We introduce a holomorphic torsion invariant of log-Enriques surfaces of index two with cyclic quotient singularities of type $\frac {1}{4}(1,1)$ . The moduli space of such log-Enriques surfaces with k singular points is a modular variety of orthogonal type associated with a unimodular lattice of signature $(2,10-k)$ . We prove that the invariant, viewed as a function of the modular variety, is given by the Petersson norm of an explicit Borcherds product. We note that this torsion invariant is essentially the BCOV invariant in the complex dimension $2$ . As a consequence, the BCOV invariant in this case is not a birational invariant, unlike the Calabi-Yau case.

Yoshikawa in [Invent. Math. 156 (2004), 53–117] introduces a holomorphic torsion invariant of $K3$ surfaces with involution. In this paper we completely determine its structure as an automorphic function on the moduli space of such $K3$ surfaces. On every component of the moduli space, it is expressed as the product of an explicit Borcherds lift and a classical Siegel modular form. We also introduce its twisted version. We prove its modularity and a certain uniqueness of the modular form corresponding to the twisted holomorphic torsion invariant. This is used to study an equivariant analogue of Borcherds’ conjecture.

Experimental studies were conducted to investigate the microwave (MW) heating behavior of soda-lime glass beads with added iron powder. These studies were intended to obtain fundamental knowledge for vitrification solidification and for the fabrication of metal-reinforced glass-matrix composites. The glass beads (0.2 mm diameter) did not heat very well by themselves at temperatures greater than 200 °C within 600 s in a multimode applicator at a power of 0.67 W. The addition of iron powder (average 70 μm, volume fraction 18%) made it possible to heat the glass beads above 700 °C within 60 s. At lower fractions of 3–11 vol%, however, a sudden temperature rise [thermal runaway (TRW)] occurred after the incubation time period. A single-mode MW applicator was used for clarifying the electric (E)-field and magnetic (H)-field contributions to the heating of each material and their mixtures. The results of this study demonstrated that the H-field contributed to the heating of the iron and then triggered the heating of the glass. The E-field component is necessary for heating the glass to a temperature higher than 800 °C. The factors determining the threshold values of the volume fraction causing TRW are discussed.

We shall show that the spectrum of Laplacian depends continuously on the parameter for one parameter degeneration of projective algebraic manifolds.