Blowing Up Polygonal Linkages

{Consider a planar linkage, consisting of disjoint polygonal arcs and cycles of rigid bars joined at incident endpoints (polygonal chains), with the property that no cycle surrounds another arc or cycle. We prove that the linkage can be continuously moved so that the arcs become straight, the cycles become convex, and no bars cross while preserving the bar lengths. Furthermore, our motion is piecewise-differentiable, does not decrease the distance between any pair of vertices, and preserves any symmetry present in the initial configuration. In particular, this result settles the well-studied carpenters rule conjecture.     

Bi- and poly-metallic cyanide-bridged complexes of the redox-active cyanomanganese nitrosyl unit [Mn(CN)(PR3)(NO)(eta-C5H4Me)

Cationic nitrile complexes and neutral halide and cyanide complexes, with the general formula [MnL1L2(NO)(eta-C5H4Me)]z, undergo one-electron oxidation at a Pt electrode in CH2Cl2. Linear plots of oxidation potential, Eo', vs. nu(NO) or the Lever parameters, EL, for L1 and L2, allow Eo' to be estimated for unknown analogues. In the presence of TlPF6, [MnIL'(NO)(eta-C5H4Me)] reacts with [Mn(CN)L(NO)(eta-C5H4Me)] to give [(eta5-C5H4Me)(ON)LMn(mu-CN)MnL'(NO)(eta5-C5H4Me)][PF6] which undergoes two reversible one-electron oxidations; DeltaE, the difference between the potentials for the two processes, differs significantly for stable cyanide-bridged linkage isomers. Novel pentametallic complexes such as [Mn[(mu-NC)Mn(CNBut)(NO)(eta5-C5H4Me)]4(OEt2)][PF6]2 and [Mn[(mu-NC)Mn(CNXyl)(NO)(eta5-C5H4Me)]4(NO3-O,O')][PF6], containing a trigonal bipyramidal and a distorted octahedral Mn(II) centre, respectively, result either from slow decomposition of the binuclear cyanide-bridged species or from the reaction of anhydrous MnI2 with four equivalents of [Mn(CN)L(NO)(eta5-C5H4Me)] in the presence of TlPF6.     

Ictal imaging using functional magnetic resonance

Magnetic resonance imaging (MRI) can now provide maps of human brain function with high spatial and temporal resolution. This noninvasive technique can also map the coritical activation that occurs during focal seizures, as demonstrated here by the results obtained using a conventional 1.5 T clinical MRI system for the investigation of a 4-year-old boy suffering from frequent partial motor seizures of his right side. FLASH images (TE = 60 ms) were acquired every 10 s over a period of 25 min, and activation images derived by subtracting baseline images from images obtained during clinical seizures. Functional MRI revealed sequential activation associated with specific gyri within the left hemisphere with each of five consecutive clinical seizures, and also during a period that was not associated with a detectable clinical seizure. The activated regions included gyri that were structurally abnormal. These results demonstrate (a) that functional MRI can potentially provide new insights into the dynamic events that occur in the epileptic brain and their relationship to brain structure; and (b) that there is the possibility of obtaining similar information in the absence of clinical seizures, suggesting the potential for studies in patients with interictal electrical disturbances.     

Random depth access full-field low-coherence interferometry applied to a small punch test

In small punch testing, with approximate preknowledge of the sample deformation, profile measurement need only be made at selected locations in depth. To date, profilometry through full-field low-coherence interferometry has not been applied to small punch testing—conventional methods typically measure the maximum displacement as the sample is deformed, ignoring useful shape and profile information. A modification of full-field low-coherence interferometry is presented, where a digital stepper motor is combined with piezoelectric transducer scanning to achieve random depth access three-dimensional micrometer profile measurement. Offering a rapid, inexpensive, and functional machine vision system, the measurement technique is applied to a small punch test.     

TWO-DISTANCE PRESERVING FUNCTIONS FROM EUCLIDEAN SPACE

Let 0 < c < s be fixed real numbers such that , and let f : E² → E ^d for d ≥ 2 be a function such that for every p, q ∈ E ² if |p − q| = c, then |f(p) − f(q)| ≤ c, and if |p − q| = s, then |f(p) − f(q)| ≥ s. Then f is a congruence. This result depends on and expands a result of Rádo et. al. [9], where a similar result holds, but for replacing . We also present a further extensions where E² is replaced by E ⁿ for n > 2 and where the range of c/s is enlarged. This revised version was published online in August 2006 with corrections to the Cover Date.     

The Dynamics of N(2)-O(3) and N(2)-SO(2) Probed by Microwave Spectroscopy

The microwave spectra of N(2)-O(3) and N(2)-SO(2) have been recorded in the 6-18 GHz range using a pulsed-nozzle, Fourier transform microwave spectrometer. C-type transitions have been observed for both complexes which are slightly shifted by internal tunneling motions of the O(3) or SO(2) moieties. In addition, unshifted a-type transitions have been observed for N(2)-O(3). The nuclear hyperfine pattern is typical of equivalent nitrogen nuclei. Two sets of rotational and hyperfine constants are required to fit the symmetric and antisymmetric nuclear spin states, indicating that the equivalence arises from tunneling rotation of the nitrogen molecule. Internal tunneling motions along three tunneling pathways have been identified, although no information on the N(2) tunneling frequency is available from the spectra. From the N(2)-O(3) data the tunneling frequencies cannot be decorrelated from the rotational parameters; however, the O(3) tunneling frequency upper limit is estimated to be 2.0 MHz and the frequency of the concerted tunneling motion of both moieties is estimated to be about 8.9 MHz. For N(2)-SO(2), the SO(2) tunneling frequency is 11.5 kHz and the concerted frequency 173.9 kHz. Both complexes are roughly T shaped with the N(2) axis approximately perpendicular to the O(3) or SO(2) plane. In the equilibrium structures of both complexes, the a-c inertial plane is a plane of symmetry. The centers of mass separations are estimated from the rotational parameters to be 3.582 ? for N(2)-O(3) and 3.875 ? for N(2)-SO(2). The angle between the symmetry axes of the O(3) or SO(2) and the line joining their centers of mass have been calculated as 130.84 degrees (or 49.16 degrees ) and 119.71 degrees (or 60.29 degrees ), respectively. From the quadrupole analysis, the average angle between the N(2) axis and the a-inertial axis is 32.12 degrees for N(2)-O(3) and 27.81 degrees for N(2)-SO(2). Model electrostatic and ab initio calculations confirm these structures. Differences between the experimental and calculated structural parameters highlight the role of tunneling dynamics in these complexes. Copyright 2000 Academic Press.     

Homobinuclear cyanide-bridged linkage isomers containing the redox-active unit [(micro-XY)Ru(CO)2L(o-O2C6Cl4)] (XY=CN or NC)

The salts [NEt4][Ru(CN)(CO)2L(o-O2C6Cl4)] {L=PPh3 or P(OPh)3}, which undergo one-electron oxidation at the catecholate ligand to give neutral semiquinone complexes [Ru(CN)(CO)2L(o-O2C6Cl4)], react with the dimers [{Ru(CO)2L(micro-o-O2C6Cl4)}2] {L=PPh3 or P(OPh)3} to give [NEt4][(o-O2C6Cl4)L(OC)2Ru(micro-CN)Ru(CO)2L'(o-O2C6Cl4)] {L or L'=PPh3 or P(OPh)3}. The cyanide-bridged binuclear anions are, in turn, reversibly oxidised to isolable neutral and cationic complexes [(o-O2C6Cl4)L(OC)2Ru(micro-CN)Ru(CO)2L'(o-O2C6Cl4)] and [(o-O2C6Cl4)L(OC)2Ru(micro-CN)Ru(CO)2L'(o-O2C6Cl4)]+ which contain one and two semiquinone ligands respectively. Structural studies on the redox pair [(o-O2C6Cl4)(Ph3P)(OC)2Ru(micro-CN)Ru(CO)2(PPh3)(o-O2C6Cl4)]- and [(o-O2C6Cl4)(Ph3P)(OC)2Ru(micro-CN)Ru(CO)2(PPh3)(o-O2C6Cl4)] confirm that the C-bound Ru(CO)2(o-O2C6Cl4) fragment is oxidised first. Uniquely, [(o-O2C6Cl4){(PhO)3P}(OC)2Ru(micro-CN)Ru(CO)2(PPh3)(o-O2C6Cl4)]- is oxidised first at the N-bound fragment, indicating that it is possible to control the site of electron transfer by tuning the co-ligands. Crystallisation of [(o-O2C6Cl4)(Ph3P)(OC)2Ru(micro-CN)Ru(CO)2{P(OPh)3}(o-O2C6Cl4)] resulted in the formation of an isomer in which the P(OPh)3 ligand is cis to the cyanide bridge, contrasting with the trans arrangement of the X-Ru-L fragment in all other complexes of the type RuX(CO)2L(o-O2C6Cl4).     

The structural role of Zr within alkali borosilicate glasses for nuclear waste immobilisation

Zirconium is a key constituent element of High Level nuclear Waste (HLW) glasses, occurring both as a fission product and a fuel cladding component. As part of a wider research program aimed at optimising the solubility of zirconium in HLW glasses, we have investigated the structural chemistry of zirconium in such materials using X-ray Absorption Spectroscopy (XAS). Zirconium K-edge XAS data were acquired from several inactive simulant and simplified waste glass compositions, including a specimen of blended Magnox/UO₂ fuel waste glass. These data demonstrate that zirconium is immobilised as (octahedral) six-fold coordinate ZrO₆ species in these glasses, with a Zr-O contact distance of 2.09 Å. The next nearest neighbours of the Zr species are Si at 3.42 Å and possibly Na at 3.44 Å, no next nearest neighbour Zr could be resolved.