Q2: Ashwin walks 10 m north, 12 m east, 3 m west and 5 m south and then stops to drink water. What is the magnitude of his displacement from his original point? Answer: We know that displacement is a vector quantity, hence the direction Ashwin walks will either be positive or negative along an axis. Vector NTI Advance 11.5.3 is free to download from our software library. The software is sometimes distributed under different names, such as 'Vector NTI'. The actual developer of the software is Invitrogen Corporation. This PC program process the following extensions: '.pq', '.rp' and '.mv'.
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A two-vector or bivector[1] is a tensor of type and it is the dual of a two-form, meaning that it is a linear functional which maps two-forms to the real numbers (or more generally, to scalars).
The tensor product of a pair of vectors is a two-vector. Then, any two-form can be expressed as a linear combination of tensor products of pairs of vectors, especially a linear combination of tensor products of pairs of basis vectors. If f is a two-vector, then[2]
where the f α β are the components of the two-vector. Notice that both indices of the components are contravariant. This is always the case for two-vectors, by definition. A bivector may operate on a one-form, yielding a vector:
- ,
although a problem might be which of the upper indices of the bivector to contract with. (This problem does not arise with mixed tensors because only one of such tensor's indices is upper.) However, if the bivector is symmetric then the choice of index to contract with is indifferent.
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An example of a bivector is the stress–energy tensor. Another one is the orthogonal complement[3] of the metric tensor.
Matrix notation[edit]
If one assumes that vectors may only be represented as column matrices and covectors as row matrices; then, since a square matrix operating on a column vector must yield a column vector, it follows that square matrices can only represent mixed tensors. However, there is nothing in the abstract algebraic definition of a matrix that says that such assumptions must be made. Then dropping that assumption matrices can be used to represent bivectors as well as two-forms. Example:
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or .
If f is symmetric, i.e., , then .
See also[edit]
- Bivector § Tensors and matrices (but note that the stress–energy tensor is symmetric, not skew-symmetric)
References[edit]
- ^Penrose, Roger (2004). The road to reality : a complete guide to the laws of the universe. New York: Random House, Inc. pp. 443–444. ISBN978-0-679-77631-4. Note: This book mentions “bivectors” (but not “two-vectors”) in the sense of tensors.
- ^Schutz, Bernard (1985). A first course in general relativity. Cambridge, UK: Cambridge University Press. p. 77. ISBN0-521-27703-5. Note: This book does not appear to mention “two-vectors” or “bivectors”, only tensors.
- ^Penrose, op. cit., §18.3
Retrieved from 'https://en.wikipedia.org/w/index.php?title=Two-vector&oldid=976045970'
A simplicial 3-complex.
In mathematics, a simplicial complex is a set composed of points, line segments, triangles, and their n-dimensional counterparts (see illustration). Simplicial complexes should not be confused with the more abstract notion of a simplicial set appearing in modern simplicial homotopy theory. The purely combinatorial counterpart to a simplicial complex is an abstract simplicial complex.
Definitions[edit]
A simplicial complex is a set of simplices that satisfies the following conditions:
- 1. Every face of a simplex from is also in .
- 2. The non-empty intersection of any two simplices is a face of both and .
See also the definition of an abstract simplicial complex, which loosely speaking is a simplicial complex without an associated geometry.
A simplicial k-complex is a simplicial complex where the largest dimension of any simplex in equals k. For instance, a simplicial 2-complex must contain at least one triangle, and must not contain any tetrahedra or higher-dimensional simplices.
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A pure or homogeneous simplicial k-complex is a simplicial complex where every simplex of dimension less than k is a face of some simplex of dimension exactly k. Informally, a pure 1-complex 'looks' like it's made of a bunch of lines, a 2-complex 'looks' like it's made of a bunch of triangles, etc. An example of a non-homogeneous complex is a triangle with a line segment attached to one of its vertices.
A facet is any simplex in a complex that is not a face of any larger simplex. (Note the difference from a 'face' of a simplex). A pure simplicial complex can be thought of as a complex where all facets have the same dimension.
Sometimes the term face is used to refer to a simplex of a complex, not to be confused with a face of a simplex.
For a simplicial complex embedded in a k-dimensional space, the k-faces are sometimes referred to as its cells. The term cell is sometimes used in a broader sense to denote a set homeomorphic to a simplex, leading to the definition of cell complex.
The underlying space, sometimes called the carrier of a simplicial complex is the union of its simplices.
Closure, star, and link[edit]
- Two simplices and their closure.
- A vertex and its star.
- A vertex and its link.
Let K be a simplicial complex and let S be a collection of simplices in K.
The closure of S (denoted Cl S) is the smallest simplicial subcomplex of K that containseach simplex in S. Cl S is obtained by repeatedly adding to S each face of every simplex in S.
The star of S (denoted St S) is the union of the stars of each simplex in S. For a single simplex s, the star of s is the set of simplices having s as a face. (Note that the star of S is generally not a simplicial complex itself).
The link of S (denoted Lk S) equals Cl St S − St Cl S.It is the closed star of S minus the stars of all faces of S.
Algebraic topology[edit]
In algebraic topology, simplicial complexes are often useful for concrete calculations. For the definition of homology groups of a simplicial complex, one can read the corresponding chain complex directly, provided that consistent orientations are made of all simplices. The requirements of homotopy theory lead to the use of more general spaces, the CW complexes. Infinite complexes are a technical tool basic in algebraic topology. See also the discussion at Polytope of simplicial complexes as subspaces of Euclidean space made up of subsets, each of which is a simplex. That somewhat more concrete concept is there attributed to Alexandrov. Any finite simplicial complex in the sense talked about here can be embedded as a polytope in that sense, in some large number of dimensions. In algebraic topology, a compacttopological space which is homeomorphic to the geometric realization of a finite simplicial complex is usually called a polyhedron (see Spanier 1966, Maunder 1996, Hilton & Wylie 1967).
Combinatorics[edit]
Combinatorialists often study the f-vector of a simplicial d-complex Δ, which is the integer sequence , where fi is the number of (i−1)-dimensional faces of Δ (by convention, f0 = 1 unless Δ is the empty complex). For instance, if Δ is the boundary of the octahedron, then its f-vector is (1, 6, 12, 8), and if Δ is the first simplicial complex pictured above, its f-vector is (1, 18, 23, 8, 1). A complete characterization of the possible f-vectors of simplicial complexes is given by the Kruskal–Katona theorem.
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By using the f-vector of a simplicial d-complex Δ as coefficients of a polynomial (written in decreasing order of exponents), we obtain the f-polynomial of Δ. In our two examples above, the f-polynomials would be and , respectively.
Combinatorists are often quite interested in the h-vector of a simplicial complex Δ, which is the sequence of coefficients of the polynomial that results from plugging x − 1 into the f-polynomial of Δ. Adobe zii 4 4 6 cc2019 universal patcher. Formally, if we write FΔ(x) to mean the f-polynomial of Δ, then the h-polynomial of Δ is
and the h-vector of Δ is
We calculate the h-vector of the octahedron boundary (our first example) as follows:
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So the h-vector of the boundary of the octahedron is (1, 3, 3, 1). It is not an accident this h-vector is symmetric. In fact, this happens whenever Δ is the boundary of a simplicial polytope (these are the Dehn–Sommerville equations). In general, however, the h-vector of a simplicial complex is not even necessarily positive. For instance, if we take Δ to be the 2-complex given by two triangles intersecting only at a common vertex, the resulting h-vector is (1, 3, −2).
A complete characterization of all simplicial polytope h-vectors is given by the celebrated g-theorem of Stanley, Billera, and Lee.
Simplicial complexes can be seen to have the same geometric structure as the contact graph of a sphere packing (a graph where vertices are the centers of spheres and edges exist if the corresponding packing elements touch each other) and as such can be used to determine the combinatorics of sphere packings, such as the number of touching pairs (1-simplices), touching triplets (2-simplices), and touching quadruples (3-simplices) in a sphere packing.
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See also[edit]
- Polygonal chain – 1 dimensional simplicial complex
References[edit]
- Spanier, Edwin H. (1966), Algebraic Topology, Springer, ISBN0-387-94426-5
- Maunder, Charles R.F. (1996), Algebraic Topology (Reprint of the 1980 ed.), Mineola, NY: Dover, ISBN0-486-69131-4, MR1402473
- Hilton, Peter J.; Wylie, Shaun (1967), Homology Theory, New York: Cambridge University Press, ISBN0-521-09422-4, MR0115161
External links[edit]
- Weisstein, Eric W.'Simplicial complex'. MathWorld.
- Norman J. Wildberger. 'Simplices and simplicial complexes'. A Youtube talk.
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