It includes the theorem of Reidemeister and Singer on common refinements ("stabilizations") of Heegaard splittings, the reducibility of splittings, the uniqueness of splittings of a given genus for Euclidean space, and the Rubinstein–Scharlemann graphic, a tool for studying Heegaard splittings. A final chapter surveys more advanced topics including the geometrization conjecture, Dehn surgery, foliations, laminations, and curve complexes. There are two appendices, on general position and Morse theory.
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There are several classes of three-manifolds where the set of Heegaard splittings is completely known. For example, Waldhausen's Theorem shows that all splittings of S^3 are standard. The same holds for lens spaces (as proved by Francis Bonahon and Otal). Splittings of Seifert fiber spaces are more subtle.
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Here, all splittings may be isotoped to be vertical or horizontal (as proved by Yoav Moriah and Jennifer Schultens). classified splittings of torus bundles (which includes all three-manifolds with Sol geometry). It follows from their work that all torus bundles have a unique splitting of minimal genus. All other splittings of the torus bundle are stabilizations of the minimal genus one.
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The result relied heavily on techniques developed for studying the topology of Heegaard splittings.
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Thompson extended David Gabai's concept of thin position from knots to 3-manifolds and Heegaard splittings..
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"JSJ-splittings for finitely presented groups over slender groups." Inventiones Mathematicae, vol. 135 (1999), no. 1, pp.
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Waldhausen conjectured that every closed orientable 3-manifold has only finitely many Heegaard splittings (up to homeomorphism) of any given genus.
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8 (2002), pp. 20–28B. H. Bowditch. "Cut points and canonical splittings of hyperbolic groups." Acta Mathematica, vol. 180 (1998), no.
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By contrast, the three ligands in the meridional isomer were chemically unequivalent, so the DFT-calculated peak of each carbon atom showed splittings.
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As of 2008, the only hyperbolic three-manifolds whose Heegaard splittings are classified are two-bridge knot complements, in a paper of Tsuyoshi Kobayashi.
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Muscovite and phlogopite splittings can be fabricated into various built-up mica products. Produced by mechanized or hand setting of overlapping splittings and alternate layers of binders and splittings, built-up mica is used primarily as an electrical insulation material. Mica insulation is used in high-temperature and fire-resistant power cables in aluminium plants, blast furnaces, critical wiring circuits (for example, defense systems, fire and security alarm systems, and surveillance systems), heaters and boilers, lumber kilns, metal smelters, and tanks and furnace wiring. Specific high-temperature mica-insulated wire and cable is rated to work for up to 15 minutes in molten aluminium, glass, and steel.
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Mica is mechanically stable in micrometer-thin sheets which are relatively transparent to radiation (such as alpha particles) while being impervious to most gases. It is therefore used as a window on radiation detectors such as Geiger-Müller tubes. In 2008, mica splittings represented the largest part of the sheet mica industry in the United States. Consumption of muscovite and phlogopite splittings was about 308 t in 2008.
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Schultens earned her Ph.D. in 1993 at the University of California, Santa Barbara. Her dissertation, Classification of Heegaard Splittings for Some Seifert Manifolds, was supervised by Martin Scharlemann.
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Splittings are considered up to isotopy. The gluing map ƒ need only be specified up to taking a double coset in the mapping class group of H. This connection with the mapping class group was first made by W. B. R. Lickorish. Heegaard splittings can also be defined for compact 3-manifolds with boundary by replacing handlebodies with compression bodies. The gluing map is between the positive boundaries of the compression bodies.
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For instance, modeling time dependent magnetization measurements of a crystal of Fe8 molecular magnets with the Landau-Zener formula, Wernsdorfer and Sessoli inferred tunneling splittings in the range of 10−7 Kelvin.
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A possible model for random 3-manifolds is to take random Heegaard splittings. The proof that this model is hyperbolic almost surely (in a certain sense) uses the geometry of the complex of curves.
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The latter formed a central ruling ministry in form of the Chief Apostle. Parallel to the progress of the new apostolic denomination there occurred splittings throughout the world on several occasions. The reasons for these splittings were different but mainly related with the central ministry of the Chief Apostle and its claim for supremacy. The ministry of a Chief Apostle cannot be proved by the Bible as well as no other church leader can derive his absolute power from the special position of Peter among the disciples.
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It categorifies the Alexander polynomial. Knot Floer homology was defined by and independently by . It is known to detect knot genus. Using grid diagrams for the Heegaard splittings, knot Floer homology was given a combinatorial construction by .
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By using experimental indices of refraction he gave an estimate of the Stark splittings. This estimate was a few orders of magnitude too low. Not deterred by this prediction, StarkJ. Stark, Beobachtungen über den Effekt des elektrischen Feldes auf Spektrallinien I. Quereffekt (Observations of the effect of the electric field on spectral lines I. Transverse effect), Annalen der Physik, vol. 43, pp. 965–983 (1914). Published earlier (1913) in Sitzungsberichten der Kgl. Preuss. Akad. d. Wiss. undertook measurements on excited states of the hydrogen atom and succeeded in observing splittings.
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The idea of JSJ-decomposition was later extended by Rips and Sela to torsion-free finitely presented groupsE. Rips, and Z. Sela. "Cyclic splittings of finitely presented groups and the canonical JSJ decomposition." Annals of Mathematics (2), vol.
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The tunneling frequency is extremely sensitive to the height and shape of the inter-conversion barrier, and thus samples the potential in the classically forbidden regions. Resolved tunneling splittings proved to be common in the spectra of weakly bound molecular dimers.
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Schultens is the author of the book Introduction to 3-Manifolds (Graduate Studies in Mathematics, 2014). With Martin Scharlemann and Toshio Saito, she is a co- author of Lecture Notes On Generalized Heegaard Splittings (World Scientific, 2016). Her dissertation research involved the classification of Heegaard splittings of three-dimensional manifolds into handlebodies, which she also published in the Proceedings of the London Mathematical Society. Other topics in her research include the behavior of knot invariants like bridge number when knots are combined by the connected sum operation, and the Kakimizu complexes of knot complements and other spaces.
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Heegaard splittings appeared in the theory of minimal surfaces first in the work of Blaine Lawson who proved that embedded minimal surfaces in compact manifolds of positive sectional curvature are Heegaard splittings. This result was extended by William Meeks to flat manifolds, except he proves that an embedded minimal surface in a flat three- manifold is either a Heegaard surface or totally geodesic. Meeks and Shing- Tung Yau went on to use results of Waldhausen to prove results about the topological uniqueness of minimal surfaces of finite genus in \R^3. The final topological classification of embedded minimal surfaces in \R^3 was given by Meeks and Frohman.
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His major contributions include results involving almost normal Heegaard splittings and the closely related joint work with Jon T. Pitts relating strongly irreducible Heegaard splittings to minimal surfaces, joint work with William Jaco on special triangulations of 3-manifolds (namely 0-efficient and 1-efficient triangulations), and joint work with Martin Scharlemann on the Rubinstein- Scharlemann graphic. He is a key figure in the algorithmic theory of 3-manifolds, and one of the initial developers of the Regina program, which implements his 3-sphere recognition algorithm. His research interests also include: shortest networks applied to underground mine design, machine learning, learning theory, financial mathematics, and stock market trading systems.
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3, pp. 561–593 motivated by the notion of a JSJ decomposition for 3-manifolds. A JSJ-decomposition is a representation of a word-hyperbolic group as the fundamental group of a graph of groups which encodes in a canonical way all possible splittings over infinite cyclic subgroups.
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The idea of a Heegaard splitting was introduced by . While Heegaard splittings were studied extensively by mathematicians such as Wolfgang Haken and Friedhelm Waldhausen in the 1960s, it was not until a few decades later that the field was rejuvenated by , primarily through their concept of strong irreducibility.
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In addition to the requirements of updating records and directories to accommodate the new numbers, for efficient conversion this requires a period of "permissive dialing" in which the new and old codes are both allowed to work. Also, many splittings involved significant technical issues, especially when the area splittings occurred over boundaries other than phone network divisions. In 1998 area code 612, which had covered the Minneapolis – Saint Paul Twin Cities, was split to create area code 651 for St. Paul and the eastern metropolitan area. The Minnesota Public Utilities Commission mandated that the new boundary exactly follow municipal boundaries, which were distinctly different from telephone exchange boundaries, and that all subscribers keep their 7-digit numbers.
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The importance of handlebodies in 3-manifold theory comes from their connection with Heegaard splittings. The importance of handlebodies in geometric group theory comes from the fact that their fundamental group is free. A 3-dimensional handlebody is sometimes, particularly in older literature, referred to as a cube with handles.
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India is the world's biggest producer of mica blocks and mica splittings. India ranks second amongst the world's largest producers of barite and chromite. The Pleistocene system is rich in minerals. India is the third-largest coal producer in the world and ranks fourth in the production of iron ore.
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Later, Scott et al. explained and clarified their results while sorting out subtle but important issues concerning the true convergence of the polarized wave function. The outcome meant that it was possible to solve for the asymptotic exchange energy splittings to any order. The Holstein–Herring method has been extended to the two-active electron case i.e.
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Algorithmically computing the Grushko decomposition for specific classes of groups is a difficult problem which primarily requires being able to determine if a given group is freely decomposable. Positive results are available for some classes of groups such as torsion-free word-hyperbolic groups, certain classes of relatively hyperbolic groups,François Dahmani and Daniel Groves. "Detecting free splittings in relatively hyperbolic groups".
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This is obviously not the true set of energy levels for the infinite chain, but it is a good approximation to the true set in the temperature range where: the further splittings caused by adding more orbitals is negligible, and we have enough orbitals in the chain to account for splittings which are relevant in this temperature range. The results of this is that the results derived for a chain of any particular length are valid only in a particular temperature range, a range which moves to lower temperatures as the chain length increases. This means that by considering the results at many different chain lengths, one can build up a picture of the behavior of the system over a wide temperature range. The Hamiltonian for a linear chain of finite length is an example of an effective Hamiltonian.
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The Holstein–Herring method, also called the surface Integral method, or Smirnov's method is an effective means of getting the exchange energy splittings of asymptotically degenerate energy states in molecular systems. Although the exchange energy becomes elusive at large internuclear systems, it is of prominent importance in theories of molecular binding and magnetism. This splitting results from the symmetry under exchange of identical nuclei (Pauli exclusion principle).
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Juvenile giant kelp grow directly on the parent female gametophyte, extending one or two primary blades, and beginning a rudimentary holdfast, which will eventually cover the gametophyte completely. Growth occurs with lengthening of the stipe, and splitting of the blades. This occurs by means of small tears where the blade meets the stipe, which splits the stipe into two. Pneumatocysts grow after the first few blade splittings.
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Wasielewski was born on the South Side of Chicago, Illinois, and spent his formative years in close proximity to both the University of Chicago and Chicago's steel making industry. He studied chemistry at University of Chicago under the direction of Leon Stock (B.S. 1971, Ph.D. 1975). His graduate research revealed how the electron-nuclear hyperfine splittings of hydrogen, carbon-13, and fluorine atoms depend on the structure of the paramagnetic molecules.
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When produced, the seed parton is expected to undergo a parton shower, which may include a series of nearly-collinear splittings before the hadronization starts. Furthermore, the jet algorithm must be robust when it comes to fluctuations in the detector response. Theoretically, If a jet algorithm is not infrared and collinear safe, it can not be guaranteed that a finite cross-section can be obtained at any order of perturbation theory.
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Suppose now that M is a closed orientable three-manifold. Reidemeister–Singer Theorem: For any pair of splittings H_1 and H_2 in M there is a third splitting H in M which is a stabilization of both. Haken's Lemma: Suppose that S_1 is an essential two-sphere in M and H is a Heegaard splitting. Then there is an essential two-sphere S_2 in M meeting H in a single curve.
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Heegaard diagrams, which are simple combinatorial descriptions of Heegaard splittings, have been used extensively to construct invariants of three- manifolds. The most recent example of this is the Heegaard Floer homology of Peter Ozsvath and Zoltán Szabó. The theory uses the g^{th} symmetric product of a Heegaard surface as the ambient space, and tori built from the boundaries of meridian disks for the two handlebodies as the Lagrangian submanifolds.
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The pouch is clearly divided between a buccal (cheek) and sublingual (below the tongue) portion. Volumetric analyses within this study attributed the differences in net cheek volume between male and female rats to the average size of the respective sexes. Due to muscle's high nutritional demand, this muscle exhibits vascularization that has been highly studied. Dissections at Boston University by Frank Brodie describe the various bifurcations (or splittings) of the common carotid.
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This is realized as zero field splitting and exchange coupling, and can be large in magnitude. # The magnetic moment of a nucleus with a non-zero nuclear spin will affect any unpaired electrons associated with that atom. This leads to the phenomenon of hyperfine coupling, analogous to J-coupling in NMR, splitting the EPR resonance signal into doublets, triplets and so forth. Additional smaller splittings from nearby nuclei is sometimes termed "superhyperfine" coupling.
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This causes a splitting in the energy levels for the isolated impurity. One then considers the effect of adding further orbitals along the chain, doing which splits the hitherto derived energy levels further. Because the couplings decrease along the chain, the successive splittings caused by adding orbitals to the chain decrease. When a particular number of orbitals have been added to the chain, we have a set of energy levels for that finite chain.
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Among his notable results is his work with Marc Culler, John Luecke, and Peter Shalen on the cyclic surgery theorem. This was an important ingredient in his work with Luecke showing that knots were determined by their complement. Gordon was also involved in the resolution of the Smith conjecture. Andrew Casson and Gordon defined and proved basic theorems regarding strongly irreducible Heegaard splittings, an important concept in the modernization of Heegaard splitting theory.
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Muscovite splittings from India accounted for essentially all US consumption. The remainder was primarily imported from Madagascar. Small squared pieces of sheet mica are also used in the traditional Japanese Kodo ceremony to burn incense: A burning piece of coal is placed inside a cone made of white ash. The sheet of mica is placed on top, acting as a separator between the heat source and the incense, in order to spread the fragrance without burning it.
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If two neighboring carbon atoms are labeled, a doublet of doublets may degenerate into a triplet if the doublet splittings are equal. The drawbacks to using NMR techniques for metabolic flux analysis purposes is that it is different from other NMR applications because it is a rather specialized discipline. An NMR spectrometer may not be directly available for all research teams. The optimization of NMR measurement parameters and proper analysis of peak structures requires a skilled NMR specialist.
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This structure is caused by the influence of the nucleus on the dynamics of orbiting electrons. The magnetic moments of supposed "nuclear electrons" should produce hyperfine spectral line splittings similar to the Zeeman effect, but no such effects were observed. It seemed that the magnetic moment of the electron vanished when it was within the nucleus. While on a visit to Utrecht University in 1928, Kronig learned of a surprising aspect of the rotational spectrum of N2+.
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Cantonese) the dark entering tone further splits into high () and low () registers, depending on the length of the nucleus, for a total of nine tone-classes. Some dialects have a complex tone splittings, and the terms dark and light are insufficient to cover the possibilities. The number of tone-classes is based on Chinese tradition, and is as much register as it is actual tone. The entering 'tones', for example, are distinct only because they are checked by a final stop consonant, not because they have a tone contour that contrasts with non-entering tones.
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An important sufficient condition for tameness in terms of splittings of the fundamental group had been obtained by Bonahon. The conjecture was proved in 2004 by Ian Agol, and independently, by Danny Calegari and David Gabai. Agol's proof relies on the use of manifolds of pinched negative curvature and on Canary's trick of "diskbusting" that allows to replace a compressible end with an incompressible end, for which the conjecture has already been proved. The Calegari–Gabai proof is centered on the existence of certain closed, non- positively curved surfaces that they call "shrinkwrapped".
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Any pair of symmetry-related X-C-C-Y fragments (where X and Y are different magnetically active nuclei) as well as XYC-CXY (cis or trans) and X2C-CY2 fragments may show magnetic inequivalence when the heteronuclear coupling constants (2JXY or 3JXY) are non-negligible. In principle, the magnetically active nuclei may also be disposed on non- carbon atoms. A classic example is the 1H-NMR spectrum of 1,1-difluoroethylene. The single 1H-NMR signal is made complex by the 2JH-H and two different 3JH-F splittings.
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3673–3686 Bowditch's work relied on extracting various discrete tree-like structures from the action of a word-hyperbolic group on its boundary. Bowditch also proved that (modulo a few exceptions) the boundary of a one- ended word-hyperbolic group G has local cut-points if and only if G admits an essential splitting, as an amalgamated free product or an HNN extension, over a virtually infinite cyclic group. This allowed Bowditch to produceB. H. Bowditch, "Cut points and canonical splittings of hyperbolic groups" Acta Mathematica, vol. 180 (1998), no.
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In general, the direct magnetic interaction between a pair of electrons (due to their electron magnetic moments) is negligibly small compared to this electric interaction. Exchange energy splittings are very elusive to calculate for molecular systems at large internuclear distances. However, analytical formulae have been worked out for the hydrogen molecular ion (see references herein). Normally, exchange interactions are very short- ranged, confined to electrons in orbitals on the same atom (intra-atomic exchange) or nearest neighbor atoms (direct exchange) but longer-ranged interactions can occur via intermediary atoms and this is termed superexchange.
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Chapter 5 discusses normal surfaces, surfaces that intersect the tetrahedra of a triangulation of a manifold in a controlled way. By parameterizing these surfaces by how many pieces of each possible type they can have within each tetrahedron of a triangulation, one can reduce many questions about manifolds such as the recognition of trivial knots and trivial manifolds to questions in number theory, on the existence of solutions to certain Diophantine equations. The book uses this tool to prove the existence and uniqueness of prime decompositions of manifolds. Chapter 6 concerns Heegaard splittings, surfaces which split a given manifold into two handlebodies.
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In the 1980s, the road was widened into a six-lane highway. During the late 1990s and early 2000s (decade), the avenue was prone to heavy traffic and accidents due to the increase in number of public transportation vehicles plying the highway, and sidewalk vendors crowding onto the road. In the late 2000s (decade), the Metropolitan Manila Development Authority cleared the sidewalk vendors, especially in the Tandang Sora area, which was prone to heavy rush hour traffic. Fairview Avenue uses stoplights and center island splittings in its intersections, while the Don Mariano Marcos Avenue uses interchanges in its intersections.
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Consumption of segment plate was about 149 t in 2008 in the US. Some types of built-up mica have the bonded splittings reinforced with cloth, glass, linen, muslin, plastic, silk, or special paper. These products are very flexible and are produced in wide, continuous sheets that are either shipped, rolled, or cut into ribbons or tapes, or trimmed to specified dimensions. Built-up mica products may also be corrugated or reinforced by multiple layering. In 2008, about 351 t of built- up mica was consumed in the US, mostly for molding plates (19%) and segment plates (42%).
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A key idea in the theory is to study a 3-manifold by considering special surfaces embedded in it. One can choose the surface to be nicely placed in the 3-manifold, which leads to the idea of an incompressible surface and the theory of Haken manifolds, or one can choose the complementary pieces to be as nice as possible, leading to structures such as Heegaard splittings, which are useful even in the non-Haken case. Thurston's contributions to the theory allow one to also consider, in many cases, the additional structure given by a particular Thurston model geometry (of which there are eight). The most prevalent geometry is hyperbolic geometry.
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The Soviet Internal Troops were formed in 1919 under the Cheka (later NKVD, and were known as "NKVD Troops", formerly the "Internal Security Forces" (Russian: Voyska vnutrenney okhrany Respubliki or VOHR)), remained there with all the mergers and splittings of Soviet state security services and ended up under the control of the police-like MVD. The most well-known of the Internal Troops divisions is OMSDON based near Moscow which traces its roots to the "OSNAZ" detachment of the VChK (formerly 1st Automobile Fighting Detachment of the VTsIK). It was later reorganized into the DON (Special-Purpose Division) of the OGPU and the NKVD.
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In the case of ferric heme proteins, MCD is capable of determining both oxidation and spin state to a remarkably exquisite degree. In regular proteins, MCD is capable of stoichiometrically measuring the tryptophan content of proteins, assuming there are no other competing absorbers in the spectroscopic system. In addition, the application of MCD spectroscopy greatly improved the level of understanding in the ferrous non-heme systems because of the direct observation of the d–d transitions, which generally can not be obtained in optical absorption spectroscopy owing to the weak extinction coefficients and are often electron paramagnetic resonance silent due to relatively large ground-state sublevel splittings and fast relaxation times.
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T NMR Magnet at HWB-NMR, Birmingham, UK NMR spectroscopy is one of the principal techniques used to obtain physical, chemical, electronic and structural information about molecules due to the chemical shift of the resonance frequencies of the nuclear spins in the sample. Peak splittings due to J- or dipolar couplings between nuclei are also useful. NMR spectroscopy can provide detailed and quantitative information on the functional groups, topology, dynamics and three-dimensional structure of molecules in solution and the solid state. Since the area under an NMR peak is usually proportional to the number of spins involved, peak integrals can be used to determine composition quantitatively.
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Thompson won the 2003 Ruth Lyttle Satter Prize in Mathematics for her research on thin position and Heegard splittings.. In 2013, she became one of the inaugural fellows of the American Mathematical Society.List of Fellows of the American Mathematical Society, retrieved 2014-12-25. In February 2020, Abigail Thompson has been recognized by the American Council of Trustees and Alumni (ACTA) as a "Hero of Intellectual Freedom." The award is due to an op-ed Thompson published in The Wall Street Journal on December 19, 2019, denouncing the use of diversity statements in faculty hiring practices in the University of California system. Thompson will deliver the keynote address at ACTA’s ATHENA Roundtable Conference on November 13, 2020.
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The Academy of Sciences of the Democratic People's Republic of Korea () or State Academy of Sciences, formerly the National Academy of Sciences, is the national academy of sciences of North Korea. It was founded in 1952, and until 1981 was responsible for all research conducted in the country before various organizational reforms and splittings of academies were conducted. Since 1980s the Academy has suffered from a long-term lack of funding, but it remains as a large and the most important scientific institution in the country. The Academy has played a role in the nuclear weapons program of North Korea until a separate military-run Academy of National Defense Science was established to take on this role.
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Dunwoody works on geometric group theory and low-dimensional topology. He is a leading expert in splittings and accessibility of discrete groups, groups acting on graphs and trees, JSJ-decompositions, the topology of 3-manifolds and the structure of their fundamental groups. Since 1971 several mathematicians have been working on Wall's conjecture, posed by Wall in a 1971 paper,Wall, C. T. C., Pairs of relative cohomological dimension one. Journal of Pure and Applied Algebra, vol. 1 (1971), no. 2, pp. 141-154 which said that all finitely generated groups are accessible. Roughly, this means that every finitely generated group can be constructed from finite and one-ended groups via a finite number of amalgamated free products and HNN extensions over finite subgroups.
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Ceperley not only applied Feynman's exact mapping of superfluid 4He onto classical ring polymers but also created the algorithms to make path integration a precise calculational tool to compare theory with experiment. This method has enabled the elucidation of superfluid in terms of winding numbers and to reveal the deep relation between superfluidity and Bose-Einstein condensation. He derived the exact expression for tunnelling splittings in complex systems and, by computing the exchange in quantum crystals, resolved the origin of magnetism in solid 3He. He introduced the restricted path integral method to treat Fermi statistics in finite-temperature many-body quantum systems and applied this method to the normal 3He liquid and to hydrogen under extreme conditions thus predicting the principal Hugoniot of compressed deuterium in agreement with shock wave experiments.
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In-depth studies of the unflavored scalar mesons began with the Crystal Ball and Crystal Barrel experiments of the mid 1990s, focusing on the mass range between 1 GeV/c² and 2 GeV/c². The scalar mesons in the mass range of 1 GeV/c² to 2 GeV/c² are generally believed to be conventional quark-antiquark states with orbital excitation L = 1 and spin excitation S = 1, although they occur at a higher mass than one would expect in the framework of mass-splittings from spin-orbit coupling.F. E. Close, "An Introduction to Quarks and Partons", Academic Press: New York (1979), pgs. 88-89 The scalar glueball is also expected to fall in this mass region, appearing in similar fashion to the conventional mesons but having very distinctive decay characteristics.
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Also, for a non-singular M-matrix, the diagonal elements of A must be positive. Here we will further characterize only the class of non-singular M-matrices. Many statements that are equivalent to this definition of non-singular M-matrices are known, and any one of these statements can serve as a starting definition of a non-singular M-matrix.. For example, Plemmons lists 40 such equivalences.. These characterizations has been categorized by Plemmons in terms of their relations to the properties of: (1) positivity of principal minors, (2) inverse-positivity and splittings, (3) stability, and (4) semipositivity and diagonal dominance. It makes sense to categorize the properties in this way because the statements within a particular group are related to each other even when matrix is an arbitrary matrix, and not necessarily a Z-matrix.
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In atomic physics, hyperfine structure is defined by small shifts and splittings in the energy levels of atoms, molecules, and ions, due to interaction between the state of the nucleus and the state of the electron clouds. In atoms, hyperfine structure arises from the energy of the nuclear magnetic dipole moment interacting with the magnetic field generated by the electrons and the energy of the nuclear electric quadrupole moment in the electric field gradient due to the distribution of charge within the atom. Molecular hyperfine structure is generally dominated by these two effects, but also includes the energy associated with the interaction between the magnetic moments associated with different magnetic nuclei in a molecule, as well as between the nuclear magnetic moments and the magnetic field generated by the rotation of the molecule. Hyperfine structure contrasts with fine structure, which results from the interaction between the magnetic moments associated with electron spin and the electrons' orbital angular momentum.
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Whereas compact matrix pseudogroups are typically versions of Drinfeld-Jimbo quantum groups in a dual function algebra formulation, with additional structure, the bicrossproduct ones are a distinct second family of quantum groups of increasing importance as deformations of solvable rather than semisimple Lie groups. They are associated to Lie splittings of Lie algebras or local factorisations of Lie groups and can be viewed as the cross product or Mackey quantisation of one of the factors acting on the other for the algebra and a similar story for the coproduct Δ with the second factor acting back on the first. The very simplest nontrivial example corresponds to two copies of R locally acting on each other and results in a quantum group (given here in an algebraic form) with generators p, K, K−1, say, and coproduct :[p, K]=h K(K-1) :\Delta p=p\otimes K+1\otimes p :\Delta K=K\otimes K where h is the deformation parameter. This quantum group was linked to a toy model of Planck scale physics implementing Born reciprocity when viewed as a deformation of the Heisenberg algebra of quantum mechanics.
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