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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]
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
In mathematics, specifically the study of differential equations, the Picard–Lindelöf theorem gives a set of conditions under which an initial value problem has a unique solution. It is also known as Picard's existence theorem, the Cauchy–Lipschitz theorem, or the existence and uniqueness theorem.
Unique primarily refers to: Uniqueness , a state or condition wherein something is unlike anything else In mathematics and logic, a unique object is the only object with a certain property, see Uniqueness quantification
Uniqueness is a consequence of the last two conditions. Basic properties Proofs ... , using the same four conditions as in our definition above.
In psychology, grandiosity is a sense of superiority, uniqueness, or invulnerability that is unrealistic and not based on personal capability.It may be expressed by exaggerated beliefs regarding one's abilities, the belief that few other people have anything in common with oneself, and that one can only be understood by a few, very special people. [1]
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.
In the theory of differential equations, Lipschitz continuity is the central condition of the Picard–Lindelöf theorem which guarantees the existence and uniqueness of the solution to an initial value problem. A special type of Lipschitz continuity, called contraction, is used in the Banach fixed-point theorem. [2]