In this note we continue the study of asymptotic invariants of Riemannian tori (e.g. see [BuI] and references there). By asymptotic invariants we mean invariants which do not change under passing to finite covers. In [BuI] we show that the asymptotic volume growth of a Riemannian torus is at least as fast as that of a flat one. One may ask what are the possible values of “asymptotic isoperimetr...

This study was conducted to assay the effects of methanol foliar application (0, 10, 20 and 30% v/v) and bed cover (control, white and black covers) on yield and some physiological traits of marigold as a factorial based on RCBD. The results revealed the significant interaction effects of bed cover and methanol foliar application on chlorophyll a content and plant height. The highest plant heig...

Abstract: We propose a reduced constrained Hamiltonian formalism for the exactly soluble B ∧ F theory of flat connections and closed two-forms over manifolds with topology Σ× (0, 1). The reduced phase space variables are the holonomies of a flat connection for loops which form a basis of the first homotopy group π1(Σ ), and elements of the second cohomology group of Σ with value in the Lie alge...

If a and b are degrees of unsolvability, a is called a minimal cover of b if b<a and no degree c satisfies b<c<a. The degree a is called a minimal cover if it is a minimal cover of some degree b. We prove by a very simple argument that 0" is not a minimal cover for any n. From this result and the axiom of Borel determinateness (BD) we show that the degrees of arithmetical sets (with their usual...

Let R be a Dedekind domain, G an affine flat R-group scheme, and B a flat R-algebra on which G acts. Let A → BG be an R-algebra map. Assume that A is Noetherian. We show that if the induced map K ⊗ A → (K ⊗ B)K⊗G is an isomorphism for any algebraically closed field K which is an R-algebra, then S ⊗A→ (S ⊗B)S⊗G is an isomorphism for any R-algebra S.

A cover (with triples) of λKn is an ordered pair (V,B) where V is an n-element set and B is a set of 3-element subsets of V called blocks such that each 2-element subset of V appears in at least λ blocks. Define E(B) = {{x, y}, {x, z}, {y, z} | {x, y, z} ∈ B}. The excess (sometimes called the padding) of a cover is defined to be E(B)−E(λKn), although it is often convenient to think of the exces...

Let φ be a function that maps any non-empty subset A of R to a non-empty subset φ(A) of R. A φ-cover of a set T = {T1, T2, . . . , Tm} of pairwise non-crossing trees in the plane is a set of pairwise disjoint connected regions such that 1. each tree Ti is contained in some region of the cover, 2. each region of the cover is either (a) φ(Ti) for some i, or (b) φ(A ∪B), where A and B are construc...

Formerly extensive on the Volcanic Plateau, frost flat heathland dominated by monoao (Dracophyllum subulatum ) is now reduced to fragments. It is threatened by mouse-ear hawkweed (Hieracium pilosella) at the only major site where it is likely to survive indefinitely (Rangitaiki Conservation Area). In the past, fire has perpetuated monoao on frost flats, but in future it may accelerate the sprea...

The Anthropocene is a geological epoch marked by major human influences on processes in the atmosphere, biosphere, hydrosphere and geosphere. One of the most dramatic features of the Anthropocene is the massive alteration of the Earth's vegetation, including forests. Here we investigate the role of topography in shaping human impacts on tree cover from local to global scales. We show that human...

We examined UV-B radiation flux and its environmental control within and among streams of northern Michigan. UV-B flux was estimated in streams by plastic dosimetry strips, which allow for the simultaneous and repeated in situ measurement of solar radiation. During the summer of 2004, UV-B flux was measured across depth gradients and along longitudinal transects in seven streams, which were cho...