Levi-Civita field

The Levi -Civita body is a body, which was invented by Tullio Levi -Civita. The real numbers or the complex numbers are a subfield of the Levi -Civita body. The Levi -Civita body has applications in efficient symbolic computation of values ​​of higher derivatives of functions.

  • 6.1 Levi -Civita field of real functions
  • 6.2 Levi -Civita field of complex functions
  • 6.3 seminorm
  • 7.1 order topology
  • 7.2 seminorm topology

Definition

Basic amount of the body

The basic amount of the Levi -Civita body are all functions (respectively) that have a finite carrier left.

Notation

  • Just as the real numbers are abbreviated, you can abbreviate the Levi -Civita body or, depending on whether the basic set of real or complex functions there.
  • If is the Levi -Civita body and a non-empty carrier has, it is referred to the minimum of the support that exists for links finitude.
  • One writes for and and that.

Addition

Addition of two elements of the base, and which is defined as follows:

The additive inverse is as follows:

The zero element is:

Multiplication

The multiplication of two elements of the base, and which is defined as follows:

One element

The unit element of the Levi -Civita body is the function

Multiplicative inverse

If an element of the Levi -Civita body, so you can get a multiplicative inverse construct as follows: One chooses, with the smallest number with and. If the support contains only the 0, then. Otherwise, for a Levi -Civita body and you will not look at for one. We define the sequence by and. Then satisfies the desired property. Then is. Now one can find the multiplicative inverse of by.

Fixed point theorem

The above definition of multiplicative inverse results (see the first source ) from the proof of the fixed point theorem, which guarantees that the limit of the sequence exists and satisfies the desired property. The fixed point theorem is as follows:

Be. Be or the set of elements, so. Further, let or a function with the properties

  • (or )

Then there is exactly one or so:

Embedding of the real or complex numbers

To embed the real or complex numbers in the Levi -Civita body, one uses the following function:

Here, the identity element is mapped from and to the unity element of or. Further, a homomorphism relation to the addition and multiplication. Therefore, the real and complex numbers can be viewed as a body of the Levi -Civita body.

Order of real Levi -Civita body

Be respectively. They say if and. This is the Levi -Civita field of real functions to a subordinate body.

With this order, for example, the number

Smaller than any positive real number.

The Archimedean axiom is not satisfied for the Levi -Civita body. For example, the following applies:

Root

With respect to multiplication defined above, each exactly different -th roots. For one, there are the following numbers of - th roots of:

Amount

Levi -Civita field of real functions

Be. The value of x is defined by:

Levi -Civita field of complex functions

Be where the imaginary number. The value of x is defined by:

Here, the root with respect to the above defined multiplication of the Levi -Civita body is meant.

Seminorm

Be. Then one can define the following seminorm on the Levi -Civita body:

Wherein the magnitude of the real or complex numbers.

Topologies

Order topology

Be respectively. Be

For the order topology is defined as an open set, given

This topology has the following characteristics:

  • She makes and non-contiguous Hausdorff spaces.
  • Are adopting this definition of open sets and not locally compact spaces.
  • In this topology agrees with the discrete topology of the same.

Semi-norm topology

Be the semi-norm of the Levi -Civita body. Be respectively. Be

For the semi-norm topology we define M as an open set, given

This topology has the following characteristics:

  • She makes and Hausdorff spaces with countable bases.
  • The topology defined by it is limited to the real or complex numbers, the standard topology.

Derivation

One can define the Levi -Civita body a derivation:

For this derivation applies:

Applications

The Levi -Civita body enables the efficient calculation of higher derivatives of functions such as

There is a system based on the Levi -Civita body program, which calculates the value of the derivative of this function 19 at the point 0 within less than one second. Mathematica required for calculating the contrast value of the derivative of this function at the 6 position 0 more than 6 minutes.

Swell

  • Martin Berz: Calculus and Numerics on Levi -Civita Fields. In: Martin Berz, Christian Bischof, George Corliss, Andreas Griewank ( Eds.): Computational differentiation. Techniques, applications, and tools. Proceedings of the 2nd International Workshop held in Santa Fe, NM, February 12-14, 1996. ISBN 0-89871-385-4, 2 (online, accessed on 6 June 2013).
  • Khodr Shamseddine, Martin Berz: The Differential Algebraic Structure of the Levi -Civita Field and Applications. (Online (PDF, 199 kB), accessed on August 15, 2013).
  • Body ( algebra)
  • Analysis
  • Body theory
Base (topology)
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