Duality (math)

In many areas of mathematics it often happens that one can construct another object for each object of the class under consideration and use it to investigate . This object is then referred to as or similar to express the dependence on . If you use the same (or a similar) construction , you get an object marked with . Often standing and in a close relationship, e.g. B. equal or isomorphic , which is why information about must contain. One then calls the too dual and the bidual object. In the associated mathematical theory of duality , one then investigates how properties of can be translated into properties of and vice versa. ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle X'}$ ${\displaystyle X}$ ${\displaystyle X'}$ ${\displaystyle X''=(X')'}$ ${\displaystyle X}$ ${\displaystyle X''}$ ${\displaystyle X'}$ ${\displaystyle X}$ ${\displaystyle X'}$ ${\displaystyle X}$ ${\displaystyle X''}$ ${\displaystyle X}$ ${\displaystyle X'}$

Duality as an overarching principle

In the most general sense, duality uses the observation of an object from a second, dual, side for the purpose of gaining knowledge. Duality is therefore a close relationship between mathematical or scientific objects that have similarities in such a way that they can be used for the (simplified) solution of problems. The purpose of this approach is that some problems can be solved more easily from one perspective, others from the second (dual) perspective or approach.

Duality is one of the most important epistemological principles of mathematics and natural sciences and plays an important role in very many completely different areas, in particular in mathematics, for example, in geometry, algebra and analysis.

Duality is not to be confused with the philosophical term dualism. In contrast to the term dualism , the focus of interest is not opposites between dual objects, but rather the ability to transform one another.

The term “construction” used above is formulated mathematically, a figure. Duality is therefore a one-to-one mapping of mathematical terms, theorems or structures to other terms, theorems and structures. In the narrower sense, the mapping used has the form of an involution (mathematics) , a self-inverse mapping: If B is the dual of A, A is again the dual of B. In the broader sense, the term "duality" can also be used for images which are not involution, if e.g. For example, the reverse mapping is based on a similar construction or is the same as the mapping on a large class of objects.

Didactic example to demonstrate the principle

Duality through logical negation

One of the simplest examples of duality is inversion, for example when using indirect proof. For every statement there is a logically inverse statement that can be easily constructed.

Let us consider the statement "All birds can fly". This has to be examined for truth. To prove this directly would, strictly speaking, mean examining all birds, more precisely, all animals that biologists assign to birds. People accept incompleteness in their everyday life by initially “believing” this sentence (e.g. at a young age) if they know a sufficient number of confirming examples and no counterexample. The technical term for this would be (incomplete) induction , an unreliable, and above all strictly logically not admissible, conclusion.

First of all, the sentence under consideration is reformulated more precisely to “All species of birds can fly”. B. to exclude cases of injury or the like. The logically reversed (inverse) statement is: "Not all bird species can fly". This can in turn be rephrased as “There is a species of bird that cannot fly”.

This statement can be seen as dual to the first. Obviously, it is much easier to prove this inverse statement by simply finding a single example of such a bird that cannot fly, such as a penguin .

Practically, the double negative results in the original statement or, applied to our example: If the inverse statement is true, the original statement must be false. The method used here is a very simple example of the mathematical method of indirect proof .

So you can view a lot of statements as a statement space and transform this into a dual space, here the space of the opposite statements. As can be seen from the examples, some questions can be solved in the dual space, others in the original space.

Duality in geometry

Duality of polytopes

The dual polyhedron to the cube is the octahedron. The centers of the side surfaces of one body form the corners of the other.

Two polytopes (ie polygons , polyhedra , etc.) and hot abstract dual if their side organizations (the inclusion of their pages, so corners, edges, faces, etc.) antiisomorph are. Here is an example: Choose the center points of the side faces of a three-dimensional convex polyhedron as corners, and connect two “new” corners if the two corresponding side faces have a common edge, i.e. H. one forms the convex hull of the "new" corners, so one gets a dual polyhedron . The number of corners of is equal to the number of areas of and vice versa, the numbers of edges are the same. Such duality is also called dimension-reversing. The following applies: The dual of the dual is the original. ${\displaystyle P}$ ${\displaystyle Q}$ ${\displaystyle P}$ ${\displaystyle P}$ ${\displaystyle Q}$ ${\displaystyle Q}$ ${\displaystyle P}$

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There are also self-dual bodies where the dual is similar to the original body . One example is the regular tetrahedron .

But this says nothing about whether the polytopes and are invariant under the same symmetry mappings. For example, a square and any quadrangle are combinatorial dual, since two edges meet at each corner and each edge has two corners. As a rule, the symmetry images of the square do not include reflections, but the square does. ${\displaystyle P}$ ${\displaystyle Q}$

For each polytope there is a special combinatorial dual polytope, the so-called polar . For this purpose, the polytope is understood as a closed subset of a Euclidean vector space. The polar then consists of all points which satisfy the inequality for all of . Assuming that the geometric center of gravity of is on zero, and its polar have the same symmetry group. The double-dual polyhedron is similar to and equal to this if the zero point is contained in its interior. ${\displaystyle P}$ ${\displaystyle P}$ ${\displaystyle y}$ ${\displaystyle x}$ ${\displaystyle P}$ ${\displaystyle \langle y,x\rangle \leq 1}$ ${\displaystyle P}$ ${\displaystyle P}$ ${\displaystyle P}$

For examples see: Platonic Solid , Archimedean Solid

Principle of duality in projective geometry and in incidence structures

→ Main article : Duality (projective geometry)

In planar projective geometry , the following principle of duality applies: If you swap the terms "point" and "straight line" in a true statement about points and straight lines of a projective plane and replace the term "straight line connecting two points" with the term "intersection of two straight lines" and vice versa, one gets a true statement about the dual projective geometry. For Desargue's projective geometries, for example all two-dimensional projective spaces over bodies , the dual projective geometry is identical to the original geometry except for isomorphism, so in such projective geometries a sentence applies precisely if the sentence applies in which the terms “point ”And“ Straight ”are swapped.

Examples of pairs of dual sentences are the Desargues theorem , which is self- dual , or the Pascal theorem and the Brianchon theorem .

→ The concrete construction of duality as an isomorphism on a projective space depends on the selected projective coordinate system and is therefore presented in the main article Projective coordinate system .

→ A generalization of the duality principle in planar projective geometry is the duality principle for incidence structures .

Geometrically dual graph

Graph theory for planar graphs has a similar definition . A graph that is geometrically dual to the graph is created by adding new nodes to each surface of the graph and creating a new edge for each edge that connects the two adjacent surfaces. ${\displaystyle G=(V,E)}$ ${\displaystyle G'=(V',E')}$ ${\displaystyle G}$ ${\displaystyle v'}$ ${\displaystyle e\in E}$ ${\displaystyle e'}$ ${\displaystyle v'}$

Is the graph not only planar but also integrally , as is also true here that the number of nodes in the number of faces in corresponding to the number of faces in those of the nodes in and the number of edges remains constant. In the connected case there are bijective mappings between the sets of edges of the two graphs and the sets of nodes and surfaces. It is also true that . ${\displaystyle G}$ ${\displaystyle G'}$ ${\displaystyle G}$ ${\displaystyle G'}$ ${\displaystyle G}$ ${\displaystyle G''=G}$

Dual space of a vector space

→ Main article : dual space

If a vector space is over a body , then the dual vector space or dual space is the vector space whose elements are the linear mappings . Is finite dimensional , then has the same dimension as , and is canonically isomorphic to . ${\displaystyle V}$ ${\displaystyle K}$ ${\displaystyle V^{*}}$ ${\displaystyle V\to K}$ ${\displaystyle V}$ ${\displaystyle V^{*}}$ ${\displaystyle V}$ ${\displaystyle V^{**}}$ ${\displaystyle V}$

In the case of a Banach space , the dual space consists of the continuous linear functionals . If infinite-dimensional, then the dual space is generally not canonically isomorphic to , but there is a canonical embedding of in the dual space . Those spaces for which this embedding is surjective (and thus an isomorphism) are called reflexive . Examples are the spaces L ^p for and all Hilbert spaces . ${\displaystyle X}$ ${\displaystyle X^{*}}$ ${\displaystyle X}$ ${\displaystyle X^{**}}$ ${\displaystyle X}$ ${\displaystyle X}$ ${\displaystyle X^{**}}$ ${\displaystyle 1<p<\infty }$

Set theory: complement formation

A duality, which is usually not denoted with this word, is the formation of the complement of a set: If a basic set is given, the complement of a subset is the set of elements of which are not in . The complement of the complement is itself again . The formation of the complement relates the union and the intersection to one another: ( see de Morgan's rules ). ${\displaystyle G}$ ${\displaystyle M\subseteq G}$ ${\displaystyle M^{\complement }=G\setminus M}$ ${\displaystyle G}$ ${\displaystyle M}$ ${\displaystyle M}$ ${\displaystyle (M_{1}\cup M_{2})^{\complement }=M_{1}^{\complement }\cap M_{2}^{\complement }}$

A generalization of this example is the negation in any Boolean algebra .

According to the principle of duality for associations , a true statement can be obtained from any true statement about subsets of a basic set if one swaps the symbols (union) and (intersection) as well as the symbols (empty set) and (basic set). ${\displaystyle G}$ ${\displaystyle \cup }$ ${\displaystyle \cap }$ ${\displaystyle \emptyset }$ ${\displaystyle G}$

• See also: complement (set theory) , Boolean algebra

Lagrangian duality in optimization

In the mathematical optimization is Lagrangian duality used. You can solve any problem of optimization of the shape

${\displaystyle {\begin{aligned}{\text{Minimiere }}&f_{0}(x)&\\{\text{unter den Nebenbedingungen }}&f_{i}(x)\leq 0&i=1,\dots ,p\\&h_{j}(x)=0&j=1\dots ,q\\&x\in X&\end{aligned}}}$.

a so-called dual problem

${\displaystyle {\begin{aligned}{\text{maximiere }}&g(\lambda ,\mu )=\inf _{x}f_{0}(x)+\sum _{i=1}^{p}\lambda _{i}f_{i}(x)+\sum _{j=1}^{q}\mu _{j}h_{j}(x)\\{\text{unter den Nebenbedingungen }}&\lambda \geq 0\end{aligned}}}$

assign. This has easier constraints than the primal problem and is a convex optimization problem , but the objective function is usually more difficult to calculate. The duality in linear optimization is a special case of the Lagrange duality. The Lagrange duality plays an important role for optimality criteria such as the Karush-Kuhn-Tucker conditions or algorithms such as interior point methods .

See also

The term duality is widely used in mathematics. The following list contains a selection of such concepts, some of which are very advanced.

• Duality in the representation set for Boolean algebras
• Dual form of differential forms
• Dual category
• Poincaré duality
• Pontryagin duality

Individual evidence

1. ↑ ^{a b} Atiyah, Michael: Duality in Mathematics and Physics, 2007. Lecture notes from the Institut de Matematica de la Universitat de Barcelona (IMUB), accessed January 18, 2017
2. ↑ ^{a b} Holger Stephan: Duality in elementary geometry. Lecture on Mathematics Day 2012, accessed on January 18, 2017
3. ↑ Duality (mathematics) in en.wikipedia.org, accessed on January 18, 2017