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Uniqueness theorem

From Wikipedia, the free encyclopedia

In mathematics, a uniqueness theorem is a theorem asserting the uniqueness of an object satisfying certain conditions, or the equivalence of all objects satisfying the said conditions.[1] Examples of uniqueness theorems include:

A theorem, also called a unicity theorem, stating the uniqueness of a mathematical object, which usually means that there is only one object fulfilling given properties, or that all objects of a given class are equivalent (i.e., they can be represented by the same model). This is often expressed by saying that the object is uniquely determined by a certain set of data. The word unique is sometimes replaced by essentially unique, whenever one wants to stress that the uniqueness is only referred to the underlying structure, whereas the form may vary in all ways that do not affect the mathematical content.[1]

A uniqueness theorem (or its proof) is, at least within the mathematics of differential equations, often combined with an existence theorem (or its proof) to a combined existence and uniqueness theorem (e.g., existence and uniqueness of solution to first-order differential equations with boundary condition[3]).

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Transcription

Suppose I come up with a differential equation: dy/dx = f (x,y) which is a perfect model for a system I'm trying to study and I have an initial-value, I know an initial state and I want to try to make a prediction by solving an initial-value problem. Now it would be pretty silly for me to study this problem if first of all, I didn't know that any solutions existed. I would spend a long time looking for solutions that don't exist. Second of all, if I didn't know that a solution to this initial-value problem was unique then it would be pretty silly for me to try to 'predict' something because my prediction would be meaningless: it would depend on which one of the many solutions I had chosen to make the prediction. So 'existence' and 'uniqueness' of solutions is an important thing to study in order to both solve problems and make predictions. The main question is this: if I have an initial-value problem like this. " When does a solution to the initial-value problem exist? " and " If it exists, is it a unique solution? " Fortunately we have easily testable, sufficient conditions that will tell us 'when' a solution exists and 'when' it is unique. First of all " if f is continuous 'near' the point (a,b) " " then a solution exists. " Remember f here is the right-hand side, defining the ordinary differential equation 'f' is a function of two variables 'x' and 'y' and we require that it be continuous near the point '(a,b)' which is the point I'm trying to get the solution to pass through in order to satisfy the initial-value. What does this look like? I have a point in the x-y plane: that's the point (a,b). So here it is, right there, this little 'orange' dot. Near (a,b) is some sort of 'area', some blob, some 'neighborhood' around that point, and I need the function f to be continuous inside that blob. Now if that's satisfied then I know that, even for just a little.. ..maybe just a for a tiny little section, I can draw my solution through that point (a,b). Now, what about 'uniqueness'? Who says I can't draw many 'little solutions' through that point (a,b)? Well, I need another hypothesis to guarantee uniqueness: if (also) the partial derivative of f with respect to y is continuous near (a,b) then the solution will be unique. So I have to check not only ..that f is continuous..but also (if I want to have uniqueness) that the partial derivative of f with respect y is continuous. I'd like to point out two subtle aspects of this discussion so far: The first is: the guarantee of 'existence' and 'uniqueness' is only 'near' (a,b). In other words I know the solution exists but I don't know how 'big' the solution [will become]: I only know that it exists for a little 'while' just around at the point (a,b). Also the uniqueness is only guaranteed near (a,b): there's nothing to say that the solution won't be unique for a while and then will split into multiple pieces somewhere else. The second subtle thing is that this is not an 'if and only if' [<=>] theorem. These conditions are not necessary for existence and uniqueness. So there's nothing to say that even if f is not continuous and even if 'partial f / partial y' is not continuous I still may have existence and uniqueness.

See also

References

  1. ^ a b Weisstein, Eric W. "Uniqueness Theorem". mathworld.wolfram.com. Retrieved 2019-11-29.
  2. ^ "The uniqueness theorem". farside.ph.utexas.edu. Retrieved 2019-11-29.
  3. ^ "Existence and Uniqueness". www.sosmath.com. Retrieved 2019-11-29.
This page was last edited on 3 November 2021, at 17:57
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