Kinematic geometry of mass-triangles and reduction of Schrödinger’s equation of three-body systems to partial differential equations solely defined on triangular parameters

Schrödinger’s equation of a three-body system is a linear partial differential equation (PDE) defined on the 9-dimensional configuration space, ℝ9, naturally equipped with Jacobi’s kinematic metric and with translational and rotational symmetries. The natural invariance of Schrödinger’s equation with respect to the translational symmetry enables us to reduce the configuration space to that of a 6-dimensional one, while that of the rotational symmetry provides the quantum mechanical version of angular momentum conservation. However, the problem of maximizing the use of rotational invariance so as to enable us to reduce Schrödinger’s equation to corresponding PDEs solely defined on triangular parameters—i.e., at the level of ℝ6/SO(3)—has never been adequately treated. This article describes the results on the orbital geometry and the harmonic analysis of (SO(3),ℝ6) which enable us to obtain such a reduction of Schrödinger’s equation of three-body systems to PDEs solely defined on triangular parameters.

The exaptive excellence of spandrels as a term and prototype

In 1979, Lewontin and I borrowed the architectural term “spandrel” (using the pendentives of San Marco in Venice as an example) to designate the class of forms and spaces that arise as necessary byproducts of another decision in design, and not as adaptations for direct utility in themselves. This proposal has generated a large literature featuring two critiques: (i) the terminological claim that the spandrels of San Marco are not true spandrels at all and (ii) the conceptual claim that they are adaptations and not byproducts. The features of the San Marco pendentives that we explicitly defined as spandrel-properties—their necessary number (four) and shape (roughly triangular)—are inevitable architectural byproducts, whatever the structural attributes of the pendentives themselves. The term spandrel may be extended from its particular architectural use for two-dimensional byproducts to the generality of “spaces left over,” a definition that properly includes the San Marco pendentives. Evolutionary biology needs such an explicit term for features arising as byproducts, rather than adaptations, whatever their subsequent exaptive utility. The concept of biological spandrels—including the examples here given of masculinized genitalia in female hyenas, exaptive use of an umbilicus as a brooding chamber by snails, the shoulder hump of the giant Irish deer, and several key features of human mentality—anchors the critique of overreliance upon adaptive scenarios in evolutionary explanation. Causes of historical origin must always be separated from current utilities; their conflation has seriously hampered the evolutionary analysis of form in the history of life.

Preassembly of interleukin 2 (IL-2) receptor subunits on resting Kit 225 K6 T cells and their modulation by IL-2, IL-7, and IL-15: A fluorescence resonance energy transfer study

Assembly and mutual proximities of α, β, and γc subunits of the interleukin 2 receptors (IL-2R) in plasma membranes of Kit 225 K6 T lymphoma cells were investigated by fluorescence resonance energy transfer (FRET) using fluorescein isothiocyanate- and Cy3-conjugated monoclonal antibodies (mAbs) that were directed against the IL-2Rα, IL-2Rβ, and γc subunits of IL-2R. The cell-surface distribution of subunits was analyzed at the nanometer scale (2–10 nm) by FRET on a cell-by-cell basis. The cells were probed in resting phase and after coculture with saturating concentrations of IL-2, IL-7, and IL-15. FRET data from donor- and acceptor-labeled IL-2Rβ-α, γ-α, and γ-β pairs demonstrated close proximity of all subunits to each other in the plasma membrane of resting T cells. These mutual proximities do not appear to represent mAb-induced microaggregation, because FRET measurements with Fab fragments of the mAbs gave similar results. The relative proximities were meaningfully modulated by binding of IL-2, IL-7, and IL-15. Based on FRET analysis the topology of the three subunits at the surface of resting cells can be best described by a “triangular model” in the absence of added interleukins. IL-2 strengthens the bridges between the subunits, making the triangle more compact. IL-7 and IL-15 act in the opposite direction by opening the triangle possibly because they associate their private specific α receptors with the β and/or γc subunits of the IL-2R complex. These data suggest that IL-2R subunits are already colocalized in resting T cells and do not require cytokine-induced redistribution. This colocalization is significantly modulated by binding of relevant interleukins in a cytokine-specific manner.

New infinite families of exact sums of squares formulas, Jacobi elliptic functions, and Ramanujan’s tau function

In this paper, we give two infinite families of explicit exact formulas that generalize Jacobi’s (1829) 4 and 8 squares identities to 4n2 or 4n(n + 1) squares, respectively, without using cusp forms. Our 24 squares identity leads to a different formula for Ramanujan’s tau function τ(n), when n is odd. These results arise in the setting of Jacobi elliptic functions, Jacobi continued fractions, Hankel or Turánian determinants, Fourier series, Lambert series, inclusion/exclusion, Laplace expansion formula for determinants, and Schur functions. We have also obtained many additional infinite families of identities in this same setting that are analogous to the η-function identities in appendix I of Macdonald’s work [Macdonald, I. G. (1972) Invent. Math. 15, 91–143]. A special case of our methods yields a proof of the two conjectured [Kac, V. G. and Wakimoto, M. (1994) in Progress in Mathematics, eds. Brylinski, J.-L., Brylinski, R., Guillemin, V. & Kac, V. (Birkhäuser Boston, Boston, MA), Vol. 123, pp. 415–456] identities involving representing a positive integer by sums of 4n2 or 4n(n + 1) triangular numbers, respectively. Our 16 and 24 squares identities were originally obtained via multiple basic hypergeometric series, Gustafson’s Cℓ nonterminating 6φ5 summation theorem, and Andrews’ basic hypergeometric series proof of Jacobi’s 4 and 8 squares identities. We have (elsewhere) applied symmetry and Schur function techniques to this original approach to prove the existence of similar infinite families of sums of squares identities for n2 or n(n + 1) squares, respectively. Our sums of more than 8 squares identities are not the same as the formulas of Mathews (1895), Glaisher (1907), Ramanujan (1916), Mordell (1917, 1919), Hardy (1918, 1920), Kac and Wakimoto, and many others.

Structure of the ripple phase in lecithin bilayers.

The phases of the x-ray form factors are derived for the ripple (Pbeta') thermodynamic phase in the lecithin bilayer system. By combining these phases with experimental intensity data, the electron density map of the ripple phase of dimyristoyl-phosphatidylcholine is constructed. The phases are derived by fitting the intensity data to two-dimensional electron density models, which are created by convolving an asymmetric triangular ripple profile with a transbilayer electron density profile. The robustness of the model method is indicated by the result that many different models of the transbilayer profile yield essentially the same phases, except for the weaker, purely ripple (0,k) peaks. Even with this residual ambiguity, the ripple profile is well determined, resulting in 19 angstroms for the ripple amplitude and 10 degrees and 26 degrees for the slopes of the major and the minor sides, respectively. Estimates for the bilayer head-head spacings show that the major side of the ripple is consistent with gel-like structure, and the minor side appears to be thinner with lower electron density.

Pars triangularis asymmetry and language dominance.

The pars triangular is a portion of Broca's area. The convolutions that form the inferior and caudal extent of the pars triangularis include the anterior horizontal and anterior ascending rami of the sylvian fissure, respectively. To learn if there are anatomic asymmetries of the pars triangularis, these convolutions were measured on volumetric magnetic resonance imaging scans of 11 patients who had undergone selective hemispheric anesthesia (Wada testing) to determine hemispheric speech and language lateralization. Of the 10 patients with language lateralized to the left hemisphere, 9 had a leftward asymmetry of the pars triangularis. The 1 patient with language lateralized to the right hemisphere had a significant rightward asymmetry of the pars triangularis. Our data suggest that asymmetries of the pars triangularis may be related to speech-language lateralization.

The hypocone as a key innovation in mammalian evolution.

The hypocone, a cusp added to the primitively triangular upper molar teeth of therian mammals, has evolved convergently > 20 times among mammals during the Cenozoic. Acquisition of the hypocone itself involves little phenotypic change, but subsequent diversification of groups possessing the hypocone may be greatly enhanced. Our analysis of the Cenozoic mammalian radiations, including the Recent fauna, shows that high species diversity of mammals with hypocones and association of the hypocone with herbivory strongly support recognition of the hypocone as a key innovation that has allowed invasion of, and diversification within, herbivorous adaptive zones. In contrast, mammals lacking hypocones show no marked increase in species diversity during the Cenozoic.