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  Mission ::Our mission is to provide always output from any
               input by making use of
               SCHWARTZ DISTRIBUTIONS                
  Vision  ::Our Vision is to provide Uncertainty free
               phenomena in Science and Technology 

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Schwartz Distributions And Its Value About A Point


The theory of distributions (generalized functions) was originally designed to solve differential equations – mostly   coming from mathematical physics – and to formalize the physicist's intuition about point particles (which are represented as δ-functions). This theory has indeed led to many important physical applications. On the other hand, there are many purely mathematical developments of this theory that has not yet led to applications. Moreover, Physicists strongly believe that many of these new developments can be interpreted in a physically meaningful way and hopefully, will lead to new efficient computer programs for solving important practical problems.

This work contains two important ideas. One is that the distributions are closer to physics and engineering than many researchers think. Where do functions appear in applications? The simplest case is a “physical function” x(t) that describes how the value of a certain physical quantity x (e.g., of temperature) depends on time t. The objective of many physical theories is to make predictions about such dependence. For example, in meteorology, we measure the values of temperature, pressure, etc., at different points in different moments of time, and then computer programs use these measurement results to predict the future values of these variables. Every time we want to measure the value x(t0) of the quantity x at a certain moment of time t0, then, due to the inevitable inertia of any measurement device, the measured value is not x(t0), but rather an “average” temperature over an interval containing t0, i.e., in precise terms, an integral ∫f(t)•x(t)dt. Different measuring instruments correspond to different functions f(t),
some with narrower support, some with wider support. From this viewpoint, when we say that we have a physical function, i.e., a quantity depending on time, in reality, what we measure is a functional f→∫f•x that maps smooth fast decreasing functions f(t) (corresponding to different measuring instruments) into real numbers. In other words, a natural computer representation of a physical function is such a functional – i.e., a distribution. From this viewpoint, not only the original definition becomes physically natural, but also physically meaningful versions of this notion can be obtained if we take into consideration which test functions f(t) make physical sense.

Similarly, physical fields – physical functions of several variables – are naturally represented as corresponding distributions. Thus, from the computing viewpoint, to study physical functions, we must study not just real numbers and traditional mathematical functions x, but also functionals that map such functions into real numbers. In computer science, such a transition is very natural: every time we have two types of objects t1 and t2, we can define a new type – functions that input objects of type t1 and output objects of type t2. The corresponding
formalism is called λ -calculus: for example, λ x.x˛ + 1 means a function that takes x as an input and returns x˛+1. In these terms, distributions can be described as λ x. ∫ f•x.
                                                      
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The theory of whole computer science & its practical uses (software) are based on theory of functions y=f(x) up till now. Moreover, Logicians since 1910 (see, Hindley & Seldon (1986)) have pointed out that the function does not provide always output for any input in a system or sub-system. These Logicians further pointed out that the operator can provide always output for any input. Keeping this view in mind,( Misra (2002),Distribution Theory In Computer Science,SCI-TEC,2002) has developed a theory of operators in terms of Schwartz Distribution to carry on our desired results as stated. In fact, the importance of the said work can be found above.
 

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