![]() ![]() Here our main goal is to find conditions when path hyperintegral has finite value or its value is a real or complex number, i.e., when path hyperintegration coincedes with path integration. This better correlates with the situation in contemporary physics, which often encounters infinitely big numbers in a form of divergent series and integrals, while there are no infinitely small number in physics. In contrast to this, the theory of hyperfunctionals and generalized distributions does not change the inner structure of spaces of real and complex numbers, but adds to them infinitely big and oscillating numbers as external objects. For example, nonstandard analysis changes spaces of real and complex numbers by injecting infinitely small numbers and other nonstandard entities. Although, the new theory resembles nonstandard analysis, there are several distinctions between these theories. ![]() The theory of hyperfunctionals and generalized distributions, as a part of hyperanalysis that includes hyperintegration, is a novel approach in functional analysis that provides flexible means for analysis in infinite dimensional spaces. It is based on hyperintegration, which extends the path integral to the path hyperintegral. In this paper, a new approach to the path integral is developed. The Feynman path integral, being very popular in physics, has not yet found a concise unified mathematical representation. Spaces of extrafunctions and hypernumbers are special cases of hyperspaces of integral vector spaces. The main constructions are put together in the context of fiber bundles over hyperspaces of integral vector spaces and integral algebras. In this paper, a method of regularization of irregular operations, functionals and operators is developed and applied to multiplication of hypernumbers and extrafunctions (Section 5) and integration of extrafunctions (Sections 6 and 7). Examples of such operations are multiplication, differentiation and integration, which are important for calculus, differential equations and many applications of mathematics, e.g., in physics. However, there are important operations with functions and operators in function spaces the extension of which by coordinates does not work because their application is not invariant with respect to representations of extrafunctions. Examples of such operations are addition of real functions and multiplication of real functions by real numbers. It is proved that it is possible to extend several basic operations with functions and operators in function spaces to regular operations with extrafunctions and operators in spaces of extrafunctions. Operations and operators performed in this manner are called regular. Hope you liked this edition of cool infographics.It is possible to perform some operations with extrafunctions and operators in spaces of extrafunctions applying these operations (operators) separately to each coordinate of the representing sequence. Cool Infographics / Charts of the Week.And this silhouette of towns does that very well.Īnd of course there is NY Times Bubble visualization of Olympic Medals that I discussed here. Most of the times, the purpose of a chart is to sell one idea, convey one message or prove one point. Silhouette of Towns in Middle ages and Now Īlthough not strictly a graph, this photo published on Winston Salem Journal makes the point very effectively. If you are a star wars freak like me, then you will love this map. The star wars galaxy and well known hyperspaces, now mapped Currently the maps are available for few cities across the world (you are seeing hong kong downtown in the image), they give interesting statistics about the city like big-mac index. Onion maps is the new kid on the mapping sites space and they have managed to come up with something unique and cool. Onion Maps – 3d outlines of buildings colored and mashed up with google maps This is an area chart modified to indicate how a movie has performed everyweek, thus the movie markers go down (except for dark knight) after 1-2 weeks Super cool visualization of 2008 Box office collection What a week it has been for the chart makers □ Time for another week of ogling at cool visualizations. Cool Infographics & Data Visualizations - 0 comments
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