

A new tripodal ligand, tris[2-(((2-thiazolyl)methylidene)amino)ethyl]amine (Tatren), has been synthesized and characterized by NMR, IR, and UV-visible absorbance spectroscopy and elemental analysis. Tatren forms stable complexes with transition metal ions (Zn(2+), 1; Mn(2+), 2; Co(2+), 3) and the alkaline earth metal ions (Ca(2+), 4; Mg(2+), 5). Single-crystal X-ray structures of 1, 2, and 5 revealed six-coordinate chelate complexes with formula [M(Tatren)](ClO(4))(2) in which the metal centers are coordinated by three thiazolyl N atoms and three acyclic imine N atoms. Crystals of 1, 2, and 5 are monoclinic, P2(1)/c space group. Crystals of 4 are triclinic, P space group. The Ca(2+) complex is eight-coordinate with all N atoms of Tatren and one water molecule coordinated to the metal ion. Spectrophotometric titrations show that formation constants for the chelates of metal ions are >1 in methanol. Free Tatren inhibits the catalytic domain of matrix metalloproteinase-13 (MMP-13, collagenase-3) with K(i) = 3.5 +/- 0.6 microM. Molecular mechanics-based docking calculations suggest that one leg of Tatren coordinates to the catalytic Zn(2+) in MMPs-2, -9, and -13 with significant hydrogen bonding to backbone amide groups. High-level DFT calculations suggest that, in the absence of nonbonded interactions between Tatren and the enzyme, the most stable first coordination sphere of the catalytic Zn(2+) is achieved with three imidazolyl groups from His residues and two imine N atoms from one leg of Tatren. While complexes (1-3) do not inhibit MMP-13 to a significant extent, 4 does (K(i) = 30 +/- 10 microM). Hence, this study shows that tripodal chelating ligands of this class and their Ca(2+) complexes have potential as active-site inhibitors for MMPs.     

An efficient synthesis of the constrained peptidomimetic 2-oxo-3-(N-9-fluorenyloxycarbonylamino)-1-azabicyclo[4.3.0]nonane-9-carboxylic acid from pyroglutamic acid

[reaction: see text] Azabicyclo[X.Y.0]alkane amino acids are rigid dipeptide mimetics that are useful tools for structure-activity studies in peptide-based drug discovery. Herein, we report an efficient synthesis of three diastereomers of 9-tert-butoxycarbonyl-2-oxo-3-(N-tert-butoxycarbonylamino)-1-azabicyclo[4.3.0]nonane (3S,6S,9S, 3S,6R,9R, and 3S,6R,9S). Methyl N-Boc-pyroglutamate is cleaved with vinylmagnesium bromide to produce an acyclic gamma-vinyl ketone. Michael addition of N-diphenylmethyleneglycine tert-butyl ester produces the N-Boc-delta-oxo-alpha,omega-diaminoazelate intermediate, which, on hydrogenloysis, gives the fused ring system. Acidolytic deprotection followed by Fmoc-protection provided building blocks suitable for solid-phase synthesis.     

Determination of DNA structure in solution: enzymatic deuteration of the ribose 2' carbon

An enzymatic solution to the problem of obtaining 13C/15N-labeled nucleotides that are deuterated uniquely at the H2' ' position within the ribose ring is presented. Selective deuteration occurs with an overall yield of >80%. The deuteron at the H2' ' position allows measurement of the scalar and residual dipolar couplings for the bond vectors attached to the C2' carbon of each ribose sugar. These data allow the accurate determination of sugar conformation. Interesting DNA double helices of 2-3 turns are now within the reach of solution NMR spectroscopy. As an example, these labeled nucleotides are incorporated uniquely at positions 6-14 in a 20-bp DNA sequence containing the adenovirus major late promoter.     

Determination of uranium by thermal and epithermal neutron activation in natural waters and in human urine

Uranium is determined via its ²³⁹U nuclide (74.0 keV, t_{$\text{1}\text{2}$} = 23.5 min) in natural waters down to 0.03 ng U ml^-1 after preconcentration with activated carbon and oxine; 30-min irradiation and counting times are used. No preconcentration is required for samples containing more than 4 ng U ml^-1 with 10-min irradiation and counting times. Uranium in urine can be determined under a boron shield at the 5 ng ml^-1 level after 30-min irradiation and counting.     

Transport behavior in isomeric alcohols. I. The conductance of tetraalkylammonium salts in isobutyl and isopentyl alcohols at 25°C

Conductance measurements at 25°C have been carried out on Pr₄NBr, Bu₄NBr,n-Pe₄NPr, Pr₄NI, Bu₄NI,n-Pe₄NI, Bu₄NClO₄, Bu₄NBF₄, and Bu₄PClO₄ in isobutyl alcohol and on Bu₄NBr and Bu₄NClO₄ in isopentyl alcohol. The main purpose of the measurements was to investigate the effect of increased branching in the alkyl portion of the alcohol on both ionic mobilities and ionic association. Comparison is made with the two isodielectric solvents butanone andn-butyl alcohol. Branching appears to produce little change in ionic mobilities and approximately a 10% increase in association.     

Determination of selected elements in bone samples by neutron activation and γ-spectrometry

A non-destructive method for the quantification of eleven elements in bone samples is described. The analytical scheme is based on short (30 s) irradiation with thermal neutrons followed immediately (decay time 10 s) by counting fluorine-20 for 30 s and, after a total waiting time of 150 s, by 10-min γ-spectrometry counting, which give data for Ca, Cl, Mg, Mn, Na, and V. Use of a boron carbide shield for a second set of irradiations with epithermal neutrons permits the additional determination of bromine and strontium and calculation of the contribution of aluminium and phosphorus to the total activity of ₂₈Al.