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The Peano existence theorem shows only existence, not uniqueness, but it assumes only that f is continuous in y, instead of Lipschitz continuous. For example, the right-hand side of the equation dy / dt = y 1 / 3 with initial condition y(0) = 0 is continuous but not Lipschitz continuous.
In mathematics and logic, the term "uniqueness" refers to the property of being the one and only object satisfying a certain condition. [1] This sort of quantification is known as uniqueness quantification or unique existential quantification, and is often denoted with the symbols "∃!" [2] or "∃ =1". For example, the formal statement
Uniqueness is a state or condition wherein someone or something is unlike anything else in comparison, or is remarkable, or unusual. [1] When used in relation to humans, it is often in relation to a person's personality, or some specific characteristics of it, signalling that it is unlike the personality traits that are prevalent in that individual's culture. [2]
Examples of uniqueness theorems include: Cauchy's rigidity theorem and Alexandrov's uniqueness theorem for three-dimensional polyhedra. Black hole uniqueness theorem; Cauchy–Kowalevski theorem is the main local existence and uniqueness theorem for analytic partial differential equations associated with Cauchy initial value problems.
A translation and dilation of a set of uniqueness is a set of uniqueness. A union of a countable family of closed sets of uniqueness is a set of uniqueness. There exists an example of two sets of uniqueness whose union is not a set of uniqueness, but the sets in this example are not Borel. It is an open problem whether the union of any two ...
Examples of non-uniqueness of extension [ edit ] There can be more than one extension of a pre-measure to the generated σ-algebra, if the pre-measure is not σ {\displaystyle \sigma } -finite, even if the extensions themselves are σ {\displaystyle \sigma } -finite (see example "Via rationals" below).
We have listed some of our favorite examples of unique and varied aircraft camo patterns. Check out the gallery above. Take a Look at the Largest Military Aircraft. More from Business Insider:
For the first set of solutions, uniqueness fails at one point, =, and for the second solution, uniqueness fails at every value of . Thus, the solution y s {\displaystyle y_{s}} is a singular solution in the stronger sense that uniqueness fails at every value of x .