In this paper, we will show that the unitary equivalence of two multiplication operators on the Bergman spaces on polygons depends on the geometry of the polygon.

Buried glacier ice is common in the McMurdo Dry Valleys and under ideal climatic and geomorphological conditions may be preserved for multimillion-year timescales. This study focuses on the analysis of ~300 m2 of buried glacier ice in lower Kennar Valley, Quartermain Range. The mapped ice is clean,<10 m thick and covered by a~25 cm sandy drift. The mouth of Kennar Valley is occupied by a lobe of Taylor Glacier, an outlet glacier from Taylor Dome. Based on ice–sediment characteristics, air bubble concentrations and stable isotopic analyses from three ice cores, the lower Kennar Valley ice is glacial in origin. These data coupled with a previously reported exposure age chronology indicate that the buried ice was deposited by a late Pleistocene advance of Taylor Glacier, probably during an interglacial interval. The surface of the buried glacier ice exhibits a patterned ground morphology characterized by small, dome-shaped polygons with deep troughs. This shape possibly reflects the final stages of ice loss, as stagnant, isolated ice pinnacles sublimate in place. This study highlights how polygon morphology can be used to infer the thickness of clean buried ice and its geomorphological stability throughout Antarctica, as well as other in cold, arid landscapes.

Ground ice is one of the most important and dynamic geologic components of permafrost; however, few studies have investigated the distribution and origin of ground ice in the McMurdo Dry Valleys of Antarctica. In this study, ice-bearing permafrost cores were collected from 18 sites in University Valley, a small hanging glacial valley in the Quartermain Mountains. Ground ice was found to be ubiquitous in the upper 2 m of permafrost soils, with excess ice contents reaching 93%, but ground ice conditions were not homogeneous. Ground ice content was variable within polygons and along the valley floor, decreasing in the centres of polygons and increasing in the shoulders of polygons towards the mouth of the valley. Ground ice also had different origins: vapour deposition, freezing of partially evaporated snow meltwater and buried glacier ice. The variability in the distribution and origin of ground ice can be attributed to ground surface temperature and moisture conditions, which separate the valley into distinct zones. Ground ice of vapour-deposition origin was predominantly situated in perennially cryotic zones, whereas ground ice formed by the freezing of evaporated snow meltwater was predominantly found in seasonally non-cryotic zones.

In this paper, we introduce cell-forms on 𝔐0,n, which are top-dimensional differential forms diverging along the boundary of exactly one cell (connected component) of the real moduli space 𝔐0,n(ℝ). We show that the cell-forms generate the top-dimensional cohomology group of 𝔐0,n, so that there is a natural duality between cells and cell-forms. In the heart of the paper, we determine an explicit basis for the subspace of differential forms which converge along a given cell X. The elements of this basis are called insertion forms; their integrals over X are real numbers, called cell-zeta values, which generate a ℚ-algebra called the cell-zeta algebra. By a result of F. Brown, the cell-zeta algebra is equal to the algebra of multizeta values. The cell-zeta values satisfy a family of simple quadratic relations coming from the geometry of moduli spaces, which leads to a natural definition of a formal version of the cell-zeta algebra, conjecturally isomorphic to the formal multizeta algebra defined by the much-studied double shuffle relations.

We prove some new upper and lower bounds for the first Dirichlet eigenvalue of triangles and quadrilaterals. In particular, we improvePólya and Szegö's [Annals of Mathematical Studies 27 (1951)] lower bound for quadrilaterals and extendHersch's [Z. Angew. Math. Phys. 17 (1966) 457–460] upper bound for parallelograms to general quadrilaterals.

We show that given a stable weighted configuration on the asymptotic boundary of a locally compact Hadamard space, there is a polygon with Gauss map prescribed by the given weighted configuration. Moreover, the same result holds for semistable configurations on arbitrary Euclidean buildings.

Let T be a triangle. P be a parallelogram, E be an ellipse, A, B be concentric circles, C, D be concentric dartboard regions, R, S be rectangles of the same orientation, U, V be two finite unions and/or differences of convex regions in the Euclidean plane. Given a function f on [0,∞), let E[/(r), U, V] denote the mean value of f(|u–v|), where |u–v| is the distance between u∊U and v∊V. Using Borel’s overlap technique, a specific distance weight function and a specific equivalence relation, we obtain formulae expressing E[f(r), U, V] in terms of triple integrals, expressing E(rn, U, V), E[f(r), A, V] and E[f(r), R, V] in terms of double integrals, expressing E[f(r), A, B], E[f(r), R, S], E[f(r), T, T], E[f(r), P, P], E(rn, C, D) and E(rn, R, V) in terms of single integrals, and expressing E(rn, R, S), E(rn, P, P), E(rn, T, T), E(rn, E, E) in terms of elementary functions, where n is an integer ≧−1. Many other related results are also given.