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A real tensor in 3D (i.e., one with a 3x3 component matrix) has as many as six independent invariants, three being the invariants of its symmetric part and three characterizing the orientation of the axial vector of the skew-symmetric part relative to the principal directions of the symmetric part.
Invariant-based programming [1] is a programming methodology where specifications and invariants are written before the actual program statements. Writing down the invariants during the programming process has a number of advantages: it requires the programmer to make their intentions about the program behavior explicit before actually implementing it, and invariants can be evaluated ...
[1] are a set of tensor invariants that span the space of real, symmetric, second-order, 3-dimensional tensors and are isomorphic with respect to principal stress space. This right-handed orthogonal coordinate system is named in honor of the German scientist Dr. Walter Lode because of his seminal paper written in 1926 describing the effect of ...
The theory of optimizing compilers, the methodology of design by contract, and formal methods for determining program correctness, all rely heavily on invariants. Programmers often use assertions in their code to make invariants explicit. Some object oriented programming languages have a special syntax for specifying class invariants.
a tensor of order k. Then T is a symmetric tensor if = for the braiding maps associated to every permutation σ on the symbols {1,2,...,k} (or equivalently for every transposition on these symbols). Given a basis {e i} of V, any symmetric tensor T of rank k can be written as
The Java Modeling Language (JML) is a specification language for Java programs, using Hoare style pre-and postconditions and invariants, that follows the design by contract paradigm. Specifications are written as Java annotation comments to the source files, which hence can be compiled with any Java compiler .
The Ada programming language has native support for type invariants (as well as pre- and postconditions, subtype predicates, etc.). A type invariant may be given on a private type (for example to define a relationship between its abstract properties), or on its full definition (typically to help in verifying the correctness of the implementation of the type). [5]
Invariant (computer science), an expression whose value doesn't change during program execution Loop invariant, a property of a program loop that is true before (and after) each iteration; A data type in method overriding that is neither covariant nor contravariant; Class invariant, an invariant used to constrain objects of a class