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.

Convex polytopes and quantization of symplectic manifolds

Quantum mechanics associate to some symplectic manifolds M a quantum model Q(M), which is a Hilbert space. The space Q(M) is the quantum mechanical analogue of the classical phase space M. We discuss here relations between the volume of M and the dimension of the vector space Q(M). Analogues for convex polyhedra are considered.

Antibodies with infinite affinity

Here we report an approach to the design and production of antibody/ligand pairs, to achieve functional affinity far greater than avidin/biotin. Using fundamental chemical principles, we have developed antibody/ligand pairs that retain the binding specificity of the antibody, but do not dissociate. Choosing a structurally characterized antibody/ligand pair as an example, we engineered complementary reactive groups in the antibody binding pocket and the ligand, so that they would be in close proximity in the antibody/ligand complex. Cross-reactions with other molecules in the medium are averted because of the low reactivity of these groups; however, in the antibody/ligand complex the effective local concentrations of the complementary reactive groups are very large, allowing a covalent reaction to link the two together. By eliminating the dissociation of the ligand from the antibody, we have made the affinity functionally infinite. This chemical manipulation of affinity is applicable to other biological binding pairs.

Structure-activity relationships derived by machine learning: the use of atoms and their bond connectivities to predict mutagenicity by inductive logic programming.

We present a general approach to forming structure-activity relationships (SARs). This approach is based on representing chemical structure by atoms and their bond connectivities in combination with the inductive logic programming (ILP) algorithm PROGOL. Existing SAR methods describe chemical structure by using attributes which are general properties of an object. It is not possible to map chemical structure directly to attribute-based descriptions, as such descriptions have no internal organization. A more natural and general way to describe chemical structure is to use a relational description, where the internal construction of the description maps that of the object described. Our atom and bond connectivities representation is a relational description. ILP algorithms can form SARs with relational descriptions. We have tested the relational approach by investigating the SARs of 230 aromatic and heteroaromatic nitro compounds. These compounds had been split previously into two subsets, 188 compounds that were amenable to regression and 42 that were not. For the 188 compounds, a SAR was found that was as accurate as the best statistical or neural network-generated SARs. The PROGOL SAR has the advantages that it did not need the use of any indicator variables handcrafted by an expert, and the generated rules were easily comprehensible. For the 42 compounds, PROGOL formed a SAR that was significantly (P < 0.025) more accurate than linear regression, quadratic regression, and back-propagation. This SAR is based on an automatically generated structural alert for mutagenicity.