Topic
Division (mathematics)
About: Division (mathematics) is a research topic. Over the lifetime, 12717 publications have been published within this topic receiving 87814 citations.
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150 citations
TL;DR: This work presents a procedure by which additions and subtractions can be performed rapidly and accurately and shows that these operations are thereby competitive with their floating-point equivalents, and presents some large-scale case studies which show that the average performance of the LNS exceeds floating- point, in terms of both speed and accuracy.
Abstract: A new European research project aims to develop a microprocessor based on the logarithmic number system, in which a real number is represented as a fixed-point logarithm. Multiplication and division therefore proceed in minimal time with no rounding error. However, the system can only offer an overall advantage over floating-point if addition and subtraction can be performed with speed and accuracy at least equal to that of floating-point, but these operations require the interpolation of a nonlinear function which has hitherto been either time-consuming or inaccurate. We present a procedure by which additions and subtractions can be performed rapidly and accurately and show that these operations are thereby competitive with their floating-point equivalents. We then present some large-scale case studies which show that the average performance of the LNS exceeds floating-point, in terms of both speed and accuracy.
148 citations
TL;DR: It is proposed that division plane orientation by tensile stress offers a general rule for symmetric cell division in plants, and simulations of tissues growing in an isotropic stress field, and dividing along maximal tension, provided division plane distributions comparable with the geometrical rule.
Abstract: Cell geometry has long been proposed to play a key role in the orientation of symmetric cell division planes. In particular, the recently proposed Besson–Dumais rule generalizes Errera’s rule and predicts that cells divide along one of the local minima of plane area. However, this rule has been tested only on tissues with rather local spherical shape and homogeneous growth. Here, we tested the application of the Besson–Dumais rule to the divisions occurring in the Arabidopsis shoot apex, which contains domains with anisotropic curvature and differential growth. We found that the Besson–Dumais rule works well in the central part of the apex, but fails to account for cell division planes in the saddle-shaped boundary region. Because curvature anisotropy and differential growth prescribe directional tensile stress in that region, we tested the putative contribution of anisotropic stress fields to cell division plane orientation at the shoot apex. To do so, we compared two division rules: geometrical (new plane along the shortest path) and mechanical (new plane along maximal tension). The mechanical division rule reproduced the enrichment of long planes observed in the boundary region. Experimental perturbation of mechanical stress pattern further supported a contribution of anisotropic tensile stress in division plane orientation. Importantly, simulations of tissues growing in an isotropic stress field, and dividing along maximal tension, provided division plane distributions comparable to those obtained with the geometrical rule. We thus propose that division plane orientation by tensile stress offers a general rule for symmetric cell division in plants.
145 citations
Book•
01 Aug 1991
TL;DR: In this paper, a generalized Riemann and variational integration in division systems and division spaces is presented, with a focus on a finite number of division systems (spaces) and integration in infinite-dimensional spaces.
Abstract: Introduction and prerequisites Division systems and division spaces Generalized Riemann and variational integration in division systems and division spaces Limits under the integral sign, functions depending on a parameter Differentiation Cartesian products of a finite number of division systems (spaces) Integration in infinite-dimensional spaces Perron-type, Ward-type, and convergence-factor integrals Functional analysis and integration theory References
143 citations
TL;DR: This work uses edges to drive the division process and introduces a nodal numbering that maximizes the trapezoid quality created by each mid-edge node.
142 citations