essentially unique in a sentence

Examples

1. This implies that is a principal ideal domain and any positive integer can be written as the products of primes in an essentially unique way.
2. In general given two isomorphic examples of an essentially unique object, there is no " natural " ( unique ) isomorphism between them.
3. Maximal compact subgroups of Lie groups are " not " in general unique, but are unique up to conjugation  they are essentially unique.
4. It has many applications, including a simplified proof of Moise's theorem, which states that every 3-manifold can be triangulated in an essentially unique way.
5. Due to loose tolerances, each aircraft was essentially unique, with workers performing the final assembly having the unenviable task of mating what often proved to be somewhat dissimilar components.
6. They exhibit many of the general properties of the concept of " irreducibility " that equally apply to irreducible polynomials, such as the essentially unique factorization into prime or irreducible factors.
7. One of the most remarkable facts about a locally compact group " G " is that it carries an essentially unique natural ?-algebra generated by the compact sets.
8. Complex structures on a closed oriented surface correspond to Poincar?) states that any Riemannian metric on an oriented, closed surface is conformally equivalent to an essentially unique metric of constant curvature.
9. The cycle decomposition is essentially unique : apart from the reordering the cycles in the product, there are no other ways to write " ? " as a product of cycles.
10. During 1949 1951 he held an appointment at the Institute for Advanced Study during which he proved Moise's theorem that every 3-manifold can be triangulated in an essentially unique way.