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위키데이터

• ID : Q1755512

말뭉치

1. Duality, in mathematics, principle whereby one true statement can be obtained from another by merely interchanging two words.
2. Linear programming (LP) duality is examined in the context of other dualities in mathematics.
3. The mathematical and economic properties of LP duality are discussed and its uses are considered.
4. The duality between homology and cohomology consists in the following.
5. For specific categories or theories the construction of this duality is realized, for example, in the following manner.
6. This theorem was strengthened by O. Veblen (1923) who formulated it for homology bases, while the use of cohomology groups imparted it a form expressive of the content of this duality.
7. Kolmogorov proved the above duality isomorphisms by way of his functional homology and cohomology groups (cf.
8. For example the Wikipedia website shows many forms of duality in Mathematics.
9. For duality we need to define two spaces of objects and an attribute (property) of those objects.
10. Any duality in mathematics can be expressed as a bijective function between two spaces of objects.
11. In the language of the first paragraph it is, since we establish a bijection between objects and, by definition of inverse, duality and inversion are linked together.
12. The duality principle is no longer valid if the formulas are understood in their constructive sense.
13. The duality principle is determined by the dual nature of the axioms of projective geometry and the theorems which follow from them.
14. This fascination has only increased with the passage of time right up to the current intense investigation of Langlands duality.
15. Such a liberation was in large measure the consequence of the discovery of the principle of duality and of the manifold extensions and applications which were made of it.
16. We can also look to an “internal epistemology” of duality, which tries to understand the gains mathematicians have found in exploiting dual situations.
17. 6One key problem to address when we confront duality is that there is no definitive agreement about what the term means.
18. This paper is devoted to the study of the relation between Osserman algebraic curvature tensors and algebraic curvature tensors which satisfy the duality principle.
19. We give a short overview of the duality principle in Osserman manifolds and extend this notion to null vectors.
20. Cat \to Cat is also known as abstract duality.
21. Instances here include linear duality, Stone duality, Pontryagin duality, and projective inversions with respect to a conic hypersurface.
22. D \to C between categories which can be termed a duality of sorts, in that concepts developed in C C are mapped to dual concepts in D D and vice-versa.
23. More general still is a concrete duality induced by a dualizing object.
24. Thus the duality of Projective Geometry: Two points determine a line; two lines determine a point.
25. Gergonne first introduced the word duality in mathematics in 1826.
26. An informal survey of some topologists has revealed the following names of duality in current use.
27. Fortunately, one has been provided by A. Dold and D. Puppe with their concept of strong duality.
28. From a category theory viewpoint, duality can also be seen as a functor, at least in the realm of vector spaces.
29. A simple, maybe the most simple, duality arises from considering subsets of a fixed set S .
30. A duality in geometry is provided by the dual cone construction.
31. This gives rise to the first example of a duality mentioned above.
32. The Princeton Companion of Mathematics tells us that "Duality is an important general theme which has manifestations in almost every area of mathematics...
33. To begin such an investigation, we propose the following questions: Is there a general theory of duality?
34. Is it the case that category theory is best adapted to make sense of duality?
35. The use of some dualities is motivated by the claim that one side of the duality is simpler.
36. This paper constructs an infinite-dimensional version of the Duality Theorem for a Linear Program (LP).
37. Under some mild nondegeneracy conditions involving strict positivity, the new Duality Theorem asserts that the optimal value of the primal LP equals the optimal value of the topological dual LP.
38. While instances of duality in mathematics have increased enormously through the twentieth century, philosophers since Nagel have paid little attention to the phenomenon.
39. Systems mapped onto themselves by a duality transformation are called self-dual and exhibit remarkable properties, as exemplified by the scale invariance of an Ising magnet at the critical point.
40. We show that these puzzling properties arise from a duality between pairs of configurations on either side of a mechanical critical point.
41. Duality gap of the conic convex constrained optimization problems in normed spaces.
42. Duality phenomena occur in nearly all mathematically formalized disciplines, such as algebra, geometry, logic and natural language semantics.
43. However, many of these disciplines use the term ‘duality’ in vastly different senses, and while some of these senses are intimately connected to each other, others seem to be entirely unrelated.
44. This article focuses exclusively on duality phenomena involving the interaction between an ‘external’ and an ‘internal’ negation of some kind, which arise primarily in logic and linguistics.
45. A well-known example from logic is the duality between conjunction and disjunction in classical propositional logic: is logically equivalent to , and hence is logically equivalent to .
46. To address the above problems, we improve the extended duality theory by adding fuzzy objective functions.
47. The most important aspect of duality is the existence of the duality gap, which is the difference between the optimal solution by solving the original problem and the lower bound of the dual problem.
48. However, for nonconvex problems, the duality gap is generally nonzero and may be large value for some problems.
49. To deal with nonconvex fuzzy optimization problems with continuous, discrete, and mixed variable, we improve the extended duality theory by adding fuzzy objective functions.

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위키데이터

• ID : Q1755512

Spacy 패턴 목록

• [{'LEMMA': 'duality'}]