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Geiringer–Laman Theorem
The Geiringer–Laman theorem gives a Structural rigidity, combinatorial characterization of Structural rigidity#Definitions, generically rigid graphs in 2-dimensional Euclidean space, with respect to Geometric constraint system#Bar-joint systems, bar-joint frameworks. This theorem was first proved by Hilda Geiringer, Hilda Pollaczek-Geiringer in 1927, and later by Gerard Laman in 1970. An efficient algorithm called the Pebble game (rigidity), pebble game is used to identify this class of graphs. This theorem has been the inspiration for many Geiringer-Laman type results for Structural rigidity#Rigidity for other types of frameworks, other types of frameworks with generalized pebble games. Statement of the theorem This theorem relies on definitions of genericity that can be found on the Structural rigidity#Definitions, structural rigidity page. Let V(E) denote the vertex set of a set of edges E. Geiringer-Laman Theorem. A graph G=(V,E) is Structural rigidity#Definitions, ge ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Structural Rigidity
In discrete geometry and mechanics, structural rigidity is a combinatorial theory for predicting the flexibility of ensembles formed by rigid bodies connected by flexible linkages or hinges. Definitions Rigidity is the property of a structure that it does not bend or flex under an applied force. The opposite of rigidity is flexibility. In structural rigidity theory, structures are formed by collections of objects that are themselves rigid bodies, often assumed to take simple geometric forms such as straight rods (line segments), with pairs of objects connected by flexible hinges. A structure is rigid if it cannot flex; that is, if there is no continuous motion of the structure that preserves the shape of its rigid components and the pattern of their connections at the hinges. There are two essentially different kinds of rigidity. Finite or macroscopic rigidity means that the structure will not flex, fold, or bend by a positive amount. Infinitesimal rigidity means that the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kernel (linear Algebra)
In mathematics, the kernel of a linear map, also known as the null space or nullspace, is the part of the domain which is mapped to the zero vector of the co-domain; the kernel is always a linear subspace of the domain. That is, given a linear map between two vector spaces and , the kernel of is the vector space of all elements of such that , where denotes the zero vector in , or more symbolically: \ker(L) = \left\ = L^(\mathbf). Properties The kernel of is a linear subspace of the domain .Linear algebra, as discussed in this article, is a very well established mathematical discipline for which there are many sources. Almost all of the material in this article can be found in , , and Strang's lectures. In the linear map L : V \to W, two elements of have the same image in if and only if their difference lies in the kernel of , that is, L\left(\mathbf_1\right) = L\left(\mathbf_2\right) \quad \text \quad L\left(\mathbf_1-\mathbf_2\right) = \mathbf. From this, it follows ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laman Graph
In graph theory, the Laman graphs are a family of sparse graphs describing the minimally rigid systems of rods and joints in the plane. Formally, a Laman graph is a graph on n vertices such that, for all k\geq 2, every k-vertex subgraph has at most 2k-3 edges, and such that the whole graph has exactly 2n-3 edges. Laman graphs are named after Gerard Laman, of the University of Amsterdam, who in 1970 used them to characterize rigid planar structures. However, this characterization, the Geiringer–Laman theorem, had already been discovered in 1927 by Hilda Geiringer. Rigidity Laman graphs arise in rigidity theory (structural), rigidity theory: if one places the vertices of a Laman graph in the Euclidean plane, in general position, there will in general be no simultaneous continuous motion of all the points, other than Congruence (geometry), Euclidean congruences, that preserves the lengths of all the graph edges. A graph is rigid in this sense if and only if it has a Laman subgraph t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generic Rigid Equi Inf Rigid
Generic or generics may refer to: In business * Generic term, a common name used for a range or class of similar things not protected by trademark * Generic brand, a brand for a product that does not have an associated brand or trademark, other than the trading name of the business providing the product * Generic trademark, a trademark that sometimes or usually replaces a common term in colloquial usage * Generic drug, a drug identified by its chemical name rather than its brand name In computer programming * Generic function, a computer programming entity made up of all methods having the same name * Generic programming, a computer programming paradigm based on method/functions or classes defined irrespective of the concrete data types used upon instantiation ** Generics in Java In linguistics *A pronoun or other word used with a less specific meaning, such as: ** generic ''you'' ** generic ''he'' or generic ''she'' ** generic ''they'' * Generic mood, a grammatical mood used ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jacobian Matrix And Determinant
In vector calculus, the Jacobian matrix (, ) of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. If this matrix is square, that is, if the number of variables equals the number of components of function values, then its determinant is called the Jacobian determinant. Both the matrix and (if applicable) the determinant are often referred to simply as the Jacobian. They are named after Carl Gustav Jacob Jacobi. The Jacobian matrix is the natural generalization to vector valued functions of several variables of the derivative and the differential of a usual function. This generalization includes generalizations of the inverse function theorem and the implicit function theorem, where the non-nullity of the derivative is replaced by the non-nullity of the Jacobian determinant, and the multiplicative inverse of the derivative is replaced by the inverse of the Jacobian matrix. The Jacobian determinant is fundamentally use ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differentiability Classes
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives (''differentiability class)'' it has over its domain. A function of class C^k is a function of smoothness at least ; that is, a function of class C^k is a function that has a th derivative that is continuous in its domain. A function of class C^\infty or C^\infty-function (pronounced C-infinity function) is an infinitely differentiable function, that is, a function that has derivatives of all orders (this implies that all these derivatives are continuous). Generally, the term smooth function refers to a C^-function. However, it may also mean "sufficiently differentiable" for the problem under consideration. Differentiability classes Differentiability class is a classification of functions according to the properties of their derivatives. It is a measure of the highest order of derivative that exists and is continuous for a function. Consider an ope ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differentiable Manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is differentiable), then computations done in one chart are valid in any other differentiable chart. In formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a vector space. To induce a global differential structure on the local coordinate systems induced by the homeomorphism ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Implicit Function Theorem
In multivariable calculus, the implicit function theorem is a tool that allows relations to be converted to functions of several real variables. It does so by representing the relation as the graph of a function. There may not be a single function whose graph can represent the entire relation, but there may be such a function on a restriction of the domain of the relation. The implicit function theorem gives a sufficient condition to ensure that there is such a function. More precisely, given a system of equations (often abbreviated into ), the theorem states that, under a mild condition on the partial derivatives (with respect to each ) at a point, the variables are differentiable functions of the in some neighborhood of the point. As these functions generally cannot be expressed in closed form, they are ''implicitly'' defined by the equations, and this motivated the name of the theorem. In other words, under a mild condition on the partial derivatives, the set of zero ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rank–nullity Theorem
The rank–nullity theorem is a theorem in linear algebra, which asserts: * the number of columns of a matrix is the sum of the rank of and the nullity of ; and * the dimension of the domain of a linear transformation is the sum of the rank of (the dimension of the image of ) and the nullity of (the dimension of the kernel of ). p. 70, §2.1, Theorem 2.3 It follows that for linear transformations of vector spaces of equal finite dimension, either injectivity or surjectivity implies bijectivity. Stating the theorem Linear transformations Let T : V \to W be a linear transformation between two vector spaces where T's domain V is finite dimensional. Then \operatorname(T) ~+~ \operatorname(T) ~=~ \dim V, where \operatorname(T) is the rank of T (the dimension of its image) and \operatorname(T) is the nullity of T (the dimension of its kernel). In other words, \dim (\operatorname T) + \dim (\operatorname T) = \dim (\operatorname(T)). This theorem can be refined via th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minor (linear Algebra)
In linear algebra, a minor of a matrix (mathematics), matrix is the determinant of some smaller square matrix generated from by removing one or more of its rows and columns. Minors obtained by removing just one row and one column from square matrices (first minors) are required for calculating matrix cofactors, which are useful for computing both the determinant and Inverse matrix, inverse of square matrices. The requirement that the square matrix be smaller than the original matrix is often omitted in the definition. Definition and illustration First minors If is a square matrix, then the ''minor'' of the entry in the -th row and -th column (also called the ''minor'', or a ''first minor'') is the determinant of the submatrix formed by deleting the -th row and -th column. This number is often denoted . The ''cofactor'' is obtained by multiplying the minor by . To illustrate these definitions, consider the following matrix, \begin 1 & 4 & 7 \\ 3 & 0 & 5 \\ -1 & 9 & 11 \\ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces'' of any positive integer dimension ''n'', which are called Euclidean ''n''-spaces when one wants to specify their dimension. For ''n'' equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes. The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of '' proving'' all properties of the space as theorems, by starting from a few fundamental properties, called '' postulates'', which either were considered as evid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Rigidity
In discrete geometry, geometric rigidity is a theory for determining if a geometric constraint system (GCS) has finitely many d-dimensional solutions, or Geometric constraint system#Definitions, frameworks, in some metric space. A framework of a GCS is rigid in d-dimensions, for a given d if it is an Isolated point, isolated solution of the GCS, factoring out the set of trivial motions, or Isometry group, isometric group, of the metric space, e.g. translations and rotations in Euclidean space. In other words, a rigid framework (G,p) of a GCS has no nearby framework of the GCS that is reachable via a non-trivial Path (topology), continuous motion of (G,p) that preserves the constraints of the GCS. Structural rigidity is another theory of rigidity that concerns Structural rigidity#Definitions, generic frameworks, i.e., frameworks whose rigidity properties are representative of all frameworks with the same Geometric constraint system#Definitions, constraint graph. Results in geomet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
