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Consider the complete bipartite graph $K_{3,3}$ in plane such that all its vertices lie on a circle. Is this framework locally rigid in plane (which I believe is the case) and if so, how to prove this?
I know that the above framework is not infinitesimally rigid. Moreover, it is known (due to Bol...
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The classical theorem of Asimow and Roth says that for a generic framework (i.e., coordinates of the nodes are algebraically independent), local rigidity and infinitesimal rigidity are equivalent. I was wondering how this theorem fails if we replace "generic'' by "(affine) general position". So m...
Rigidity theory may refer to
Study of the concept of rigidity (mathematics)
Mathematical theory of structural rigidity
Rigidity theory (physics), or topological constraints theory, describes or predicts the mechanical properties of glass
In mathematics, a rigid collection C of mathematical objects (for instance sets or functions) is one in which every c ∈ C is uniquely determined by less information about c than one would expect.
The above statement does not define a mathematical property; instead, it describes in what sense the adjective "rigid" is typically used in mathematics, by mathematicians.
== Examples ==
Some examples include:
Harmonic functions on the unit disk are rigid in the sense that they are uniquely determined by their boundary values.
Holomorphic functions are determined by the set of all derivatives at a single...
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...
Rigidity theory, or topological constraint theory, is a tool for predicting properties of complex networks (such as glasses) based on their composition. It was introduced by James Charles Phillips in 1979 and 1981, and refined by Michael Thorpe in 1983. Inspired by the study of the stability of mechanical trusses as pioneered by James Clerk Maxwell, and by the seminal work on glass structure done by William Houlder Zachariasen, this theory reduces complex molecular networks to nodes (atoms, molecules, proteins, etc.) constrained by rods (chemical constraints), thus filtering out microscopic details...
In combinatorics, a branch of mathematics, a matroid is a structure that abstracts and generalizes the notion of linear independence in vector spaces. There are many equivalent ways to define a matroid axiomatically, the most significant being in terms of: independent sets; bases or circuits; rank functions; closure operators; and closed sets or flats. In the language of partially ordered sets, a finite simple matroid is equivalent to a geometric lattice.
Matroid theory borrows extensively from the terminology of both linear algebra and graph theory, largely because it is the abstraction of various...
In the mathematical theory of matroids, a graphic matroid (also called a cycle matroid or polygon matroid) is a matroid whose independent sets are the forests in a given finite undirected graph. The dual matroids of graphic matroids are called co-graphic matroids or bond matroids. A matroid that is both graphic and co-graphic is sometimes called a planar matroid (but this should not be confused with matroids of rank 3, which generalize planar point configurations); these are exactly the graphic matroids formed from planar graphs.
== Definition ==
A matroid may be defined as a family of finite sets...
In mathematics, a uniform matroid is a matroid in which the independent sets are exactly the sets containing at most r elements, for some fixed integer r. An alternative definition is that every permutation of the elements is a symmetry.
== Definition ==
The uniform matroid
U
n
r
{\displaystyle U{}_{n}^{r}}
is defined over a set of
n
{\displaystyle n}
elements. A subset of the elements is independent if...
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I am reading about Functional Completeness in Wikipedia. In the "Formal Definition:
"Since every Boolean function of at least one variable can be expressed in terms of binary Boolean functions, F is functionally complete if and only if every binary Boolean function can be expressed in terms of t...
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In logic, a functionally complete set of logical connectives or Boolean operators is one which can be used to express all possible truth tables by combining members of the set into a Boolean expression. A well-known complete set of connectives is { AND, NOT }. Each of the singleton sets { NAND } and { NOR } is functionally complete. However, the set { AND, OR } is incomplete, due to its inability to express NOT.
A gate or set of gates which is functionally complete can also be called a universal gate / gates.
A functionally complete set of gates may utilise or generate 'garbage bits' as part of...
