Contractions of Degenerate Quadratic Algebras, Abstract and Geometric

Quadratic algebras are generalizations of Lie algebras which include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. Distinct superintegrable systems and their quadratic algebras can be related by geometric contractions, induced by B\^ocher contractions of the conformal Lie algebra $\mathfrak{so}(4,\mathbb {C})$ to itself. In 2 dimensions there are two kinds of quadratic algebras, nondegenerate and degenerate. In the geometric case these correspond to 3 parameter and 1 parameter potentials, respectively. In a previous paper we classified all abstract parameter-free nondegenerate quadratic algebras in terms of canonical forms and determined which of these can be realized as quadratic algebras of 2D nondegenerate superintegrable systems on constant curvature spaces and Darboux spaces, and studied the relationship between B\^ocher contractions of these systems and abstract contractions of the free quadratic algebras. Here we carry out an analogous study of abstract parameter-free degenerate quadratic algebras and their possible geometric realizations. We show that the only free degenerate quadratic algebras that can be constructed in phase space are those that arise from superintegrability. We classify all B\^ocher contractions relating degenerate superintegrable systems and, separately, all abstract contractions relating free degenerate quadratic algebras. We point out the few exceptions where abstract contractions cannot be realized by the geometric B\^ocher contractions

Structure Relations and Darboux Contractions for 2D 2nd Order Superintegrable Systems

Two-dimensional quadratic algebras are generalizations of Lie algebras that include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. Distinct superintegrable systems and their quadratic algebras can be related by geometric contractions, induced by In\"on\"u-Wigner type Lie algebra contractions. These geometric contractions have important physical and geometric meanings, such as obtaining classical phenomena as limits of quantum phenomena as ${\hbar}\to 0$ and nonrelativistic phenomena from special relativistic as $c\to \infty$, and the derivation of the Askey scheme for obtaining all hypergeometric orthogonal polynomials as limits of Racah/Wilson polynomials. In this paper we show how to simplify the structure relations for abstract nondegenerate and degenerate quadratic algebras and their contractions. In earlier papers we have classified contractions of 2nd order superintegrable systems on constant curvature spaces and have shown that all results are derivable from free quadratic algebras contained in the enveloping algebras of the Lie algebras $e(2,{\mathbb C})$ in flat space and $o(3,{\mathbb C})$ on nonzero constant curvature spaces. The quadratic algebra contractions are induced by generalizations of In\"on\"u-Wigner contractions of these Lie algebras. As a special case we obtained the Askey scheme for hypergeometric orthogonal polynomials. Here we complete this theoretical development for 2D superintegrable systems by showing that the Darboux superintegrable systems are also characterized by free quadratic algebras contained in the symmetry algebras of these spaces and that their contractions are also induced by In\"on\"u-Wigner contractions. We present tables of the contraction results

Energy allocation trade-offs as a function of age in fungiid corals

To compete effectively, living organisms must adjust the allocation of available energy resources for growth, survival, maintenance, and reproduction throughout their life histories. Energy demands and allocations change throughout the life history of an organism, and understanding their energy allocation strategies requires determination of the relative age of individuals. As most scleractinian corals are colonial, the relationship between age and mass/size is complicated by colony fragmentation, partial mortality, and asexual reproduction. To overcome these limitations, solitary mushroom corals, Herpolitha limax from Okinawa, Japan and Fungia fungites from Okinawa and the Great Barrier Reef (GBR), Australia, were used to investigate how energy allocation between these fundamental processes varies as a function of age. Measurements of the relative growth, biochemical profiles, fecundity of individuals of different sizes, and the settlement success of their progeny have revealed physiological trade-offs between growth and reproduction, with increasing body mass ultimately leading to senescence. The importance of energy allocation for reproduction led us to examine the reproductive strategies and sex allocation in the two studied species. In the present study, the smallest individuals of both species studied were found to invest most of their energy in relative growth, showing higher lipid and carbohydrate content than the later stages. In medium-sized corals, this pattern was overturned in favour of reproduction, manifesting in terms of both the highest fecundity and settlement success of the resulting brooded larvae. Finally, a phase of apparent senescence was observed in the largest individuals, characterized by a decrease in most of the parameters measured. In addition, complex reproductive plasticity has been revealed in F. fungites in the GBR, with individual females releasing eggs, embryos, planulae, or a combination of these. These data provide the most direct estimates currently available for physiological, age-related trade-offs during the life history of a coral. The unusual reproductive characteristics of the GBR F. fungites indicate previously unknown layers of complexity in the reproductive biology of corals and have implications for their adaptive potential across a wide geographical scale

the unusual metal ion binding ability of histidyl tags and their mutated derivatives

Peptides that consist of repeated sequences of alternating histidines and alanines strongly bind Cu(ii) and form α-helical structures

Structure relations and darboux contractions for 2D 2nd order superintegrable systems

Two-dimensional quadratic algebras are generalizations of Lie algebras that include the symmetry algebras of 2nd order superintegrable systems in 2 dimensions as special cases. The superintegrable systems are exactly solvable physical systems in classical and quantum mechanics. Distinct superintegrable systems and their quadratic algebras can be related by geometric contractions, induced by Inönü-Wigner type Lie algebra contractions. These geometric contractions have important physical and geometric meanings, such as obtaining classical phenomena as limits of quantum phenomena as h → 0 and nonrelativistic phenomena from special relativistic as c → ∞, and the derivation of the Askey scheme for obtaining all hypergeometric orthogonal polynomials as limits of Racah/Wilson polynomials. In this paper we show how to simplify the structure relations for abstract nondegenerate and degenerate quadratic algebras and their contractions. In earlier papers we have classif ied contractions of 2nd order superintegrable systems on constant curvature spaces and have shown that all results are derivable from free quadratic algebras contained in the enveloping algebras of the Lie algebras e(2, C) in flat space and o(3, C) on nonzero constant curvature spaces. The quadratic algebra contractions are induced by generalizations of Inönü-Wigner contractions of these Lie algebras. As a special case we obtained the Askey scheme for hypergeometric orthogonal polynomials. After constant curvature spaces, the 4 Darboux spaces are the 2D manifolds admitting the most 2nd order Killing tensors. Here we complete this theoretical development for 2D superintegrable systems by showing that the Darboux superintegrable systems are also characterized by free quadratic algebras contained in the symmetry algebras of these spaces and that their contractions are also induced by Inönü-Wigner contractions. We present tables of the contraction results

Cerebral monitoring with transcranial Doppler ultrasonography improves neurologic outcome during repairs of acute type A aortic dissection

ObjectiveNeurologic complications after repair of acute type A aortic dissection remain significant. The use of power M-mode transcranial Doppler monitoring to verify cerebral blood flow during these repairs might decrease cerebral ischemia by correcting malperfusion. The purpose of this study was to analyze the use of power M-mode transcranial Doppler monitoring during repairs of acute type A dissection with regard to neurologic outcome.MethodsWe performed a prospective study of patients undergoing repairs of acute type A aortic dissection. Repairs included profound hypothermic circulatory arrest and retrograde cerebral perfusion. Patients in whom transcranial Doppler monitoring was used to monitor cerebral blood flow and modify operative technique during repair (study group) were compared with those without monitoring and modification (control group).ResultsBetween September 2001 and October 2003, we repaired 56 cases of acute type A dissection. Power M-mode transcranial Doppler monitoring was used in 50% (28/56) of cases. Power M-mode transcranial Doppler monitoring altered operative cannulation and guided retrograde cerebral perfusion flow in 28.5% (8/28) and 78.6% (22/28) of cases, respectively. Two patients presented with preoperative stroke, one in each group. One operative death occurred in each group. In-hospital mortality and the occurrence of new stroke were not significantly different between the 2 groups. Temporary neurologic dysfunction occurred less often in the study group (14.8% [4/27] vs 51.8% [14/27], P = .008).ConclusionsIdentification of cerebral malperfusion requires cerebral monitoring. By ensuring cerebral blood flow by using power M-mode transcranial Doppler monitoring and correcting cerebral malperfusion by modifying operative technique, neurologic outcome was improved during repairs of acute type A aortic dissection