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A standard proof relies on transforming the differential equation into an integral equation, then applying the Banach fixed-point theorem to prove the existence of a solution, and then applying Grönwall's lemma to prove the uniqueness of the solution.
In turn, it was the California Practice Act that served as the foundation of the California Code of Civil Procedure. New York never enacted Field's proposed civil or political codes, and belatedly enacted his proposed penal and criminal procedure codes only after California, but they were the basis of the codes enacted by California in 1872. [11]
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
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]
The theorem and its proof are valid for analytic functions of either real or complex variables. Let K denote either the fields of real or complex numbers, and let V = K m and W = K n . Let A 1 , ..., A n −1 be analytic functions defined on some neighbourhood of (0, 0) in W × V and taking values in the m × m matrices, and let b be an ...
The disjunction property is satisfied by a theory if, whenever a sentence A ∨ B is a theorem, then either A is a theorem, or B is a theorem.; The existence property or witness property is satisfied by a theory if, whenever a sentence (∃x)A(x) is a theorem, where A(x) has no other free variables, then there is some term t such that the theory proves A(t).
Bernard Witkin's Summary of California Law, a legal treatise popular with California judges and lawyers. The Constitution of California is the foremost source of state law. Legislation is enacted within the California Statutes, which in turn have been codified into the 29 California Codes.
The Peano existence theorem cannot be straightforwardly extended to a general Hilbert space: for an open subset of , the continuity of : alone is insufficient for guaranteeing the existence of solutions for the associated initial value problem.