The structure proposed for baddeleyite by NAray-Szab6 (1936) The structure was determined and refined by use of Patterson and Fourier projections on the three faces of the unit cell. The seven shortest Zr-O separations in the coordination polyhedron around each zirconium range from 2.04 to 2.26 A; since the next nearest Zr-O separation is 3.77 ~, the coordination number of zirconium is clearly seven. The structure is an interesting combination of fluorite-like layers parallel to (100) in which the oxide ions are tetrahedrally coordinated, with layers in which the oxide ions are in triangular coordination. The strong tendency to twin on (100) is explained in terms of this feature of the structure.

The isomorphism of Sb2Se 3 with Sb2S 3 has been confirmed through a re-determination and refinement of the structure of Sb2Se 3. Antimony selenide is orthorhombic, Pbnm, with a---11.62:t:0.01, b ~-11.77~0.01, c = 3"962:t:0"007/k and z ----4. The 'bonded' Sb-Se distances range from 2.576/~ to 2.777 A, while the 'non-bonded' Sb-Se separations start at 2-98 A. The Se-Sb-Se angles range from 86.6 ° to 96"0 ° and the Sb-Se-Sb angles from 91.0 ° to 98-9 °. The structure consists of chain.s parallel to the c or needle axis. The strongest bonds (shortest separations) are within the chains; however, there are strong interactions between the chains. All important features of the structure reported by Hofmann for Sb2S 3 (stibnite) have been confirmed.

Our main theorem shows that the regularity of non-degenerate homogeneous prime ideals is not bounded by any polynomial function of the degree; this holds over any field k. In particular, we provide counterexamples to the longstanding Regularity Conjecture, also known as the Eisenbud-Goto Conjecture (1984). We introduce a method which, starting from a homogeneous ideal I, produces a prime ideal whose projective dimension, regularity, degree, dimension, depth, and codimension are expressed in terms of numerical invariants of I. The method is also related to producing bounds in the spirit of Stillman's Conjecture, recently solved by Ananyan-Hochster.

The crystal structure of fl selenium as reported by Burbank (1952) is shown to be in error. From reanalysis of intensity data of the type h/c0 the Ses molecule is shown to be a puckered ring rather than a chain. This ring has approximate symmetry 82m, as in a selenium. Satisfactory packing between Se 8 molecules is also obtained with the revised parameters, and there appears to be very little structural difference between a selenium and fl selenium.Evidence is presented that a single crystal of fl selenitun may transform into an aggregate of crystals of hexagonal selenium rather than into a single crystal as proposed by Burbank.

Abstract. Motivated by Stillman's question, we show that the projective dimension of an ideal generated by four quadric forms in a polynomial ring is at most 6; moreover, this bound is tight. We achieve this bound, in part, by giving a characterization of the low degree generators of ideals primary to height three primes of multiplicities one and two.