Abstract: A technique for determining the number of constraints on a set of input data, or equivalently the topological dimension, especially when such data are produced by a nonlinear system, such as a pathological vocal system or econometric data and the like. The technique characterizes the tangent space about a predetermined base point by identifying a maximal set of non-redundant nonlinear fits to the data. It needs only a modest number of data points and does not assume prior knowledge of the functional form of the true constraints, other than smoothness. Each fit is equivalent to a set of contours (including curves, surfaces, and other manifolds), with the data themselves all lying along the zero-value contour of the fit. For each fit, the gradient of the fit at the base point in the uphill direction across the contours identifies the constraint direction.

Abstract: A method of evaluating the exposure property of data to a wafer in which errors in the production of a photomask and the formation of patterns caused by defocus in the transfer of data to the wafer are considered. Accordingly, errors in the production of the photomask and deformation of patterns caused by defocus can be evaluated in the stage of design data.