Product (mathematics)

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

General Framework for Consistent Sampling in Hilbert Spaces

Abstract: We introduce a general framework for consistent linear reconstruction in infinite-dimensional Hilbert spaces. We study stable reconstructions in terms of Riesz bases and frames, and generalize the notion of oblique dual frames to infinite-dimensional frames. As we show, the linear reconstruction scheme coincides with the so-called oblique projection, which turns into an ordinary orthogonal projection when adapting the inner product. The inner product of interest is, in general, not unique. We characterize the inner products and corresponding positive operators for which the new geometrical interpretation applies.

Products having rfid tags to provide information to product consumers

Abstract: RFID tags attached to a product contain information regarding the product and communicate this information to a reader. This information can include the age of the product, the amount of product left and preparation or care instructions for the product. To record the age of a product, a timer is initiated in the RFID tag. Thereafter, the elapsed time on the timer can be read and displayed by the reader. For other products, the RFID tag can have contacts monitoring the level of remaining product. Once the contacts sense that the amount of product is low, this information can be transmitted and communicated by the reader. Preparation and care instructions are prerecorded on the tag. Instructions can also be communicated between a product and a base, such as a heater.

Intersections and transformations of complexes and manifolds

Abstract: In writing this paper my first objective has been to prove certain formulas on fixed points and coincidences of continuous transformations of manifolds. To this proof for orientable manifolds without boundary is devoted most of the second part, the remainder of which is taken up by a study of product complexes in the sense of E. Steinitz, as they are the foundation on which the proof rests. With suitable restrictions the formulas derived are susceptible of extension to a wider range of manifolds, but this will be reserved for a later occasion. It may be stated that our formulas include and completely generalize the early results due to Brouwer and whatever has been obtained since along the same line.t No such generality would have been possible without that powerful instrument, the product complex. The principle of the method is best explained by means of a very simple example. Letf(x) and so(x) be continuous and uni-valued functions over the interval 0, 1, and let their values on the interval also lie between 0 and 1. It is required to find the number of solutions of f(x) = (x), 0 < x < 1. Graphically the problem is solved by plotting the curvilinear arcs y =(x), y = (x), 0 < x<1 and taking their intersections. A slight modification of the functions may change tlle number of solutions, even make them become infinite in number. However, the difference between the numbers of positive and negative crossings of sufficiently close polygonal approximations to the arcs is a fixed number, their Kronecker index. Its determination is then a partial answer to the question, and indeed seemingly the only possible general answer.