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In this paper, we propose a unified architecture for computation of double-precision floating-point division, reciprocal, square root, inverse square root and multiplication with a significant area reduction. First, a double-precision multiplication-based divider, the common datapath shared with these arithmetic computations, is optimized by a modified Goldschmidt algorithm to achieve better area efficiency. In this algorithm, a linear-degree minimax approximation instead of second-degree is used to obtain a 15-bit precision estimate of the reciprocal so that we can get a rather small lookup table (LUT) as well as reduced amount of computation when accumulating the partial products. Two Goldschmidt iterations specially designed for hardware reuse are performed to gain the final accurate result of division. By virtue of the pipelined processing, the time cost for the two iterations is minimized. Second, a reconfigurable datapath with a little extra area cost is introduced to dynamically support multiple double-precision computations by executing the optimized divider iteratively. The design is finally implemented and synthesized in SMIC 0.13-μm CMOS process. The experimental results show that the proposed design can achieve a speed of 400 MHz with area of 61.6 K logic gates and 9-Kb LUT. Compared with other works, the area efficiency (performance/area ratio) of the proposed unified architecture is increased by about 20% in average, which is a better performance-area trade-off for embedded microprocessors.

The firing rate gain of neurons, defined as the slope of the relation between input to a neuron and its firing rate, has received considerable attention in the past few years. This has been largely motivated by the many experimental demonstrations of behavior related gain changes in a variety of neural circuits of the CNS. A surprising result was that a prime candidate, shunting inhibition, apparently does not change the firing rate gain of neurons. However, in this paper, we show a physiologically plausible mechanism by which shunting inhibition in the dendritic tree does, in a simple and direct manner, modulate the firing gain of neurons. The effect is due to a strong attenuation of the dendritic current arriving at the soma by shunting dendritic inhibition. Increasing the dendritic inhibitory conductance enhances the attenuation of current flowing from the dendritic to the somatic compartment and thus reduces firing gain. This mechanism relies on known physiological and anatomical properties of CNS neurons and does not require special features such as tunable neural noise inputs. Gain control by the proposed mechanism may prove to be a ubiquitous feature of neural circuit operations and it is readily verifiable experimentally.

A new method for identification of stages in arithmetic task solving has been presented. Solving of addition tasks induced formation of EEG foci in delta–theta-band were formed in fronto-central and parietal regions of the left hemisphere and in parietal-temporal and frontal areas of right hemisphere. In multiplication tasks, delta–theta-foci were formed in frontal, central and parietal regions of the left hemisphere as well as in temporal areas of the right hemisphere. When solving addition tasks, the coherence asymmetry was increased in left hemisphere between frontal, parietal and temporal areas in delta–theta-band as well as between frontal and parietal regions in alpha-band. The neurophysiological mechanisms of complex addition and multiplication arithmetic task solving is discussed.

Through an analysis of a popular Chinese elementary school mathematics teacher manual, this chapter discusses a significant feature of teacher manual, namely, the manual is designed to help teachers achieve a thorough understanding of student text mathematically and pedagogically. Using multi-digit number multiplication, this chapter discusses the in-depth analyses and detailed suggestions for the core part of a lesson with concrete examples. In particular, it argues that an in-depth analysis of student text is the main focus of the guide. It also argues that concerted efforts are made to help create an environment that helps teachers professionally and teachers need to study the student text carefully in order to conduct classroom instruction effectively.