Dual Vector Spaces and Physical Singularities

We assume the reader has a strong background in elementary linear algebra. In this section we familiarize the reader with the algebraic notation used in quantum mechanics, remind the reader of some basic facts about complex vector spaces, and introduce some notions that might not have been covered in an elementary linear algebra course. The linear algebra notation used in quantum computing will likely be familiar to the student of physics, but may be alien to a student of mathematics or computer science. It is the Dirac notation, which was invented by Paul Dirac and which is used often in quantum mechanics. In mathematics and physics textbooks, vectors are often distinguished from scalars by writing an arrow over the identifying symbol: e.g a⃗. Sometimes boldface is used for this purpose: e.g. a. In the Dirac notation, the symbol identifying a vector is written inside a ‘ket’, and looks like |a⟩. We denote the dual vector for a (defined later) with a ‘bra’, written as ⟨a|. Then inner products will be written as ‘bra-kets’ (e.g. ⟨a|b⟩). We now carefully review the definitions of the main algebraic objects of interest, using the Dirac notation. The vector spaces we consider will be over the complex numbers, and are finite-dimensional, which significantly simplifies the mathematics we need. Such vector spaces are members of a class of vector spaces called Hilbert spaces. Nothing substantial is gained at this point by defining rigorously what a Hilbert space is, but virtually all the quantum computing literature refers to a finite-dimensional complex vector space by the name ‘Hilbert space’, and so we will follow this convention. We will use H to denote such a space. Since H is finite-dimensional, we can choose a basis and alternatively represent vectors (kets) in this basis as finite column vectors, and represent operators with finite matrices. As you see in Section 3, the Hilbert spaces of interest for quantum computing will typically have dimension 2n, for some positive integer n. This is because, as with classical information, we will construct larger state spaces by concatenating a string of smaller systems, usually of size two.

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Cohomology operations and duality

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s030500410004250x ◽

1968 ◽

Vol 64 (1) ◽

pp. 15-30 ◽

Cited By ~ 1

Author(s):

C. R. F. Maunder

Keyword(s):

Vector Spaces ◽

Kronecker Products ◽

Dual Vector ◽

Alexander Duality ◽

Cohomology Operation ◽

Yield Information ◽

Stable Cohomology ◽

Steenrod Square ◽

Cohomology Operations ◽

Image Position

Since Thom first introduced the notion of the ‘dual’ of a Steenrod square, in (12), it has become apparent that calculation with such duals in the cohomology of, say, a simplicial complex X will often yield information about the impossibility of embedding X in Sn, for certain values of n. For example, the celebrated theorem that cannot be embedded in can easily be proved in this way. In this paper, we seek to generalize this method to any pair of extraordinary cohomology theories h* and k*, and natural stable cohomology operation θ: h* → k*. We show in section 3 that a simplicial embeddingf: X → Sn gives rise via the Alexander duality isomorphism to a homology operationwhich is independent of n, the particular embedding f, and even the particular triangulations of X and Sn. If h* and k* are multiplicative cohomology theories, there are Kronecker productsif h0(S0) = k0(S0) = G, a field, and the Kronecker products make h*, h* and k*, k* into dual vector spaces over G, then can be turned into a cohomology operation c(θ): k*(X)→h*(X), by using this duality. This is certainly true if h* = k* = H*(;Zp), p prime, and in this case we have the original situation considered by Thom, who showed, for example, that

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Duality of the spaces of linear functionals on dual vector spaces

Bulletin de la Société mathématique de France ◽

10.24033/bsmf.1432 ◽

1952 ◽

Vol 79 ◽

pp. 233-235

Author(s):

H.S. Allen

Keyword(s):

Vector Spaces ◽

Linear Functionals ◽

Dual Vector

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Dual Vector Spaces and Physical Singularities

Unified Field Mechanics ◽

10.1142/9789814719063_0008 ◽

2015 ◽

Cited By ~ 3

Author(s):

PETER ROWLANDS

Keyword(s):

Vector Spaces ◽

Dual Vector

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On the Relationship between Primal/Dual Cell Complexes of the Cell Method and Primal/Dual Vector Spaces: an Application to the Cantilever Elastic Beam with Elastic Inclusion

Curved and Layered Structures ◽

10.1515/cls-2019-0007 ◽

2019 ◽

Vol 6 (1) ◽

pp. 77-89 ◽

Cited By ~ 1

Author(s):

Elena Ferretti

Keyword(s):

Vector Space ◽

Elastic Inclusion ◽

Elastic Beam ◽

Cell Complex ◽

Vector Spaces ◽

Dual Algebra ◽

Cell Method ◽

Dual Vector ◽

Global Variables ◽

Primal Dual

AbstractThe Cell Method (CM) is an algebraic numerical method based on the use of global variables: the configuration, source and energetic global variables. The configuration variables with their topological equations, on the one hand, and the source variables with their topological equations, on the other hand, define two vector spaces that are a bialgebra and its dual algebra. The operators of these topological equations are generated by the outer product of the geometric algebra, for the primal vector space, and by the dual product of the dual algebra, for the dual vector space. The topological equations in the primal cell complex are coboundary processes on even exterior discrete p−forms, whereas the topological equations in the dual cell complex are coboundary processes on odd exterior discrete p−forms. Being expressed by coboundary processes in two different vector spaces, compatibility and equilibrium can be enforced at the same time, with compatibility enforced on the primal cell complex and equilibrium enforced on the dual cell complex. By way of example, in the present paper compatibility and equilibrium are enforced on a cantilever elastic beam with elastic inclusion. In effect, the CM shows its maximum potentialities right in domains made of several materials, as, being an algebraic approach, can treat any kind of discontinuities of the domain easily.

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