O(N) and O(N) and O(N)

Three related analyses of $\phi^4$ theory with $O(N)$ symmetry are presented. In the first, we review the $O(N)$ model over the $p$-adic numbers and the discrete renormalization group transformations which can be understood as spin blocking in an ultrametric context. We demonstrate the existence of a Wilson-Fisher fixed point using an $\epsilon$ expansion, and we show how to obtain leading order results for the anomalous dimensions of low dimension operators near the fixed point. Along the way, we note an important aspect of ultrametric field theories, which is a non-renormalization theorem for kinetic terms. In the second analysis, we employ large $N$ methods to establish formulas for anomalous dimensions which are valid equally for field theories over the $p$-adic numbers and field theories on $\mathbb{R}^n$. Results for anomalous dimensions agree between the first and second analyses when they can be meaningfully compared. In the third analysis, we consider higher derivative versions of the $O(N)$ model on $\mathbb{R}^n$, the simplest of which has been studied in connection with spatially modulated phases. Our general formula for anomalous dimensions can still be applied. Analogies with two-derivative theories hint at the existence of some interesting unconventional field theories in four real Euclidean dimensions.Comment: 44 pages, 8 figure

Tripodal O-N-O bis-Phenolato Amine Titanium(IV) Complexes Show High In Vitro Anti-Cancer Activity

The octahedral titanium(IV) complexes trans,mer‐[Ti{R3N(CH2C6H2‐2‐O‐4‐R2‐6‐R1)2}2] (R1 = Me, OMe, Cl; R2 = Me, OMe, F, Cl; R3 = Me, Et; not all combinations) are synthesised in two steps from simple phenols in 36‐53% overall yield. The highly crystalline (4 X‐ray structures) complexes are active against MCF‐7 (breast) and HCT‐116 (colon) cancer cell lines showing widely varying GI50 values in the range 1‐100 µM depending on R1‐R3. Highest activities are realised when R1 = OMe and R2, R3 = Me (GI50 ~1 µM for MCF‐7 and 2‐3 µM for HCT‐116). These are respectively 8× and 3× times greater than the activities of cisplatin in the same cell lines. These titanium complexes show some significant selectivity for cancer cell lines; up to 7× higher in MCF‐7 compared to non‐cancer (MRC‐5) fibroblast cells. Details of cellular mode of action indicators (cell cycle perturbation, Annexin V, γ‐H2AX, and caspase studies) that point to an apoptosis mode for the most active compound (R1 = OMe and R2, R3 = Me) are also reported

An In-Place Sorting with O(n log n) Comparisons and O(n) Moves

We present the first in-place algorithm for sorting an array of size n that performs, in the worst case, at most O(n log n) element comparisons and O(n) element transports. This solves a long-standing open problem, stated explicitly, e.g., in [J.I. Munro and V. Raman, Sorting with minimum data movement, J. Algorithms, 13, 374-93, 1992], of whether there exists a sorting algorithm that matches the asymptotic lower bounds on all computational resources simultaneously

Insertion Sort is O(n log n)

Traditional Insertion Sort runs in O(n^2) time because each insertion takes O(n) time. When people run Insertion Sort in the physical world, they leave gaps between items to accelerate insertions. Gaps help in computers as well. This paper shows that Gapped Insertion Sort has insertion times of O(log n) with high probability, yielding a total running time of O(n log n) with high probability.Comment: 6 pages, Latex. In Proceedings of the Third International Conference on Fun With Algorithms, FUN 200

Collision integrals and high temperature transport properties for N-N, O-O, and N-O

Accurate collision integrals are reported for the interactions of N(4 S 0) + N(4 S 0), O(3 P), and N(4 S 0) + O(3 P). These are computed from a semiclassical formulation of the scattering using the best available representations of all of the potential energy curves needed to describe the collisions. Experimental RKR curves and other accurate measured data are used where available; the results of accurate ab initio electronic structure calculations are used to determine the remaining potential curves. The high-lying states are found to give the largest contributions to the collision cross sections. The nine collision integrals, needed to determine transport properties to second order, are tabulated for translational temperatures in the range 250 K to 100,000 K. These results are intended to reduce the uncertainty in future predictions of the transport properties of nonequilibrium air, particularly at high temperatures. The viscosity, thermal conductivity, diffusion coefficient, and thermal diffusion factor for a gas composed of nitrogen and oxygen atoms in thermal equilibrium are calculated. It was found that the second order contribution to the transport properties is small. Graphs of these transport properties for various mixture ratios are presented for temperatures in the range 5000 to 15000 K

Critical exponent omega at O(1/N) in O(N) x O(m) spin models

We compute the O(1/N) correction to the stability critical exponent, omega, in the Landau-Ginzburg-Wilson model with O(N) x O(m) symmetry at the stable chiral fixed point and the stable direction at the unstable antichiral fixed point. Several constraints on the O(1/N) coefficients of the four loop perturbative beta-functions are computed.Comment: 16 latex pages, 3 postscript figure

Chiral exponents in O(N) x O(m) spin models at O(1/N^2)

The critical exponents corresponding to chirality are computed at O(1/N^2) in d-dimensions at the stable chiral fixed point of a scalar field theory with an O(N) x O(m) symmetry. Pade-Borel estimates for the exponents are given in three dimensions for the Landau-Ginzburg-Wilson model at m = 2.Comment: 8 latex page

Tetra-n-butylamine(carbonato-kappa O-2,O')cobalt(III)n-butylcarbamate dihydrate

The title compound, [Co(CO3)(C4H11N)4](C5H10NO2)·2H2O, is a coordination complex with an N4O2 coordination sphere around the central CoIII ion. The small bite angle of the chelating carbonate causes a distortion of the octahedral geometry to an approximately C2v local symmetry. Hydrogen-bonding between the carbonate, carbamate and amine groups, and the water of crystallization, results in a complex two-dimensional network