From Filters to Wavelets via Direct Limits II: Wavelet-Packet Bases

In recent joint work with Nadia Larsen, we gave a new proof of a theorem of Mallat which describes how to construct wavelets from quadrature mirror filters. Our main innovation was to show how the scaling function associated to the filter can be used to identify a particular direct limit of Hilbert spaces with L ²(ℝ). Here we show that wavelet-packet bases for L ²(ℝ) also fit naturally into the same direct-limit framework.     

Dynamics of Decision Making in Animal Group Motion

We analyze a continuous-time model of a multi-agent system motivated by simulation studies on dynamics of decision making in animal groups in motion. Each individual moves at constant speed in the plane and adjusts its heading in response to relative headings of others in the population. The population includes two subgroups that are “informed” such that individuals in each subgroup have a preferred direction of motion. The model exhibits fast and slow time scales allowing for a reduction in the dimension of the problem. The stable solutions for the reduced model correspond to compromise by individuals with conflicting preferences. We study the global phase space for the proposed reduced model by computing equilibria and exploring stability and bifurcations. B. Nabet’s and N.E. Leonard’s work is supported in part by AFOSR grant FA9550-07-1-0-0528 and ONR grants N00014-02-1-0826 and N00014-04-1-0534. I.D. Couzin’s work was supported by the Royal Society, Balliol College and EPSRC grants GR/S04765/01 and GR/T11234/01, a Searle Scholar Award and DARPA grant HR001-05-1-0057. S.A. Levin’s work was supported in part by DARPA grant HR0011-05-1-0057 and NSF grant EF-0434319.     

Cyclic voltammetry of Th(IV) in the room-temperature ionic liquid [Me3NnBu][N(SO2CF3)2

A Th(IV) compound, [Th(TFSI)4(HTFSI)].2H2O [where TFSI = N(SO2CF3)2], has been synthesized and characterized using elemental analysis, thermogravimetric analysis, and vibrational spectroscopy. The analysis suggests that the TFSI anion coordinates to the metal center via the sulfonyl oxygens as well as provides evidence for the coordination of HTFSI. The voltammetric behavior of this compound has been studied in the room-temperature ionic liquid [Me3NnBu][TFSI], and results show that Th(IV) is reduced to Th(0) in this ionic liquid in a single reduction step. Analysis of cyclic voltammograms shows that an insoluble product is being formed at the electrode surface, which is attributed to the formation of ThO2 by reaction with water. The E0 value for the reduction of Th(IV) to Th(0) has been determined to be -2.20 V (vs Fc+/Fc; -1.80 V vs SHE). A comparison of this E0 value with those obtained for Th(IV) reduction in a LiCl-KCl eutectic (400 degrees C), water, and nonaqueous solvents shows that the reduction in [Me3NnBu][TFSI] is easier to accomplish than that in these other solvents.     

Forces between mica surfaces, prepared in different ways, across aqueous and nonaqueous liquids confined to molecularly thin films

We have measured normal and lateral interactions across a range of different liquids between mica surfaces using a surface force balance (SFB). The mica surfaces were prepared either by melt cutting using Pt wire and standard procedures in our laboratories or by tearing sheets (that had not been exposed to Pt) off from a freshly cleaved sheet of mica. AFM micrographs revealed the substantial absence of Pt nanoparticles on the melt cut and torn-off mica surfaces. Normal-force versus surface-separation (D) profiles and shear force versus D measurements for purified water (no added salt), for concentrated aqueous NaCl solutions, and for cyclohexane revealed that in all cases the behavior of the highly confined liquids between melt-cut and between torn-off mica sheets was identical within experimental scatter. These results demonstrate directly that interactions measured between melt-cut mica surfaces as routinely prepared using established procedures in our laboratories and in other laboratories are free of the effect of any Pt contamination.     

Molecular structures of arachno-decaborane derivatives 6,9-X2B8H10 (X = CH2, NH, Se) including a gas-phase electron-diffraction study of 6,9-C2B8H14

The molecular structures of the three heterodecaboranes arachno-6,9-C2B8H14, arachno-6,9-N2B8H12, and arachno-6,9-Se2B8H10 have been determined by ab initio MO theory. In addition, the structure of arachno-6,9-C2B8H14 was experimentally determined using gas-phase electron diffraction (GED). The accuracy of all four of these structures has been confirmed by the good agreement of the (11)B chemical shifts calculated at the GIAO-MP2 level with the experimental values. A comparison of the GIAO-HF and GIAO-MP2 methods shows that for these heteroborane clusters, electron correlation effects on the computed delta((11)B) values are quite substantial and that it is necessary to go beyond the HF level in the NMR computation.     

Rate equation modelling of erbium luminescence dynamics in erbium-doped silicon-rich-silicon-oxide

Erbium doped silicon-rich silica offers broad band and very efficient excitation of erbium photoluminescence (PL) due to a sensitization effect attributed to silicon nanocrystals (Si-nc), which grow during thermal treatment. PL decay lifetime measurements of sensitised Er³⁺ ions are usually reported to be stretched or multi exponential, very different to those that are directly excited, which usually show a single exponential decay component.In this paper, we report on SiO₂ thin films doped with Si-nc's and erbium. Time resolved PL measurements reveal two distinct 1.54 μm Er decay components; a fast microsecond component, and a relatively long lifetime component (10 ms). We also study the structural properties of these samples through TEM measurements, and reveal the formation of Er clusters. We propose that these Er clusters are responsible for the fast μs decay component, and we develop rate equation models that reproduce the experimental transient observations, and can explain some of the reported transient behaviour in previously published literature.     

On the Potts Model Partition Function in an External Field

We study the partition function of the Potts model in an external (magnetic) field, and its connections with the zero-field Potts model partition function. Using a deletion-contraction formulation for the partition function Z for this model, we show that it can be expanded in terms of the zero-field partition function. We also show that Z can be written as a sum over the spanning trees, and the spanning forests, of a graph G. Our results extend to Z the well-known spanning tree expansion for the zero-field partition function that arises though its connections with the Tutte polynomial.     

Bonding Trends Traversing the Tetravalent Actinide Series: Synthesis, Structural, and Computational Analysis of An(IV)((Ar)acnac)(4) Complexes (An = Th, U, Np, Pu; (Ar)acnac = ArNC(Ph)CHC(Ph)O; Ar = 3,5-(t)Bu(2)C(6)H(3))

A series of tetravalent An(IV) complexes with a bis-phenyl β-ketoiminate N,O donor ligand has been synthesized with the aim of identifying bonding trends and changes across the actinide series. The neutral molecules are homoleptic with the formula An((Ar)acnac)(4) (An = Th (1), U (2), Np (3), Pu (4); (Ar)acnac = ArNC(Ph)CHC(Ph)O; Ar = 3,5-(t)Bu(2)C(6)H(3)) and were synthesized through salt metathesis reactions with actinide chloride precursors. NMR and electronic absorption spectroscopy confirm the purity of all four new compounds and demonstrate stability in both solution and the solid state. The Th, U, and Pu complexes were structurally elucidated by single-crystal X-ray diffraction and shown to be isostructural in space group C2/c. Analysis of the bond lengths reveals shortening of the An-O and An-N distances arising from the actinide contraction upon moving from 1 to 2. The shortening is more pronounced upon moving from 2 to 4, and the steric constraints of the tetrakis complexes appear to prevent the enhanced U-O versus Pu-O orbital interactions previously observed in the comparison of UI(2)((Ar)acnac)(2) and PuI(2)((Ar)acnac)(2) bis-complexes. Computational analysis of models for 1, 2, and 4 (1a, 2a, and 4a, respectively) concludes that both the An-O and the An-N bonds are predominantly ionic for all three molecules, with the An-O bonds being slightly more covalent. Molecular orbital energy level diagrams indicate the largest 5f-ligand orbital mixing for 4a (Pu), but spatial overlap considerations do not lead to the conclusion that this implies significantly greater covalency in the Pu-ligand bonding. QTAIM bond critical point data suggest that both U-O/U-N and Pu-O/Pu-N are marginally more covalent than the Th analogues.