What is most fascinating about McElheny's exhibition, in addition to the formal brilliance and flat out gorgeousness of his works (all 2019), is how they function as concrete manifestations of twilight and nocturnal consciousness, when the mind is free to be sinuous and fluxional, speculative and dreamy.
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It is fluxional in solution, as the result of rapid cis–trans equilibrium.
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The Cope rearrangement of 3-methyl-hexa-1,5-diene The Cope rearrangement causes the fluxional states of the molecules in the bullvalene family.
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In bullvalones one vinyl group in one of the arms in bullvalene is replaced by a keto group on a methylene bridge. In this way it is possible to activate the fluxional state by adding base and deactivate it again by removing the base: Scheme 2. A bullvalone Compound 1 in scheme 2 is not a fluxional molecule but by adding base (sodium methoxide in methanol) the ketone converts to the enolate 2 and the fluxional state is switched on. Deuterium labeling is possible forming first 3 a then a complex mixture with up to 7 deuterium atoms, compound 4 being just one of them.
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Earl Muetterties (June 23, 1927 – January 12, 1984), was an American inorganic chemist born in Illinois, who is known for his experimental work with boranes, homogeneous catalysis, heterogeneous catalysis, fluxional processes in organometallic complexes and apicophilicity.
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The hydrogen atoms in cyclopentadiene undergo rapid [1,5]-sigmatropic shifts as indicated by 1H NMR spectra recorded at various temperatures. Even more fluxional are the derivatives C5H5E(CH3)3 (E = Si, Ge, Sn), wherein the heavier element migrates from carbon to carbon with a low activation barrier.
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NMR studies indicate that the cis isomer is slightly more abundant than the trans isomer at room temperature, while the amount of the open form is small. The fluxional process is not fast enough to produce averaging in the IR spectrum. Thus, three absorptions are seen for each isomer.
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Trimethylsilyl cyclopentadiene is an organosilicon compound with the chemical formula C5H5Si(CH3)3. It exists as a colorless liquid. It is used in the synthesis of some metal cyclopentadienyl complexes and has attracted interest for its fluxional structure. Trimethylsilyl cyclopentadiene is an example of a molecule that undergoes rapid sigmatropic rearrangement.
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Careful laser spectroscopic investigations have shed useful light on the JT interactions. In particular they reveal that the barrier to pseudorotation almost vanishes (the system is highly "fluxional") which can be attributed to the fact that the 2nd-order coupling terms vanish by symmetry and the leading higher-order terms are of 4th order.
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Bullvalene is a hydrocarbon with the chemical formula C10H10. The molecule has a cage-like structure formed by the fusion of one cyclopropane and three cycloheptadiene rings. Bullvalene is unusual as an organic molecule due to the C−C and C=C bonds forming and breaking rapidly on the NMR timescale; this property makes it a fluxional molecule.
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Some other notable work include the synthesis of fulvalene, the discoveries of the Doering-LaFlamme allene synthesis and the Parikh-Doering oxidation, prediction of the existence of bullvalene as a fluxional molecule, and elucidation of the mechanism of the Baeyer–Villiger oxidation.Klärner, F.-G. (2011), William von Eggers Doering (1917–2011). Angewandte Chemie International Edition, 50: 2885–2886.
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P. J. Krusic, and P. Meakin, "An Electron Spin Resonance Study of the Fluxional Nature of Paramagnetic (π-Alkenyl)tris(trimethyl phosphite)iron Complexes," J. Am. Chem. Soc., 100, 3264 (1978). π-allyl organometallic complexes bearing M-H-C agostic interactions. He was responsible for a series of C-H activation reactions based upon fleeting zero-valent iron complexes bearing bidentate phosphorus ligands.
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The dynamics of conformational (and other kinds of) isomerism can be monitored by NMR spectroscopy at varying temperatures. The technique applies to barriers of 8–14 kcal/mol, and species exhibiting such dynamics are often called "fluxional". Besides NMR spectroscopy, IR spectroscopy is used to measure conformer ratios. For the axial and equatorial conformer of bromocyclohexane, νCBr differs by almost 50 cm−1.
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In chemistry, pyramidal inversion is a fluxional process in compounds with a pyramidal molecule, such as ammonia (NH3) "turns inside out". It is a rapid oscillation of the atom and substituents, the molecule or ion passing through a planar transition state. For a compound that would otherwise be chiral due to a stereocenter, pyramidal inversion allows its enantiomers to racemize.
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Andrew Warwick (2003) Masters of Theory: Cambridge and the Rise of Mathematical Physics, pages 114,5,9, University of Chicago Press "... [N]ew books which appeared in the mid-eighteenth century offered a systematic introduction to the fundamental operations of the fluxional calculus and showed how it could be applied to a wide range of mathematical and physical problems. ... The strongly problem-oriented presentation in the treatises ... made it much easier for university students to master the fluxional calculus and its applications [and] helped define a new field of mixed mathematical studies..." An adventurous expression of physical mathematics is found in A Treatise on Electricity and Magnetism which used partial differential equations. The text aspired to describe phenomena in four dimensions but the foundation for this physical world, Minkowski space, trailed by forty years. String theorist Greg Moore said this about physical mathematics in his vision talk at Strings 2014.
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The structure of XeF6 required several years to establish in contrast to the cases of and . In the gas phase the compound is monomeric. VSEPR theory predicts that due to the presence of six fluoride ligands and one lone pair of electrons the structure lacks perfect octahedral symmetry, and indeed electron diffraction combined with high-level calculations indicate that the compound's point group is C3v. It is a fluxional molecule.
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Chlorinated tap water decomposes rhodamine B. Rhodamine B solutions adsorb to plastics and should be kept in glass. Rhodamine B is tunable around 610 nm when used as a laser dye. Its luminescence quantum yield is 0.65 in basic ethanol, 0.49 in ethanol, 1.0, and 0.68 in 94% ethanol. The fluorescence yield is temperature dependent; the compound is fluxional in that its excitability is in thermal equilibrium at room temperature.
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Spectroscopic evidence indicates that the two carbonyl groups may be unsymmetrical, in which case the idealized C2v symmetry is reduced to C2. In solution Fe3(CO)12 is fluxional, resulting in equivalencing all 12 CO groups. Overall, it can be appreciated that these three clusters formally arise from condensation of three 16-electron M(CO)4 fragments, akin to the theoretical condensation of three methylene (:CH2) molecules into cyclopropane.
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Especially for molecules exhibiting slowly relaxing (T1) signals, the technique spin saturation transfer (SST) provides information on chemical exchange reactions. The method is widely applicable to fluxional molecules. This magnetization transfer technique provides rates, provided that they exceed 1/T1.Jarek, R. L., Flesher, R. J., Shin, S. K., "Kinetics of Internal Rotation of N,N-Dimethylacetamide: A Spin-Saturation Transfer Experiment", Journal of Chemical Education 1997, volume 74, page 978. .
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Iron-pentacarbonyl-Berry-mechanism Pentacoordinate molecules of trigonal pyramidal geometry typically exhibit a particular kind of low energy fluxional behavior called Berry pseudorotation. Famous examples of such molecules are iron pentacarbonyl (Fe(CO)5) and phosphorus pentafluoride (PF5). At higher temperatures, only one signal is observed for the ligands (e.g., by 13C or 19F NMR) whereas at low temperatures, two signals in a 2:3 ratio can be resolved.
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They can be stronger than graphene, and more flexible, in some configurations. For example, boron nanotubes have a higher 2D Young's modulus than any other known carbon and noncarbon nanostructures. Borophenes undergo novel structural phase transition under in-plane tensile loading due to the fluxional nature of their multi-center in-plane bonding. Borophene has potential as an anode material for batteries due to high theoretical specific capacities, electronic conductivity and ion transport properties.
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Steric factors therefore dominate, with coordinative saturation of the metal being balanced against inter-ligand repulsion. This results in a diverse range of coordination geometries, many of which are irregular, and also manifests itself in the highly fluxional nature of the complexes. As there is no energetic reason to be locked into a single geometry, rapid intramolecular and intermolecular ligand exchange will take place. This typically results in complexes that rapidly fluctuate between all possible configurations.
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The prototypical fluxional molecule is phosphorus pentafluoride. Its 19F NMR spectrum consists of a 31P-coupled doublet, indicating that the equatorial and axial fluorine centers interchange rapidly on the NMR timescale. Fluorine-19 NMR spectroscopy, even at temperatures as low as −100 °C, fails to distinguish the axial from the equatorial fluorine environments. The apparent equivalency arises from the low barrier for pseudorotation via the Berry mechanism, by which the axial and equatorial fluorine atoms rapidly exchange positions.
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Treatment of TiCl4 with acetylacetone gives cis-TiCl2(acac)2, a red-coloured, octahedral complex with C2 symmetry: :TiCl4 \+ 2 Hacac → TiCl2(acac)2 \+ 2 HCl This reaction requires no base. The complex TiCl2(acac)2 is fluxional in solution, the NMR spectrum exhibiting a single methyl resonance at room temperature. Unlike Ti(IV), Zr(IV) and Hf(IV) bind four bidentate acetylacetonates, reflecting the larger radius of these metals. Hafnium acetylacetonate and zirconium acetylacetonate adopt square antiprismatic structures.
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Armilenium is a C11H11 carbocation and was originally proposed as the first entirely organic sandwich compound. Named for its resemblance to an armillary sphere, NMR evidence for the carbocation was first described by Melvin J. Goldstein and Stanley A. Klein at Cornell University in 1973. In subsequent 13C NMR experiments by Goldstein and Joseph P. Dinnocenzo in 1984, the C11H11 carbocation was generated under stable ion conditions at lower temperature and at higher magnetic field than previously possible. These experiments revealed the carbocation to be fluxional.
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Newton completed no definitive publication formalizing his fluxional calculus; rather, many of his mathematical discoveries were transmitted through correspondence, smaller papers or as embedded aspects in his other definitive compilations, such as the Principia and Opticks. Newton would begin his mathematical training as the chosen heir of Isaac Barrow in Cambridge. His aptitude was recognized early and he quickly learned the current theories. By 1664 Newton had made his first important contribution by advancing the binomial theorem, which he had extended to include fractional and negative exponents.
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It is suspected that within solid titanium(IV) hydride, the molecules form aggregations (polymers), being connected by covalent bonds. Calculations suggest that TiH4 is prone to dimerisation. This largely attributed to the electron deficiency of the monomer and the small size of the hydride ligands; which allows dimerisation to take place with a very low energy barrier as there is a negligible increase in inter-ligand repulsion. The dimer is a calculated to be a fluxional molecule rapidly inter-converting between a number of forms, all of which display bridging hydrogens.
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Methanium can be visualised as a carbenium ion with a molecule of hydrogen interacting with the empty orbital in a 3-center-2-electron bond. The bonding electron pair in the H2 molecule is shared between the two hydrogen and one carbon atoms making up the 3-center-2-electron bond. The two hydrogen atoms in the H2 molecule can continuously exchange positions with the three hydrogen atoms in the ion (a conformation change called pseudorotation, specifically the Berry mechanism). The methanium ion is therefore considered a fluxional molecule.
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As for most amides, the spectroscopic evidence indicates partial double bond character for the C-N and C-O bonds. Thus, the infrared spectrum shows a C=O stretching frequency at only 1675 cm−1, whereas a ketone would absorb near 1700 cm−1. DMF is a classic example of a fluxional molecule. 300px The ambient temperature 1H NMR spectrum shows two methyl signals, indicative of hindered rotation about the (O)C-N bond. At temperatures near 100 °C, the 500 MHz NMR spectrum of this compound shows only one signal for the methyl groups.
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Sir Isaac Newton, in responding to questions from Leibniz in a letter in 1677, concealed the details of his "fluxional technique" with an anagram: :The foundations of these operations is evident enough, in fact; but because I cannot proceed with the explanation of it now, I have preferred to conceal it thus: 6accdae13eff7i3l9n4o4qrr4s8t12ux. On this foundation I have also tried to simplify the theories which concern the squaring of curves, and I have arrived at certain general Theorems. Trusted digital timestamping has first been discussed in literature by Haber and Stornetta.
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Temperature dependent changes in the NMR spectra result from dynamics associated with the fluxional molecules when those dynamics proceed at rates comparable to the frequency differences observed by NMR. The experiment is called DNMR and typically involves recording spectra at various temperatures. In the ideal case, low temperature spectra can be assigned to the "slow exchange limit", whereas spectra recorded at higher temperatures correspond to molecules at "fast exchange limit". Typically, high temperature spectra are simpler than those recorded at low temperatures, since at high temperatures, equivalent sites are averaged out.
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He was among the earliest to use carbon-14 as a tracer in the study of reaction mechanisms and molecular rearrangements. He has been a leader in the use of NMR to establish mechanisms and structures particularly in demonstrating the fluxional nature of Grignard reagents and related molecules and in the observation of a symmetry in organic chemistry." # Elias J. Corey 1984 "For outstanding research in the synthesis of organic compounds, particularly the total synthesis of highly complex natural products such as prostaglandins, alkaloids, giberellic acid, hormones, toxins, triterpenes, and others. Many of these syntheses are considered to be milestones in their fields.
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G. V. Coyne, p. 112; Rupert Hall, Philosophers at War, pages 106–107; David Brewster, The Life of Sir Isaac Newton, p. 185 It was not until the 1704 publication of an anonymous review of Newton's tract on quadrature, a review implying that Newton had borrowed the idea of the fluxional calculus from Leibniz, that any responsible mathematician doubted that Leibniz had invented the calculus independently of Newton. With respect to the review of Newton's quadrature work, all admit that there was no justification or authority for the statements made therein, which were rightly attributed to Leibniz.
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Charles Matthew Whish (1794-1833) was an English civil servant in the Madras Establishment of the East India Company. Whish was the first to bring to the notice of the western mathematical scholarship the achievements of the Kerala school of astronomy and mathematics. Whish wrote in his historical paper: (This paper has been reproduced as an Appendix in "") Kerala mathematicians had ... laid the foundation for a complete system of fluxions ... and their works ... abound with fluxional forms and series to be found in no work of foreign countries. Whish was also a linguist and had prepared a grammar and a dictionary of the Malayalam language.
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It is said in the preface that Newton revised and corrected these lectures, adding matter of his own, but it seems probable from Newton's remarks in the fluxional controversy that the additions were confined to the parts which dealt with optics. This, which is his most important work in mathematics, was republished with a few minor alterations in 1674. In 1675 he published an edition with numerous comments of the first four books of the On Conic Sections of Apollonius of Perga, and of the extant works of Archimedes and Theodosius of Bithynia. In the optical lectures many problems connected with the reflection and refraction of light are treated with ingenuity.
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In statistical thermodynamics, the symmetry number corrects for any overcounting of equivalent molecular conformations in the partition function. In this sense, the symmetry number depends upon how the partition function is formulated. For example, if one writes the partition function of ethane so that the integral includes full rotation of a methyl, then the 3-fold rotational symmetry of the methyl group contributes a factor of 3 to the symmetry number; but if one writes the partition function so that the integral includes only one rotational energy well of the methyl, then the methyl rotation does not contribute to the symmetry number. Symmetry Numbers for Rigid, Flexible and Fluxional Molecules: Theory and Applications.
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Ittel worked on hydride activation of lanthanides for Systems for Nuclear Auxiliary Power (SNAP) at Monsanto's Mound Laboratories for a short time. Upon receiving his PhD from Northwestern, he joined DuPont’s Central Research Department at the Experimental Station in Wilmington, Delaware. Ittel is best known for his contributions to organometallic chemistry and homogeneous catalysis. He discovered fluxional processes in both diamagneticJack M. Williams, Richard K. Brown, Arthur J. Schultz, Galen D. Stucky, and Steven D. Ittel, "Interaction of an Aliphatic Hydrogen Atom with a Transition Metal. The First Direct Observation of a Strong C-H...Metal Interaction Derived from a Single Crystal Neutron Diffraction Study of Fe[P(OCH3)3]3(h3-C8H13)]+[BF4]-," J. Am. Chem. Soc.
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Varman was a close acquaintance of C.M. Whish a civil servant of East India Company attached to its Madras establishment. Whish spoke of him and his work thus: "The author of Sadratnamalah is SANCARA VARMA, the younger brother of the present Raja of Cadattanada near Tellicherry, a very intelligent man and acute mathematician. This work, which is a complete system of Hindu astronomy, is comprehended in two hundred and eleven verses of different measures, and abounds with fluxional forms and series, to be found in no work of foreign or other Indian countries." Whish was the first to bring to the notice of the western mathematical scholarship the achievements of the Kerala school of astronomy and mathematics.
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Similarly to the Berry mechanism in square planar molecules, the symmetry of the intermediary phase of the vibrational mode is "chimeric"WJ Adams, HB Thompson, LS Bartell, 1970, J. Chem. Phys. 53:4040-6.LS Bartell, MJ Rothman & A Gavezzotti, 1982, , J. Chem. Phys. 76:4136-4413.M Cass, KK Hii & HS Rzepa, 2005, "Mechanisms that interchange axial and equatorial atoms in fluxional processes: Illustration of the Berry pseudorotation, the turnstile and the lever mechanisms via animation of transition state normal vibrational modes", J. Chem. Educ. (online), 2005; see , accessed 28 May 2014 of other mechanisms; it displays characteristics of the Berry mechanism, a "lever" mechanism seen in pseudorotation of disphenoidal molecules, and a "turnstile" mechanism (which can be seen in trigonal bipyramidal molecules under certain conditions).
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In fact he criticised those British mathematicians who adhered too strictly to geometry. It is indicative that he was one of the subscribers of John Landen's Residual Analysis (1764), one of the works in which the tradition of the Newtonian fluxional calculus was more severely criticised. In the preface of Meditationes Analyticae Waring showed a good knowledge of continental mathematicians such as Alexis Clairaut, Jean le Rond d'Alembert, and Euler. He lamented the fact that in Great Britain mathematics was cultivated with less interest than on the continent, and clearly desired to be considered as highly as the great names in continental mathematics—there is no doubt that he was reading their work at a level never reached by any other eighteenth-century British mathematician.
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Most notably, at the end of chapter three of Meditationes Analyticae Waring presents some partial fluxional equations (partial differential equations in Leibnizian terminology); such equations are a mathematical instrument of great importance in the study of continuous bodies which was almost completely neglected in Britain before Waring's researches. One of the most interesting results in Meditationes Analyticae is a test for the convergence of series generally attributed to d'Alembert (the 'ratio test'). The theory of convergence of series (the object of which is to establish when the summation of an infinite number of terms can be said to have a finite 'sum') was not much advanced in the eighteenth century. Waring's work was known both in Britain and on the continent, but it is difficult to evaluate his impact on the development of mathematics.
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In 1825 John Warren published a memoir on the division of time in southern India,John Warren (1825) A Collection of Memoirs on Various Modes According to which Nations of the Southern Part of India Divide Time from Google Books called the Kala Sankalita, which briefly mentions the discovery of infinite series by Kerala astronomers. The works of the Kerala school were first written up for the Western world by Englishman C. M. Whish in 1835. According to Whish, the Kerala mathematicians had "laid the foundation for a complete system of fluxions" and these works abounded "with fluxional forms and series to be found in no work of foreign countries". However, Whish's results were almost completely neglected, until over a century later, when the discoveries of the Kerala school were investigated again by C. T. Rajagopal and his associates.
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Meanwhile, Newton, though he explained his (geometrical) form of calculus in Section I of Book I of the Principia of 1687,Section I of Book I of the Principia, explaining "the method of first and last ratios", a geometrical form of infinitesimal calculus, as recognized both in Newton's time and in modern times – see citations above by L'Hospital (1696), Truesdell (1968) and Whiteside (1970) – is available online in its English translation of 1729, at page 41. did not explain his eventual fluxional notation for the calculusMarquis de l'Hôpital's original words about the 'Principia': "lequel est presque tout de ce calcul": see the preface to his Analyse des Infiniment Petits (Paris, 1696). The Principia has been called "a book dense with the theory and application of the infinitesimal calculus" also in modern times: see Clifford Truesdell, Essays in the History of Mechanics (Berlin, 1968), at p.99; for a similar view of another modern scholar see also in print until 1693 (in part) and 1704 (in full).
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The Analytical Society was a group of individuals in early-19th-century Britain whose aim was to promote the use of Leibnizian notation for differentiation in calculus as opposed to the Newton notation for differentiation.Carl B. Boyer (1989) A History of Mathematics, 2nd edition, page 592, John Wiley & Sons J. M. Dubbey (1963) "The Introduction of Differential Notation into Great Britain", Annals of Science 19: 35–48 The latter system came into being in the 18th century as a convention of Sir Isaac Newton, and was in use throughout Great Britain. According to a mathematical historian:Harvey W. Becher (1980) "Woodhouse, Babbage, Peacock and Modern Algebra", Historia Mathematica 7(4): 389–400 :In 1800, English mathematics was trapped in the doldrums of fluxional notation and of an intuitive geometric- physical approach to mathematics designed to prepare the student for reading Newton's Principia...The study of any mathematics not pertinent to the traditional questions of Tripos was not only ignored, but actually discouraged. Cambridge was isolated, and its students remained ignorant of continental developments.
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His contributions have provided the basis for both the experimental techniques and the theoretical apparatus used to interpret the experiments. Through his students and through his extensive writings, he has established a school of molecular spectroscopy that has profoundly altered the way in which we think about molecular structures." # Frank Albert Cotton 1980 "For his discovery of the fluxional behavior of transition metal organometallic compounds; for his discovery of metal-metal multiple bonded metal complexes; for his pioneering research on these systems; for the fact that these discoveries and studies are responsible for the development of a revolutionary new understanding of the bonding forces in organometallic and inorganic molecules; and for his outstanding achievements as scientist, author, and teacher." # Bert Lester Vallee 1981 "For his pioneering discoveries in the field of metalloenzyme chemistry; for his discovery and characterization of metalloenzymes and other metalloproteins; for his fundamental work with reverse transcriptase, a zinc- containing enzyme, that advanced understanding of genetic control and cancer; for his development of microwave excitation spectroscopy as a new analytical tool; and for his application of these fundamental discoveries directly to problems in human health.
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