Enriques Surfaces, Analytic Discriminants, and Borcherds's Φ Function

In [Bor 96], Borcherds constructed a non-vanishing weight 4 modular form Φ on the moduli space of marked, polarized Enriques surface of degree 2 by considering the twisted denominator function of the fake monster Lie algebra associated to an automorphism of order 2 of the Leech lattice fixing an 8-dimensional subspace. In [JT 94] and [JT 96], we defined and studied a meromorphic (multi-valued) modular form of weight 2, which we call the K3 analytic discriminant, on the moduli space of marked, polarized, K3 surfaces of degree 2d; in certain cases, including when , where p _k are distinct primes, our meromorphic form is actually a holomorphic form. Our construction involves a determinant of the Laplacian on a polarized K3 surface with respect to the Calabi-Yau metric together with the L ² norm of the image of the period map with respect to a properly scaled holomorphic two form. Since the universal cover of any Enriques surface is a K3 surface, we can restrict the K3 analytic discriminant to the moduli space of degree 2 Enriques surfaces. The main result of this paper is the observation that the square of our degree 2 analytic discriminant, viewed as a function on the moduli space of degree 2 Enriques surfaces, is equal to the Borcherd's Φ function, up to a universal multiplicative constant. This result generalizes known results in the study of generalized Kac-Moody algebras and elliptic curves, and suggests further connections with higher dimensional Calabi-Yau varieties, specifically those which can be realized as complete intersections in some, possibly weighted, projective space. Received: 24 July 1995 / Accepted: 21 March 1997     

Nuclear reactions induced by stopped negative pions in133Cs,93Nb and45Sc

The energy and angular distributions ofp, d, t,³He and⁴He from the three targets Cu, Ag and Au were measured at five different angles for bremsstrahlung with peak energy 500MeV. The measurements were made using a telescope consisting of four surface-barrier detectors. The experimental data are compared with cascade-evaporation calculations. For the⁴He-distributions the calculations were extended to include the contribution from knock-out of surface alphas by the cascade nucleons. The comparison shows that the main contribution comes from evaporation but that there is a direct component of the order of 10%.     

On the covariant structure of the two-point function

The spectral representation of the two-point function for arbitrary fields proposed recently [1] is rigorously proved and analyzed. The problem is treated in momentum space where the covariant structure is simpler because of the spectrum conditions. For finite-component fields the explicit matrix structure is found in coordinate space too and is applied to the definition of time-ordered Green functions for arbitrary spin. The decomposition of the two-point function into kernels of definite spin is carried out in the general case, a necessary and sufficient condition for the growth of the coefficients in this decomposition being given. The positive-definiteness condition (in the case of Hermitian conjugate fields) is fulfilled automatically by the elementary kernels.The formalism of homogeneous distributions in two dimensional complex domain [2] is used throughout the paper.On leave of absence from Joint Institute for Nuclear Research, Dubna, USSR and from Physical Institute of the Bulgarian Academy of Sciences, Sofia, Bulgaria.     

On the possible multi-valuedness of quantum wavefunctions

A theorem generalizing a result due to Schrödinger on the possible multi-valuedness of quantum wavefunctions is proved. Relevant examples show that double-valued wavefunctions do exist, at least in problems of a theoretical interest.     

Reducing-difference infrared spectral analysis of cis- and trans-configurated lactams

The conventional (non-polarized) and linear-polarized IR-spectra of 2-pyrrolidone and laurolactam as typical representatives of lactams with cis- and trans-amide configuration, respectively, were studied by means of difference IR-spectroscopy. The application of this method to treatment of the non-polarized IR-spectra permits additional conclusions about the likely self-associates of these compounds dissolved in aprotic solvents. By means of the difference linear-dichroic IR-spectroscopy, a more precise assignment of the characteristic bands of the lactam cis- and trans-peptide forms as well as their stereo structures were accomplished. A comparative analysis with the linear-dichroic IR-spectra of epsilon-caprolactam and caprylolactam, as similar compounds, with cis- and trans-conformation, respectively, was also carried out.     

Synthesis and characterization of novel (9H‐fluoren‐9‐ylamino) carbonylaminomethylphosphonic acid

The new α‐aminophosphonic acids are synthesized, reacting (9H‐fluoren‐9‐yl)urea with formaldehyde and phosphorus trichloride. (9H‐Fluoren‐9‐yl)urea was prepared from spiro(fluoren‐9,4′‐imidazolidine)‐2′,5′‐dione by alkaline hydrolysis with Ba(OH)₂. The structure of the title compounds was proved by means of IR, ¹H, ¹³C, and ³¹P NMR spectroscopy. © 2008 Wiley Periodicals, Inc. Heteroatom Chem 19:719–722, 2008; Published online in Wiley InterScience ( www.interscience.wiley.com ). DOI 10.1002/hc.20500     

New Silver Tellurates – The Crystal Structures of a Third Modification of Ag2Te2O6 and of Ag4TeO5

Single crystals of a third modification of Ag₂Te₂O₆ (denoted as Ag₂Te₂O₆–III) and of Ag₄TeO₅ have been obtained as minor by‐products during hydrothermal phase formation experiments in the system Ag‐Hg‐Te‐O. The crystal structure of Ag₂Te₂O₆–III (P2₁/c, Z = 4, a = 6.4255(10), b = 6.9852(11), c = 13.204(2) Å, β = 90.090(3)°, 1885 independent reflections, R[F² > 2σ(F²)] = 0.0334, wR2(F² all) = 0.0817) comprises tellurium in oxidation states +IV and +VI and is topologically related to the structure of the Ag₂Te₂O₆–I modification, which consists of similar layers and interjacent layers of Ag⁺ cations. Ag₄TeO₅ (C2/c, Z = 8, a = 16.271(2), b = 6.0874(10), c = 11.4373(16) Å, β = 106.730(10)°, 2372 independent reflections, R[F² > 2σ(F²)] = 0.0288, wR2(F² all) = 0.0737) is made up of a layer‐like arrangement of isolated [Te^VI₂O₁₀] double octahedra and of Ag⁺ cations situated both in layers parallel and inside the layers of the anionic moieties.     

A simple automatic system using the HP-85 microcomputer for programmed potentiometric titration

A simple way of constructing an automatic titration system is proposed, based on combination of a digital pH-meter and an automatic burette with an HP-85 microcomputer. An inexpensive interface unit based on a MOTOROLA 6800 has been developed, which permits communication of the computer with the pH-meter and the burette. The system constructed is very flexible and can find wide application in potentiometric titration and direct potentiometry with different kinds of ion-selective electrodes.     

Synthetic Activators of Cell Migration Designed by Constructive Machine Learning

Constructive machine learning aims to create examples from its learned domain which are likely to exhibit similar properties. Here, a recurrent neural network was trained with the chemical structures of known cell-migration modulators. This machine learning model was used to generate new molecules that mimic the training compounds. Two top-scoring designs were synthesized, and tested for functional activity in a phenotypic spheroid cell migration assay. These computationally generated small molecules significantly increased the migration of medulloblastoma cells. The results further corroborate the applicability of constructive machine learning to the de novo design of druglike molecules with desired properties.