High Order Methods for a Class of Volterra Integral Equations with Weakly Singular Kernels

The solution of the Volterra integral equation, $$( * )\qquad x(t) = g_1 (t) + \sqrt {t}g_2 (t) + \int _0^t \frac {K(t,s,x(s))} {\sqrt {t - s} } ds, \quad 0 \leqq t \leqq T,$$ where $g_1 (t)$, $g_2 (t)$ and $K(t,s,x)$ are smooth functions, can be represented as $x(t) = u(t) + \sqrt {t}v(t)$,$0 \leqq t \leqq T$, where $u(t)$, $v(t)$ are, smooth and satisfy a system of Volterra integral equations. In this paper, numerical schemes for the solution of (*) are suggested which calculate $x(t)$ via $u(t)$, $v(t)$ in a neighborhood of the origin and use (*) on the rest of the interval $0 \leqq t \leqq T$. In this way, methods of arbitrarily high order can be derived. As an example, schemes based on the product integration analogue of Simpson's rule are treated in detail. The schemes are shown to be convergent of order $h^{{7 / 2}}$. Asymptotic error estimates are derived in order to examine the numerical stability of the methods

Effects of Ascorbic Acid Deficiencies on Larvae of \u3ci\u3eLymantria Dispar\u3c/i\u3e (Lepidoptera: Lymantriidae)

We assessed the effects of ascorbic acid and total vitamin deficiencies on growth, food processing efficiencies and survival of larval gypsy moths. Artificial diet lacking ascorbic acid did not alter performance of fourth instars, whereas diet lacking a total vitamin mix margmally reduced growth. All vita- min deficient diets substantially reduced survival of fourth-fifth instars. Mortality occurred primarily during molting periods. providing further evidence of the putative role of ascorbic acid in cuticle formation

The norm of a Ree group

We give an explicit construction of the Ree groups of type $G_2$ as groups acting on mixed Moufang hexagons together with detailed proofs of the basic properties of these groups contained in the two fundamental papers of Tits on this subject. We also give a short proof that the norm of a Ree group is anisotropic

Column tessellations

A new class of random spatial tessellations is introduced -- the so-called column tessellations of three-dimensional space. The construction is based on a stationary planar tessellation. Each cell of the spatial tessellation is a prism whose base facet is congruent to a cell of the planar tessellation. Thus intensities, topological and metric mean values of the spatial tessellation can be calculated by suitably chosen parameters of the planar tessellation. A column tessellation is not facet-to-facet.Comment: 18 pages, 3 figure

Ammonia assimilation in Bacillus polymyxa. 15N NMR and enzymatic studies

Pathways of ammonia assimilation into glutamic acid and alanine in Bacillus polymyxa were investigated by 15N NMR spectroscopy in combination with measurements of the specific activities of glutamate dehydrogenase, glutamine synthetase, glutamate synthetase, alanine dehydrogenase, and glutamic-alanine transaminase. Ammonia was found to be assimilated into glutamic acid predominantly by NADPH-dependent glutamate dehydrogenase with a Km of 2.9 mM for NH4+ not only in ammonia-grown cells but also in nitrate-grown and nitrogen-fixing cells in which the intracellular NH4+ concentrations were 11.2, 1.04, and 1.5 mM, respectively. In ammonia-grown cells, the specific activity of alanine dehydrogenase was higher than that of glutamic-alanine transaminase, but the glutamate dehydrogenase/glutamic-alanine transaminase pathway was found to be the major pathway of 15NH4+ assimilation into [15N]alanine. The in vitro specific activities of glutamate dehydrogenase and glutamine synthetase, which represent the rates of synthesis of glutamic acid and glutamine, respectively, in the presence of enzyme-saturating concentrations of substrates and coenzymes are compared with the in vivo rates of biosynthesis of [15N]glutamic acid and [alpha,gamma-15N]glutamine observed by NMR, and implications of the results for factors limiting the rates of their biosynthesis in ammonia- and nitrate-grown cells are discussed