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It is common convention to use greek indices when writing expressions involving tensors in Minkowski space, while Latin indices are reserved for Euclidean space. Well-formulated expressions are constrained by the rules of Einstein summation: any index may appear at most twice and furthermore a raised index must contract with a lowered index ...
In computational number theory, the index calculus algorithm is a probabilistic algorithm for computing discrete logarithms.Dedicated to the discrete logarithm in (/) where is a prime, index calculus leads to a family of algorithms adapted to finite fields and to some families of elliptic curves.
The following are some examples: The term λx. λy. x, sometimes called the K combinator, is written as λ λ 2 with de Bruijn indices. The binder for the occurrence x is the second λ in scope. The term λx. λy. λz. x z (y z) (the S combinator), with de Bruijn indices, is λ λ λ 3 1 (2 1). The term λz. (λy. y (λx. x)) (λx.
Some solutions of a differential equation having a regular singular point with indicial roots = and .. In mathematics, the method of Frobenius, named after Ferdinand Georg Frobenius, is a way to find an infinite series solution for a linear second-order ordinary differential equation of the form ″ + ′ + = with ′ and ″.
The free indices in a tensor expression always appear in the same (upper or lower) position throughout every term, and in a tensor equation the free indices are the same on each side. Dummy indices (which implies a summation over that index) need not be the same, for example:
The power series method will give solutions only to initial value problems (opposed to boundary value problems), this is not an issue when dealing with linear equations since the solution may turn up multiple linearly independent solutions which may be combined (by superposition) to solve boundary value problems as well. A further restriction ...
The index problem is the following: compute the (analytical) index of D using only the symbol s and topological data derived from the manifold and the vector bundle. The Atiyah–Singer index theorem solves this problem, and states: The analytical index of D is equal to its topological index.
Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory of distributions, by generalising the concept of an integer index to an ordered tuple of indices.