Skill Rating by Bayesian Inference

Systems Engineering often involves computer modelling the behaviour of proposed systems and their components. Where a component is human, fallibility must be modelled by a stochastic agent. The identification of a model of decision-making over quantifiable options is investigated using the game-domain of Chess. Bayesian methods are used to infer the distribution of players’ skill levels from the moves they play rather than from their competitive results. The approach is used on large sets of games by players across a broad FIDE Elo range, and is in principle applicable to any scenario where high-value decisions are being made under pressure.

Two new mu-(1,3-azido)-bridged polymers: alternating single and double bridges in a 1D nickel(II) complex and uniform bridge in a 2D copper(II) complex: Syntheses, single-crystal structures and magnetic studies

Two polymeric azido bridged complexes [Ni2L2(N-3)(3)](n)(ClO4). (1) and [Cu(bpdS)(2)(N-3)],(ClO4),(H2O)(2.5n) (2) [L = Schiff base, obtained from the condensation of pyridine-2-aldehyde with N,N,2,2-tetramethyl-1,3-propanediamine; bpds = 4,4'-bipyridyl disulfide] have been synthesized and their crystal structures have been determined. Complex 1, C26H42ClN15Ni2O4, crystallizes in a triclinic system, space group P1 with a 8.089(13), b = 9.392(14), c = 12.267(18) angstrom, a = 107.28(l), b 95.95(1), gamma = 96.92(1)degrees and Z = 2; complex 2, C20H21ClCuN7O6.5S4, crystallizes in an orthorhombic system, space group Pnna with a = 10.839(14), b = 13.208(17), c = 19.75(2) angstrom and Z = 4. The crystal structure of I consists of 1D polymers of nickel(L) units, alternatively connected by single and double bridging mu-(1,3-N-3) ligand with isolated perchlorate anions. Variable temperature magnetic susceptibility data of the complex have been measured and the fitting,of magnetic data was carried out applying the Borris-Almenar formula for such types of alternating one-dimensional S = 1 systems, based on the Hamiltonian H = -J Sigma(S2iS2i-1 + aS(2i)S(2i+1)). The best-fit parameters obtained are J = -106.7 +/- 2 cm(-1); a = 0.82 +/- 0.02; g = 2.21 +/- 0.02. Complex 2 is a 2D network of 4,4 topology with the nodes occupied by the Cu-II ions, and the edges formed by single azide and double bpds connectors. The perchlorate anions are located between pairs of bpds. The magnetic data have been fitted considering the complex as a pseudo-one-dimensional system, with all copper((II)) atoms linked by [mu(1,3-azido) bridging ligands at axial positions (long Cu...N-3 distances) since the coupling through long bpds is almost nil. The best-fit parameters obtained with this model are J = -1.21 +/- 0.2 cm(-1), g 2.14 +/- 0.02. (c) Wiley-VCH Verlag GmbH & Co. KGaA, 69451 Weinheim, Germany, 2005).

A ferromagnetic linear trinuclear Ni(II)-Schiff base complex supported by phenoxo and cinnamato bridges

A new linear trinuclear nickel(II) complex, [Ni-3(salme)(2)(OCn)(4)] (Hsalme = 2-[(3-methylamino-propylimino)-methyl]-phenol, OCn = cinnamate), showing weak ferromagnetic coupling (J = 1.8(1) cm(-1)) through phenoxo and a novel tridentate bridging mode (1 kappa(OO)-O-2':2 kappa O') of the cinnamate ligand has been synthesized and structurally characterized by X-ray crystallography. (C) 2009 Elsevier B.V. All rights reserved.

Facile strategies for the synthesis and crystallization of linear trinuclear nickel(II)-Schiff base, complexes with carboxylate bridges: Tuning of coordination geometry and magnetic properties

Three new linear trinuclear nickel(II) complexes, [Ni-3(salpen)(2)(OAc)(2)(H2O)(2)]center dot 4H(2)O (1) (OAc = acetate, CH3COO-), [Ni-3(salpen)(2)(OBz)(2)] (2) (OBz=benzoate, PhCOO-) and [Ni-3(salpen)(2)(OCn)(2)(CH3CN)(2)] (4) (OCn = cinnamate, PhCH=CHCOO-), H(2)salpen = tetradentate ligand, N,N'-bis(salicylidene)-1,3-pentanediamine have been synthesized and characterized structurally and magnetically. The choice of solvent for growing single crystal was made by inspecting the morphology of the initially obtained solids with the help of SEM study. The magnetic properties of a closely related complex, [Ni-3(salpen)(2)(OPh)(2)(EtOH)] (3) (OPh = phenyl acetate, PhCH2COO-) whose structure and solution properties have been reported recently, has also been studied here. The structural analyses reveal that both phenoxo and carboxylate bridging are present in all the complexes and the three Ni(II) atoms remain in linear disposition. Although the Schiff base ligand and the syn-syn bridging bidentate mode of the carboxylate group remain the same in complexes 1-4, the change of alkyl/aryl group of the carboxylates brings about systematic variations between six- and five-coordination in the geometry of the terminal Ni(II) centres of the trinuclear units. The steric demand as well as hydrophobic nature of the alkyl/aryl group of the carboxylate is found to play a crucial role in the tuning of the geometry. Variable-temperature (2-300 K) magnetic susceptibility measurements show that complexes 1-4 are antiferromagnetically coupled (J = -3.2(1), -4.6(1). -3.2(1) and -2.8(1) cm(-1) in 1-4, respectively). Calculations of the zero-field splitting parameter indicate that the values of D for complexes 1-4 are in the high range (D = +9.1(2), +14.2(2), +9.8(2) and +8.6(1) cm(-1) for 1-4, respectively). The highest D value of +14.2(2) and +9.8(2) cm(-1) for complexes 2 and 3, respectively, are consistent with the pentacoordinated geometry of the two terminal nickel(II) ions in 2 and one terminal nickel(II) ion in 3. (C) 2009 Elsevier Ltd. All rights reserved.

Trinuclear cu(ll) complexes containing peripheral ketonic oxygen bridges and a mu(3)-OH core: Syntheses, crystal structures, spectroscopic and magnetic properties

Four new trinuclear copper(II) complexes, [(CuL1)(3)(mu(3)-OH)](ClO4)(2)center dot H2O (1), [(CuL2)(3)(mu(3)-OH)](CIO4)(2) (2), [(CuL3)(3)-(mu(3)-OH)](ClO4)(4)center dot H2O (3), and [(CuL4)(3)(mu(3)-OH)](ClO4)(2)center dot H2O (4), where HL1 = 8-amino-4,7,7-trimethyl-5-azaoct-3-en-2-one, HL2 = 7-amino-4-methyl-5-azaoct-3-en-2-one, HL3 = 7(ethylamino)-4-methyl-5-azahept-3-en-2-one, and HL4 = 4-methyl-7-(methylamino)-5-azahept-3-en-2-one, have been derived from the four tridentate Schiff bases (HL1, HL2, HL3, and HL4) and structurally characterized by X-ray crystallography. For all compounds, the cationic part is trinuclear with a CU3OH core held by three carbonyl oxygen bridges between each pair of copper(II) atoms. The copper atoms are five-coordinate with a distorted square-pyramidal geometry; the equatorial plane consists of the bridging oxygen atom of the central OH group together with three atoms (N, N, O) from one ligand whereas an oxygen atom of a second ligand occupies the axial position. Magnetic measurements have been performed in the 2-300 K temperature range. The experimental data could be satisfactorily reproduced by using an isotropic exchange model, H = -J(S1S2+S2S3+S1S3) yielding as best-fit parameters: J = -66.7 and g = 2.19 for 1, J = -36.6 and g = 2.20 for 2, J = -24.5 and g = 2.20 for 3, and J = -14.9 and g = 2.05 for 4. EPR spectra at low temperature show the existence of spin frustration in complexes 3 and 4, but it has not been possible to carry out calculations of the antisymmetric exchange parameter, G, from magnetic data. In frozen methanolic solution, at 4 K, hyperfine splitting in all complexes and spin frustration in complex 4 seem to be confirmed. ((c) Wiley-VCH Verlag GmbH & Co. KGaA, 69451 Weinheim, Germany, 2005)

Trinuclear Cu(II) complexes containing peripheral ketonic oxygen bridges and a mu(3)-OH core: steric influence on their structures and existence

Two tridentate Schiff bases, HL1(6-amino-3-methyl-1-phenyl-4-azahex-2-en-1-one), and HL2 (6-atnino-3,6-dimethyl-1-phenyl-4-azahex-2-en-1-one) on reaction with Cu(II) perchlorate in the presence of triethyl amine yielded two new trinuclear copper(II) complexes, [(CuL1)(3)(mu(3)-OH)](ClO4)(2) (1) and [(CuL2)(3)(mu(3)-OH)](ClO4)(2) center dot 0.75H(2)O (2), whereas another tridentate ligand HL3 (7-amino-3-methyl-1-phenyl-4-azahept-2-en-1-one) underwent hydrolysis under the same reaction conditions to result in the formation of a mononuclear complex, [Cu(bn)(pn)ClO4] (3) [where bn = 1-benzoylacetonate and pn = 1,3-propanediamine]. All three complexes have been characterized by X-ray crystallography. For both 1 and 2 the cationic part is trinuclear with a [Cu3OH] core held by three carbonyl oxygen bridges between each pair of copper(II) atoms. The structure of 3 is a monomer with a chelating 1,3-propanediamine and a benzoyl acetone moiety. Magnetic measurements of I and 2 have been performed in the 2-300 K temperature range. The experimental data could be satisfactorily reproduced by using an isotropic exchange model, H = -J(S1S2 + S2S3 + S1S3), yielding as best fit parameters: J = -25.6 cm(-1), g = 2.21 for 1 and J = 11.2 cm(-1), g = 2.10 for 2. The EPR spectra at low temperature could be indicative of spin frustration in complex 1. (C) 2006 Elsevier Ltd. All rights reserved.

Borg's rating of perceived exertion scales: do the verbal anchors mean the same for different clinical groups?

Objective: To examine the interpretation of the verbal anchors used in the Borg rating of perceived exertion (RPE) scales in different clinical groups and a healthy control group. Design: Prospective experimental study. Setting: Rehabilitation center. Participants: Nineteen subjects with brain injury, 16 with chronic low back pain (CLBP), and 20 healthy controls. Interventions: Not applicable. Main Outcome Measures: Subjects used a visual analog scale (VAS) to rate their interpretation of the verbal anchors from the Borg RPE 6-20 and the newer 10-point category ratio scale. Results: All groups placed the verbal anchors in the order that they occur on the scales. There were significant within-group differences (P > .05) between VAS scores for 4 verbal anchors in the control group, 8 in the CLBP group, and 2 in the brain injury group. There was no significant difference in rating of each verbal anchor between the groups (P > .05). Conclusions: All subjects rated the verbal anchors in the order they occur on the scales, but there was less agreement in rating of each verbal anchor among subjects in the brain injury group. Clinicians should consider the possibility of small discrepancies in the meaning of the verbal anchors to subjects, particularly those recovering from brain injury, when they evaluate exercise perceptions.

Use of a reduced Schiff-Base ligand to prepare novel chloro-bridged chains of rare Cu(II) trinuclear complexes with mixed azido/oxo and chloro/oxo bridges

Two mixed bridged one-dimensional (1D) polynuclear complexes, [Cu3L2(mu(1,1)-N-3)(2)(mu-Cl)Cl](n) (1) and {[Cu3L2(mu-Cl)(3)Cl]center dot 0.46CH(3)OH}(n), (2), have been synthesized using the tridentate reduced Schiff-base ligand HL (2-[(2-dimethylamino-ethylamino)-methyl]-phenol). The complexes have been characterized by X-ray structural analyses and variable-temperature magnetic susceptibility measurements. In both complexes the basic trinuclear angular units are joined together by weak chloro bridges to form a 1D chain. The trinuclear structure of 1 is composed of two terminal square planar [Cu(L)(mu(1,1)-N-3)] units connected by a central Cu(II) atom through bridging nitrogen atoms of end-on azido ligands and the phenoxo oxygen atom of the tridentate ligand. These four coordinating atoms along with a chloride ion form a distorted trigonal bipyramidal geometry around the central Cu(II). The structure of 2 is similar; the only difference being a Cl bridge replacing the mu(1,1)-N-3 bridge in the trinuclear unit. The magnetic properties of both trinuclear complexes can be very well reproduced with a simple linear symmetrical trimer model (H = JS(i)S(i+1)) with only one intracluster exchange coupling (J) including a weak intertrimer interaction (.j) reproduced with the molecular field approximation. This model provides very satisfactory fits for both complexes in the whole temperature range with the following parameters: g = 2.136(3), J = 93.9(3) cm(-1) and zj= -0.90(3) cm(-1) (z = 2) for 1 and g = 2.073(7), J = -44.9(4) cm(-1) and zJ = -1.26(6) cm(-1) (z = 2) for 2.

Synthesis, crystal structure and magnetic properties of three unprecedented tri-nuclear and one very rare tetra-nuclear copper(II) Schiff-base complexes supported by mixed azido/phenoxo/nitrato or acetato bridges

Three novel mixed bridged trinuclear and one tetranuclear copper(II) complexes of tridentate NNO donor Schiff base ligands [Cu-3(L-1)(2)(mu(LI)-N-3)(2)(CH3OH)(2)(BF2)(2)] (1), [Cu-3(L-1)(2)(mu(LI)-NO3-I kappa O.2 kappa O')(2)] (2), [Cu-3(L-2)(2)(mu(LI)-N-3)(2)(mu-NOI-I kappa O 2 kappa O')(2)] (3) and [Cu-4(L-3)(2)(mu(LI)-N-3)(4)(mu-CH3COO-I kappa O 2 kappa O')(2)] (4) have been synthesized by reaction of the respective tridentate ligands (L-1 = 2[1-(2-dimethylamino-ethylimino)-ethyl]-phenol, L-2 = 2[1-(2-diethylamino-ethylimino)-ethyl]-phenol, L-3 = 2-[1-(2-dimethylamino-ethylimino)-methyl]-phenol) with the corresponding copper(II) salts in the presence of NaN3 The complexes are characterized by single-crystal X-ray diffraction analyses and variable-temperature magnetic measurements Complex 1 is composed of two terminal [Cu(L-1)(mu(LI)-N-3)] units connected by a central [Cu(BF4)(2)] unit through nitrogen atoms of end-on azido ligands and a phenoxo oxygen atom of the tridentate ligand The structures of 2 and 3 are very similar, the only difference is that the central unit is [Cu(NO1)(2)] and the nitrate group forms an additional mu-NO3-I kappa O 2 kappa O' bridge between the terminal and central copper atoms In complex 4, the central unit is a di-mu(L1)-N-3 bridged dicopper entity, [Cu-2(mu(L1)-N-3)(2)(CH3COO)(2)] that connects two terminal [Cu(L-3)(mu(L1)-N-3)] units through end-on azido; phenoxo oxygen and mu-CH3COO-1 kappa O center dot 2 kappa O' triple bridges to result in a tetranuclear unit Analyses of variable-temperature magnetic susceptibility data indicates that there is a global weak antiferromagnetic interaction between the copper(II) ions in complexes 1-3, with the exchange parameter J of -9 86, -11 6 and -19 98 cm(-1) for 1-3, respectively In complex 4 theoretical calculations show the presence of an antiferromagnetic coupling in the triple bridging ligands (acetato, phenoxo and azido) while the interaction through the double end-on azido bridging ligand is strongly ferromagnetic.

Magnetic coupling in trinuclear partial cubane copper(II) complexes with a hydroxo bridging core and peripheral phenoxo bridges from NNO donor Schiff base ligands

Three new trinuclear copper(II) complexes, [(CuL1)(3)(mu(3)-OH)](ClO4)(2)center dot 3.75H(2)O (1), [(CuL2)(3)(mu(3)-OH)](ClO4)(2) (2) and [(CuL3)(3)(mu(3)-OH)](BF4)(2)center dot 0.5CH(3)CN (3) have been synthesized from three tridentate Schiff bases HL1, HL2, and HL3 (HL1 = 2-[(2-amino-ethylimino)-methyl]-phenol, HL2 = 2-[(2-methylamino-ethylimino)-methyl]-phenol and HL3 = 2-[1-(2-dimethylamino-ethylimino)-ethyl]-phenol). The complexes are characterized by single-crystal X-ray diffraction analyses, IR, UV-vis and EPR spectroscopy, and variable-temperature magnetic measurements. All the compounds contain a partial cubane [Cu3O4] core consisting of the trinuclear unit [(CuL)(3)(mu(3)-OH)](2+) together with perchlorate or fluoroborate anions. In each of the complexes, the three copper atoms are five-coordinated with a distorted square-pyramidal geometry except in complex 1, in which one of the Cu-II ions of the trinuclear unit is six-coordinate being in addition weakly coordinated to one of the perchlorate anions. Variable-temperature magnetic measurements and EPR spectra indicate an antiferromagnetic exchange coupling between the CuII ions of complexes 1 and 2, while this turned out to be ferromagnetic for complex 3. Experimental values have been fitted according to an isotropic exchange Hamiltonian. Calculations based on Density Functional Theory have also been performed in order to estimate the exchange coupling constants in these three complexes. Both sets of values indicate similar trends and specially calculated J values establish a magneto-structural correlation between them and the Cu-O-Cu bond angle, in that the coupling is more ferromagnetic for smaller bond angle values.

Intrinsic chess rating

This paper develops and tests formulas for representing playing strength at chess by the quality of moves played, rather than by the results of games. Intrinsic quality is estimated via evaluations given by computer chess programs run to high depth, ideally so that their playing strength is sufficiently far ahead of the best human players as to be a `relatively omniscient' guide. Several formulas, each having intrinsic skill parameters s for `sensitivity' and c for `consistency', are argued theoretically and tested by regression on large sets of tournament games played by humans of varying strength as measured by the internationally standard Elo rating system. This establishes a correspondence between Elo rating and the parameters. A smooth correspondence is shown between statistical results and the century points on the Elo scale, and ratings are shown to have stayed quite constant over time. That is, there has been little or no `rating inflation'. The theory and empirical results are transferable to other rational-choice settings in which the alternatives have well-defined utilities, but in which complexity and bounded information constrain the perception of the utility values.