The maximum voltage drop in an on-chip power distribution network: analysis of square, triangular and hexagonal power pad arrangements

A mathematical model of the voltage drop which arises in on-chip power distribution networks is used to compare the maximum voltage drop in the case of different geometric arrangements of the pads supplying power to the chip. These include the square or Manhattan power pad arrangement, which currently predominates, as well as equilateral triangular and hexagonal arrangements. In agreement with the findings in the literature and with physical and SPICE models, the equilateral triangular power pad arrangement is found to minimize the maximum voltage drop. This headline finding is a consequence of relatively simple formulas for the voltage drop, with explicit error bounds, which are established using complex analysis techniques, and elliptic functions in particular.

Nonlinear principal and canonical directions from continuous extensions of multidimensional scaling

A continuous random variable is expanded as a sum of a sequence of uncorrelated random variables. These variables are principal dimensions in continuous scaling on a distance function, as an extension of classic scaling on a distance matrix. For a particular distance, these dimensions are principal components. Then some properties are studied and an inequality is obtained. Diagonal expansions are considered from the same continuous scaling point of view, by means of the chi-square distance. The geometric dimension of a bivariate distribution is defined and illustrated with copulas. It is shown that the dimension can have the power of continuum.

A Comparative Study of Desktop, Fishtank, and Cave Systems for the Exploration of Volume Rendered Confocal Data Sets

We present a participant study that compares biological data exploration tasks using volume renderings of laser confocal microscopy data across three environments that vary in level of immersion: a desktop, fishtank, and cave system. For the tasks, data, and visualization approach used in our study, we found that subjects qualitatively preferred and quantitatively performed better in the cave compared with the fishtank and desktop. Subjects performed real-world biological data analysis tasks that emphasized understanding spatial relationships including characterizing the general features in a volume, identifying colocated features, and reporting geometric relationships such as whether clusters of cells were coplanar. After analyzing data in each environment, subjects were asked to choose which environment they wanted to analyze additional data sets in - subjects uniformly selected the cave environment.