Product (mathematics)

About: Product (mathematics) is a research topic. Over the lifetime, 44382 publications have been published within this topic receiving 377809 citations.

An algorithm for associating the features of two images

Abstract: In this paper we describe an algorithm that operates on the distances between features in the two related images and delivers a set of correspondences between them. The algorithm maximizes the inner product of two matrices, one of which is the desired \`pairing matrix' and the other a \`proximity matrix' with elements exp (-r$\_{ij}^{2}$/2$\sigma ^{2}$), where r$\_{ij}$ is the distance between two features, one in each image, and $\sigma $ is an adjustable scale parameter. The output of the algorithm may be compared with the movements that people perceive when viewing two images in quick succession, and it is found that an increase in $\sigma $ affects the computed correspondences in much the same way as an increase in interstimulus interval alters the perceived displacements. Provided that $\sigma $ is not too small the algorithm will recover the feature mappings that result from image translation, expansion or shear deformation -- transformations of common occurrence in image sequences -- even when the displacements of individual features depart slightly from the general trend.

An extension of Kakutani’s theorem on infinite product measures to the tensor product of semifinite *-algebras

Abstract: Suppose that (s a?1 is the (Ii)-incomplete direct product of (k): see [9], [12] or [1].) Given a normal state vi of 4 for each i E I, a normal state v of a? is written ?iE, vi when V(J A)7 = 11 vi(A2) ieF / ieF for all Ai E E and all finite subsets F of L Our main result (Theorem 4.1) is that (g?iI vi exists on ,d if and only if >icI [d(i, vi)]2 converges, or, equivalently, if and only if lfie P(ip(, vi) converges. Here d is a metric on the set of normal states of a w*-algebra X. d is defined essentially by d(yt, v) =inf { 1x -y 1}, the infimum being taken over all vectors x and y inducing 1 and v relative to a representation of I as a von Neumann algebra. p is a kind of inner product defined by

Product units: a computationally powerful and biologically plausible extension to backpropagation networks

Abstract: We introduce a new form of computational unit for feedforward learning networks of the backpropagation type. Instead of calculating a weighted sum this unit calculates a weighted product, where each input is raised to a power determined by a variable weight. Such a unit can learn an arbitrary polynomial term, which would then feed into higher level standard summing units. We show how learning operates with product units, provide examples to show their efficiency for various types of problems, and argue that they naturally extend the family of theoretical feedforward net structures. There is a plausible neurobiological interpretation for one interesting configuration of product and summing units.

Product information storage, display, and coupon dispensing system

Abstract: A system for generating product coupons at remote locations. The system includes an apparatus for receiving and storing product information signals broadcast from a central location. The apparatus is connected to a television at the remote location and has a coupon generating unit. The coupon generating unit can generate a product coupon having selectively entered product information thereon. The system further comprises a switch such that stored product information signals can be displayed on the television.