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"disjoint" Definitions
  1. [obsolete] DISJOINTED
  2. having no elements in common
  3. to disturb the orderly structure or arrangement of
  4. to take apart at the joints
  5. to come apart at the joints

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866 Sentences With "disjoint"

How to use disjoint in a sentence? Find typical usage patterns (collocations)/phrases/context for "disjoint" and check conjugation/comparative form for "disjoint". Mastering all the usages of "disjoint" from sentence examples published by news publications.

So this disjoint has presented a challenge to my identity.
But there's no disjoint between being more righteous and more hateful.
"There is now a critical disjoint between Americans' political institutions and their loyalties," writes Hopkins.
In order for this to happen, several huge shake-ups had to permanently disjoint the status quo.
There always seems to be a certain awkward disjoint between that colossal money machine and everything else Google does.
Those numbers are basically a perfect match for my own experience … and they are way disjoint from the claims of both both TIOBE and PYPL.
This disjoint has created what's known as the memory bottleneck, in which faster processors matter less and less as it becomes impossible to serve them data at corresponding speeds.
Living in communities where everyone knows one another's business is natural, and arguably healthier than the disjoint dysfunction of, say, an apartment building whose dozens of inhabitants don't even know each other's names.
The disjoint between the Trump Show and the outbreak of comity in Congress is a reminder of one of the fundamental dynamics of Trump's Washington: Most of the political system keeps on operating even when the president of the United States is completely disengaged and focused on other things, as he was this week.
"Theres been a bit of a disjoint with the way the equity market has been trading in the last couple of sessions and the way the FX markets have been moving, and I think to some degree were seeing a bit of catchup on the FX side of things," said Bipan Rai, North American head of FX strategy at CIBC Capital Markets in Toronto.
Two disjoint sets. In mathematics, two sets are said to be disjoint sets if they have no element in common. Equivalently, two disjoint sets are sets whose intersection is the empty set.. For example, {1, 2, 3} and {4, 5, 6} are disjoint sets, while {1, 2, 3} and {3, 4, 5} are not disjoint. A collection of more than two sets is called disjoint if any two distinct sets of the collection are disjoint.
The red shapes are not interior-disjoint with the blue Triangle. The green and the yellow shapes are interior-disjoint with the blue Triangle, but only the yellow shape is entirely disjoint from the blue Triangle. Two shapes and are called interior- disjoint if the intersection of their interiors is empty. Interior-disjoint shapes may or may not intersect in their boundary.
Since disjoint cycles commute, the inverse of a product of disjoint cycles is the result of reversing each of the cycles separately.
A cluster graph, the disjoint union of complete graphs In graph theory, a branch of mathematics, the disjoint union of graphs is an operation that combines two or more graphs to form a larger graph. It is analogous to the disjoint union of sets, and is constructed by making the vertex set of the result be the disjoint union of the vertex sets of the given graphs, and by making the edge set of the result be the disjoint union of the edge sets of the given graphs. Any disjoint union of two or more nonempty graphs is necessarily disconnected.
NBMOs of non-disjoint (top) and disjoint (bottom) Non-Kekulé molecules Non-Kekulé molecules with two formal radical centers (non-Kekulé diradicals) can be classified into non-disjoint and disjoint by the shape of their two non-bonding molecular orbitals (NBMOs). Both NBMOs of molecules with non-disjoint characteristics such as trimethylenemethane have electron density at the same atom. According to Hund's rule, each orbital is filled with one electron with parallel spin, avoiding the Coulomb repulsion by filling one orbital with two electrons. Therefore, such molecules with non-disjoint NBMOs are expected to prefer a triplet ground state.
Additionally, while a collection of less than two sets is trivially disjoint, as there are no pairs to compare, the intersection of a collection of one set is equal to that set, which may be non-empty. For instance, the three sets have an empty intersection but are not disjoint. In fact, there are no two disjoint sets in this collection. Also the empty family of sets is pairwise disjoint.
This is because it cannot be written as a union of two disjoint open sets, but it can be written as a union of two (non-disjoint) closed sets.
Descriptively close sets contain elements that have matching descriptions. Such sets can be either disjoint or non-disjoint sets. Spatially near sets are also descriptively near sets. Figure 2.
MakeSet creates 8 singletons. After some operations of Union, some sets are grouped together. In computer science, a disjoint-set data structure, also called a union–find data structure or merge–find set, is a data structure that stores a collection of disjoint (non-overlapping) sets. Equivalently, it stores a partition of a set into disjoint subsets.
Disjoint-set data structures play a key role in Kruskal's algorithm for finding the minimum spanning tree of a graph. The importance of minimum spanning trees means that disjoint-set data structures underlie a wide variety of algorithms. In addition, disjoint- set data structures also have applications to symbolic computation, as well in compilers, especially for register allocation problems.
To perform a sequence of addition, union, or find operations on a disjoint-set forest with nodes requires total time , where is the extremely slow-growing inverse Ackermann function. Disjoint-set forests do not guarantee this performance on a per-operation basis. Individual union and find operations can take longer than a constant times time, but each operation causes the disjoint-set forest to adjust itself so that successive operations are faster. Disjoint-set forests are both asymptotically optimal and practically efficient.
Disjointness of two sets, or of a family of sets, may be expressed in terms of intersections of pairs of them. Two sets A and B are disjoint if and only if their intersection A\cap B is the empty set. It follows from this definition that every set is disjoint from the empty set, and that the empty set is the only set that is disjoint from itself.. If a collection contains at least two sets, the condition that the collection is disjoint implies that the intersection of the whole collection is empty. However, a collection of sets may have an empty intersection without being disjoint.
In general topology and related areas of mathematics, the disjoint union (also called the direct sum, free union, free sum, topological sum, or coproduct) of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology. Roughly speaking, two or more spaces may be considered together, each looking as it would alone. The name coproduct originates from the fact that the disjoint union is the categorical dual of the product space construction.
In the undirected edge- disjoint paths problem, we are given an undirected graph and two vertices and , and we have to find the maximum number of edge-disjoint s-t paths in . The Menger's theorem states that the maximum number of edge-disjoint s-t paths in an undirected graph is equal to the minimum number of edges in an s-t cut-set.
The disjoint union is also called the graph sum, and may be represented either by a plus sign or a circled plus sign: If G and H are two graphs, then G+H or G\oplus H denotes their disjoint union.
A linear-time algorithm that area-bisects two disjoint convex polygons is described by .
In mathematics, an edge cycle cover (sometimes called simply cycle coverCun- Quan Zhang, Integer flows and cycle covers of graphs, Marcel Dekker,1997.) of a graph is a family of cycles which are subgraphs of G and contain all edges of G. If the cycles of the cover have no vertices in common, the cover is called vertex-disjoint or sometimes simply disjoint cycle cover. In this case the set of the cycles constitutes a spanning subgraph of G. If the cycles of the cover have no edges in common, the cover is called edge-disjoint or simply disjoint cycle cover.
The version of Suurballe's algorithm as described above finds paths that have disjoint edges, but that may share vertices. It is possible to use the same algorithm to find vertex-disjoint paths, by replacing each vertex by a pair of adjacent vertices, one with all of the incoming adjacencies of the original vertex, and one with all of the outgoing adjacencies . Two edge-disjoint paths in this modified graph necessarily correspond to two vertex-disjoint paths in the original graph, and vice versa, so applying Suurballe's algorithm to the modified graph results in the construction of two vertex-disjoint paths in the original graph. Suurballe's original 1974 algorithm was for the vertex-disjoint version of the problem, and was extended in 1984 by Suurballe and Tarjan to the edge-disjoint version.. By using a modified version of Dijkstra's algorithm that simultaneously computes the distances to each vertex in the graphs , it is also possible to find the total lengths of the shortest pairs of paths from a given source vertex to every other vertex in the graph, in an amount of time that is proportional to a single instance of Dijkstra's algorithm.
Additionally, a disjoint union of two graphs that have covers will also have a cover, formed as the disjoint union of the covering graphs. If the two covers have the same ply as each other, then this will also be the ply of their union.
Scrape out the gungy meat from the head, disjoint the claws and carefully remove the meat.
Two of the seven non- isomorphic solutions to this problem can be stated in terms of structures in the Fano 3-space, PG(3,2), known as packings. A spread of a projective space is a partition of its points into disjoint lines, and a packing is a partition of the lines into disjoint spreads. In PG(3,2), a spread would be a partition of the 15 points into 5 disjoint lines (with 3 points on each line), thus corresponding to the arrangement of schoolgirls on a particular day. A packing of PG(3,2) consists of seven disjoint spreads and so corresponds to a full week of arrangements.
We say G is a (\tau, \kappa) -disjoint collection if G is the union of at most \tau subcollections G_\alpha, where for each \alpha, G_\alpha is a disjoint collection of cardinality at most \kappa It was proven by Petr Simon that X is a Boolean space with the generating set G of CO(X) being (\tau, \kappa) -disjoint if and only if X is homeomorphic to a closed subspace of \alpha \kappa ^ \tau. The Ramsey-like property for polyadic spaces as stated by Murray Bell for Boolean spaces is then as follows: every uncountable clopen collection contains an uncountable subcollection which is either linked or disjoint.
See answers to the question ″Is the empty family of sets pairwise disjoint?″ A Helly family is a system of sets within which the only subfamilies with empty intersections are the ones that are pairwise disjoint. For instance, the closed intervals of the real numbers form a Helly family: if a family of closed intervals has an empty intersection and is minimal (i.e. no subfamily of the family has an empty intersection), it must be pairwise disjoint..
Clustering methods applied can be K-means clustering, forming disjoint groups or Hierarchical clustering, forming nested partitions.
The resulting binomial trees are edge-disjoint and therefore fulfill the requirements for the ESBT- broadcasting algorithm.
From a spatial point of view, nearness (a.k.a. proximity) is considered a generalization of set intersection. For disjoint sets, a form of nearness set intersection is defined in terms of a set of objects (extracted from disjoint sets) that have similar features within some tolerance (see, e.g., §3 in).
When comparing two different transition systems (S', Λ', →') and (S", Λ", →"), the basic notions of simulation and similarity can be used by forming the disjoint composition of the two machines, (S, Λ, →) with S = S' ∐ S", Λ = Λ' ∪ Λ" and → = →' ∪ →", where ∐ is the disjoint union operator between sets.
In topology, the exterior of a subset S of a topological space X is the union of all open sets of X which are disjoint from S. It is itself an open set and is disjoint from S. The exterior of S is denoted by :ext S or :Se.
The problems of finding a vertex disjoint and edge disjoint cycle covers with minimal number of cycles are NP- complete. The problems are not in complexity class APX. The variants for digraphs are not in APX either.Complexity and Approximation: Combinatorial Optimization Problems and Their Approximability Properties (1999) p.
It is assumed for convenience that the sets V1,...,Vm are pairwise-disjoint. In general the sets may intersect, but this case can be easily reduced to the case of disjoint sets: for every vertex x, form a copy of x for each i such that Vi contains x. In the resulting graph, connect all copies of x to each other. In the new graph, the Vi are disjoint, and each ISR corresponds to an ISR in the original graph.
Even number of intersections correspond to exterior points, and odd number of intersections correspond to interior points. The assumption of boundaries as manifold cell complexes forces any boundary representation to obey disjointedness of distinct primitives, i.e. there are no self-intersections that cause non- manifold points. In particular, the manifoldness condition implies all pairs of vertices are disjoint, pairs of edges are either disjoint or intersect at one vertex, and pairs of faces are disjoint or intersect at a common edge.
In topology, a topological space is said to be resolvable if it is expressible as the union of two disjoint dense subsets. For instance, the real numbers form a resolvable topological space because the rationals and irrationals are disjoint dense subsets. A topological space that is not resolvable is termed irresolvable.
Every Turán graph is a cograph; that is, it can be formed from individual vertices by a sequence of disjoint union and complement operations. Specifically, such a sequence can begin by forming each of the independent sets of the Turán graph as a disjoint union of isolated vertices. Then, the overall graph is the complement of the disjoint union of the complements of these independent sets. Chao and Novacky (1982) show that the Turán graphs are chromatically unique: no other graphs have the same chromatic polynomials.
Every contractible space is simply connected. ;Coproduct topology: If {Xi} is a collection of spaces and X is the (set-theoretic) disjoint union of {Xi}, then the coproduct topology (or disjoint union topology, topological sum of the Xi) on X is the finest topology for which all the injection maps are continuous. ;Cosmic space: A continuous image of some separable metric space. ;Countable chain condition: A space X satisfies the countable chain condition if every family of non-empty, pairswise disjoint open sets is countable.
In this network, the maximum flow is k iff there are k edge-disjoint paths. 2\. The paths must be independent, i.e., vertex-disjoint (except for s and t). We can construct a network N = (V, E) from G with vertex capacities, where the capacities of all vertices and all edges are 1.
It's just a matter of partitioning the cells in the shell differently, i.e. picking another set of disjoint great circles.
In set theory, an amorphous set is an infinite set which is not the disjoint union of two infinite subsets..
In matroid theory, an Eulerian matroid is a matroid whose elements can be partitioned into a collection of disjoint circuits.
So this situation is easier for Breaker than the general case, but harder than the case of disjoint winning-sets.
In mathematics, two sets are almost disjoint Kunen, K. (1980), "Set Theory; an introduction to independence proofs", North Holland, p. 47Jech, R. (2006) "Set Theory (the third millennium edition, revised and expanded)", Springer, p. 118 if their intersection is small in some sense; different definitions of "small" will result in different definitions of "almost disjoint".
Equitable resolvable coverings. Journal of Combinatorial Designs, 11(2), 113-123. The Oberwolfach problem, of decomposing a complete graph into edge-disjoint copies of a given 2-regular graph, also generalizes Kirkman's schoolgirl problem. Kirkman's problem is the special case of the Oberwolfach problem in which the 2-regular graph consists of five disjoint triangles.
A partition of a set X is any collection of mutually disjoint non-empty sets whose union is X., p. 28. Every partition can equivalently be described by an equivalence relation, a binary relation that describes whether two elements belong to the same set in the partition. Disjoint-set data structures. and partition refinement.
As the figure illustrates, the first frequency concept considers all matches of a graph in original network. This definition is similar to what we have introduced above. The second concept is defined as the maximum number of edge-disjoint instances of a given graph in original network. And finally, the frequency concept entails matches with disjoint edges and nodes.
Two events, A and B are said to be mutually exclusive or disjoint if the occurrence of one implies the non-occurrence of the other, i.e., their intersection is empty. This is a stronger condition than the probability of their intersection being zero. If A and B are disjoint events, then P(A∪B) = P(A) + P(B).
The graph with one node per 6-cycle, and one edge for each disjoint pair of 6-cycles, is the Coxeter graph..
Transforming back, we get . Because contains all disjoint transpositions in , . Hence, . Since , we have demonstrated that there is a third element in .
The disjoint union of a countable family of n-manifolds is a n-manifold (the pieces must all have the same dimension).
If the initial scheme is not irreducible, the normalization is defined to be the disjoint union of the normalizations of the irreducible components.
The cultural spheres are not mutually disjoint and can even overlap, representing the innate diversity and syncretism of human cultures and historical influences.
In any case, the disjoint vote was not sufficient to ensure Borsellino to win the election, and Cuffaro was confirmed president of Sicily.
A topological space X is a normal space if, given any disjoint closed sets E and F, there are neighbourhoods U of E and V of F that are also disjoint. More intuitively, this condition says that E and F can be separated by neighbourhoods. The closed sets E and F, here represented by closed disks on opposite sides of the picture, are separated by their respective neighbourhoods U and V, here represented by larger, but still disjoint, open disks. A T4 space is a T1 space X that is normal; this is equivalent to X being normal and Hausdorff.
Alternative linear time algorithms based on path planning are known. A similar definition can also be given for the relative convex hull of two disjoint simple polygons. This type of hull can be used in algorithms for testing whether the two polygons can be separated into disjoint halfplanes by a continuous linear motion, and in data structures for collision detection of moving polygons.
The Hoffman-Singleton graph has exactly 100 independent sets of size 15 each. Each independent set is disjoint from exactly 7 other independent sets. The 100-vertex graph that connects disjoint independent sets can be partitioned into two 50-vertex subgraphs, each of which is isomorphic to the Hoffman-Singleton graph, in an unusual case of self-replicating + multiplying behavior.
Most local government areas are a single contiguous area (possibly including islands). However, Aboriginal Shires are often defined as a number of disjoint areas each containing an Indigenous community. In the case of the Aboriginal Shire of Napranum, it consists of several disjoint parts of the locality of Mission River (remainder in Shire of Cook) with the town of Napranum as its seat.
Fig 2: Link disjoint backup tree. Several protection schemes have been proposed in the literature to protect the multicast connections. The simplest idea to protect the multicast tree from single fiber failure is to compute a link disjoint backup tree. In Fig 2 a multicast session from source node F to destination nodes A,B,C,D and E forms a light tree.
Figure F illustrates both path P1 and path P2. Figure G finds the shortest pair of disjoint paths by combining the edges of paths P1 and P2 and then discarding the common reversed edges between both paths (B–D). As a result, we get the two shortest pair of disjoint paths (A–B–E–F) and (A–C–D–F).
The Lin–Kernighan heuristic is a special case of the V-opt or variable-opt technique. It involves the following steps: # Given a tour, delete k mutually disjoint edges. # Reassemble the remaining fragments into a tour, leaving no disjoint subtours (that is, don't connect a fragment's endpoints together). This in effect simplifies the TSP under consideration into a much simpler problem.
The unsigned Stirling number of the first kind, s(k, j) counts the number of permutations of k elements with exactly j disjoint cycles.
The A33 is a major road in England, situated in the counties of Berkshire and Hampshire. The road currently runs in three disjoint sections.
Let M be a matroid with an underlying set of elements E, and let N be another matroid on an underlying set F. The direct sum of matroids M and N is the matroid whose underlying set is the disjoint union of E and F, and whose independent sets are the disjoint unions of an independent set of M with an independent set of N. The union of M and N is the matroid whose underlying set is the union (not the disjoint union) of E and F, and whose independent sets are those subsets that are the union of an independent set in M and one in N. Usually the term "union" is applied when E = F, but that assumption is not essential. If E and F are disjoint, the union is the direct sum.
Every hyperconnected space is both connected and locally connected (though not necessarily path-connected or locally path-connected). Note that in the definition of hyper-connectedness, the closed sets don't have to be disjoint. This is in contrast to the definition of connectedness, in which the open sets are disjoint. For example, the space of real numbers with the standard topology is connected but not hyperconnected.
One of the interesting property of this notion lies in the fact that two isotypical representations are either quasi-equivalent or disjoint (in analogy with the fact that irreducible representations are either unitarily equivalent or disjoint). This can be understood through the correspondence between factor representations and minimal central projection (in a von Neumann algebra),.Dixmier Two minimal central projections are then either equal or orthogonal.
In topology and related branches of mathematics, a normal space is a topological space X that satisfies Axiom T4: every two disjoint closed sets of X have disjoint open neighborhoods. A normal Hausdorff space is also called a T4 space. These conditions are examples of separation axioms and their further strengthenings define completely normal Hausdorff spaces, or T5 spaces, and perfectly normal Hausdorff spaces, or T6 spaces.
Let X and Y be sets. Then apply the axiom of regularity to the set {X,Y} (which exists by the axiom of pairing). We see there must be an element of {X,Y} which is also disjoint from it. It must be either X or Y. By the definition of disjoint then, we must have either Y is not an element of X or vice versa.
Then the value of the maximum flow is equal to the maximum number of independent paths from s to t. 3\. In addition to the paths being edge-disjoint and/or vertex disjoint, the paths also have a length constraint: we count only paths whose length is exactly k, or at most k. Most variants of this problem are NP-complete, except for small values of k.
In descriptive set theory and mathematical logic, Lusin's separation theorem states that if A and B are disjoint analytic subsets of Polish space, then there is a Borel set C in the space such that A ⊆ C and B ∩ C = ∅.. It is named after Nikolai Luzin, who proved it in 1927.. The theorem can be generalized to show that for each sequence (An) of disjoint analytic sets there is a sequence (Bn) of disjoint Borel sets such that An ⊆ Bn for each n. An immediate consequence is Suslin's theorem, which states that if a set and its complement are both analytic, then the set is Borel.
The disjoint segment of MD 109 south of Poolesville was transferred to county maintenance in 1957. MD 109 was extended from Comus to Hyattstown in 1960.
The number of permutations of distinct objects is !. The number of -permutations with disjoint cycles is the signless Stirling number of the first kind, denoted by .
Adult slender madtoms are weak dispersers, with poor swimming abilities, a characteristic that may have contributed to the existence of two disjoint populations of the species.
In the language of category theory, a transversal of a collection of mutually disjoint sets is a section of the quotient map induced by the collection.
Another significant contribution was the analysis of the disjoint-set data structure; he was the first to prove the optimal runtime involving the inverse Ackermann function.
A less combinatorial example is the operad of little intervals: The space A(n) consists of all embeddings of n disjoint intervals into the unit interval.
In the mathematical field of knot theory, an unlink is a link that is equivalent (under ambient isotopy) to finitely many disjoint circles in the plane.
Porat and Rothschild (2008) present a deterministic O(nt)-time algorithm for constructing a d-disjoint matrix with n columns and \Theta(d^2\ln n) rows.
Coolkerry () is a civil parish in the barony of Clarmallagh in County Laois. It is separated into two disjoint areas by an arm of Aghaboe civil parish.
A topological space with a countable dense subset is called separable. A topological space is a Baire space if and only if the intersection of countably many dense open sets is always dense. A topological space is called resolvable if it is the union of two disjoint dense subsets. More generally, a topological space is called κ-resolvable for a cardinal κ if it contains κ pairwise disjoint dense sets.
For a topological space X the following conditions are equivalent: #X is connected, that is, it cannot be divided into two disjoint non-empty open sets. #X cannot be divided into two disjoint non-empty closed sets. #The only subsets of X which are both open and closed (clopen sets) are X and the empty set. #The only subsets of X with empty boundary are X and the empty set.
In contrast, the NBMOs of the molecules with disjoint characteristics such as tetramethyleneethane can be described without having electron density at the same atom. With such MOs, the destabilization factor by the Coulomb repulsion becomes much smaller than with non-disjoint type molecules, and therefore the relative stability of the singlet ground state to the triplet ground state will be nearly equal, or even reversed because of exchange interaction.
In another version, if both disjoint convex sets are open, then there is a hyperplane in between them, but not necessarily any gap. An axis which is orthogonal to a separating hyperplane is a separating axis, because the orthogonal projections of the convex bodies onto the axis are disjoint. The hyperplane separation theorem is due to Hermann Minkowski. The Hahn–Banach separation theorem generalizes the result to topological vector spaces.
Disjoint-set data structures support three operations: Making a new set containing new element, finding the representative of the set containing a given element, and merging two sets.
A circle packing for a five-vertex planar graph The circle packing theorem (also known as the Koebe–Andreev–Thurston theorem) describes the possible tangency relations between circles in the plane whose interiors are disjoint. A circle packing is a connected collection of circles (in general, on any Riemann surface) whose interiors are disjoint. The intersection graph of a circle packing is the graph having a vertex for each circle, and an edge for every pair of circles that are tangent. If the circle packing is on the plane, or, equivalently, on the sphere, then its intersection graph is called a coin graph; more generally, intersection graphs of interior-disjoint geometric objects are called tangency graphs or contact graphs.
For example, the sets {1, 2} and {3, 4} are disjoint, while the set of even numbers intersects the set of multiples of 3 at the multiples of 6.
If each Xi is homeomorphic to a fixed space A, then the disjoint union X is homeomorphic to the product space A × I where I has the discrete topology.
There are three possible types of bicycle: a theta graph has two vertices that are connected by three internally disjoint paths, a figure 8 graph consists of two cycles sharing a single vertex, and a handcuff graph is formed by two disjoint cycles connected by a path.Glossary of Signed and Gain Graphs and Allied Areas A graph is a pseudoforest if and only if it does not contain a bicycle as a subgraph.
If a countably infinite graph G has no odd-degree vertices, then it may be written as a union of disjoint (finite) simple cycles if and only if every finite subgraph of G can be extended (by adding more edges and vertices of G) to a finite Eulerian graph. In particular, every countably infinite graph with only one end and with no odd vertices can be written as a union of disjoint cycles .
In matroid theory, a field within mathematics, a gammoid is a certain kind of matroid, describing sets of vertices that can be reached by vertex-disjoint paths in a directed graph. The concept of a gammoid was introduced and shown to be a matroid by , based on considerations related to Menger's theorem characterizing the obstacles to the existence of systems of disjoint paths.. Gammoids were given their name by . and studied in more detail by ..
A regular edge-transitive graph G cannot admit perfect state transfer between a pair of adjacent vertices, unless it is a disjoint union of copies of the complete graph K_2. A strongly regular graph admits perfect state transfer if and only if it is the complement of the disjoint union of an even number of copies of K_2. The only cubic distance-regular graph that admits perfect state transfer is the cubical graph.
Create a vertex for each of the 30 designs, and for each row of every design (there are 70 such rows in total, each row being a 4-set of 8 and appearing in 6 designs). Connect each design to its 14 rows. Connect disjoint designs to each other (each design is disjoint with 8 others). Connect rows to each other if they have exactly one element in common (there are 4x4 = 16 such neighbors).
A topological space is said to be connected if it is not the union of two disjoint nonempty open sets. A set is open if it contains no point lying on its boundary; thus, in an informal, intuitive sense, the fact that a space can be partitioned into disjoint open sets suggests that the boundary between the two sets is not part of the space, and thus splits it into two separate pieces.
A topological space is said to be pseudonormal if given two disjoint closed sets in it, one of which is countable, there are disjoint open sets containing them. Every normal space is pseudonormal, but not vice versa. Counterexamples to some variations on these statements can be found in the lists above. Specifically, Sierpinski space is normal but not regular, while the space of functions from R to itself is Tychonoff but not normal.
However, at present there exists no theoretical underpinning to explain these empirical findings. Figure 2: A geomagic square using consecutively-sized pieces. Figure 3: A panmagic 3 × 3 geomagic square The pieces in a geomagic square may also be disjoint, or composed of separated islands, as seen in Figure 3. Since they can be placed so as to mutually overlap, disjoint pieces are often able to tile areas that connected pieces cannot.
Stated precisely, in any graph G its maximal pseudoforests consist of every tree component of G, together with one or more disjoint 1-trees covering the remaining vertices of G.
Bernard A. Galler and Michael J. Fischer. An improved equivalence algorithm. Communications of the ACM, Volume 7, Issue 5 (May 1964), pages 301–303. The paper originating disjoint-set forests.
Such data are referred to as datagrams (Datagram Sockets). UDP address space, the space of UDP port numbers (in ISO terminology, the TSAPs), is completely disjoint from that of TCP ports.
His recent work has included approximate counting and volume computation via random walks; finding edge disjoint paths in expander graphs, and exploring anti- Ramsey theory and the stability of routing algorithms.
According to the Erdős–Pósa theorem, the size of a minimum feedback vertex set is within a logarithmic factor of the maximum number of vertex-disjoint cycles in the given graph.
Two subsets A and B of a topological space X are said to be separated by neighbourhoods if there are neighbourhoods U of A and V of B that are disjoint. In particular A and B are necessarily disjoint. Two plain subsets A and B are said to be separated by a function if there exists a continuous function f from X into the unit interval [0,1] such that f(a) = 0 for all a in A and f(b) = 1 for all b in B. Any such function is called a Urysohn function for A and B. In particular A and B are necessarily disjoint. It follows that if two subsets A and B are separated by a function then so are their closures.
We say that A intersects (meets) B at an element x if x belongs to A and B. We say that A intersects (meets) B if A intersects B at some element. A intersects B if their intersection is inhabited. We say that A and B are disjoint if A does not intersect B. In plain language, they have no elements in common. A and B are disjoint if their intersection is empty, denoted A\cap B=\varnothing.
Alon and Erdős initiated the investigation of the generalization of the above question to the case where the forbidden configuration consists of k disjoint edges (k > 2). They proved that the number of edges of a geometric graph of n vertices, containing no 3 disjoint edges is O(n). The optimal bound of roughly 2.5n was determined by Černý . For larger values of k, the first linear upper bound, O(k^4n), was established by Pach and Töröcsik .
A trio is a set of 3 disjoint octads of the Golay code. The subgroup fixing a trio is the trio group 26:(PSL(2,7) x S3), order 64512, transitive and imprimitive.
Many deflated documents are divided into disjoint parts and are cleared from the markup. #Index is a database compiled by search engine indexing robots. Documents are searched in the index. #Search engine.
In mathematics, a decomposable measure is a measure that is a disjoint union of finite measures. This is a generalization of σ-finite measures, which are the same as those that are a disjoint union of countably many finite measures. There are several theorems in measure theory such as the Radon–Nikodym theorem that are not true for arbitrary measures but are true for σ-finite measures. Several such theorems remain true for the more general class of decomposable measures.
PG(3,2) arises as a background in some solutions of Kirkman's schoolgirl problem. Two of the seven non-isomorphic solutions to this problem can be embedded as structures in the Fano 3-space. In particular, a spread of PG(3,2) is a partition of points into disjoint lines, and corresponds to the arrangement of girls (points) into disjoint rows (lines of a spread) for a single day of Kirkman's schoolgirl problem. There are 56 different spreads of 5 lines each.
Also it follows that if two subsets A and B are separated by a function then A and B are separated by neighbourhoods. A normal space is a topological space in which any two disjoint closed sets can be separated by neighbourhoods. Urysohn's lemma states that a topological space is normal if and only if any two disjoint closed sets can be separated by a continuous function. The sets A and B need not be precisely separated by f, i.e.
There are four kinds. One is a balanced circle. Two other kinds are a pair of unbalanced circles together with a connecting simple path, such that the two circles are either disjoint (then the connecting path has one end in common with each circle and is otherwise disjoint from both) or share just a single common vertex (in this case the connecting path is that single vertex). The fourth kind of circuit is a theta graph in which every circle is unbalanced.
Adding two knots Two knots can be added by cutting both knots and joining the pairs of ends. The operation is called the knot sum, or sometimes the connected sum or composition of two knots. This can be formally defined as follows : consider a planar projection of each knot and suppose these projections are disjoint. Find a rectangle in the plane where one pair of opposite sides are arcs along each knot while the rest of the rectangle is disjoint from the knots.
The inverse Ackermann function grows extraordinarily slowly, so this factor is 4 or less for any n that can actually be written in the physical universe. This makes disjoint-set operations practically constant time.
Visualization of the lemma in \R^1. On the top: a collection of balls; the green balls are the disjoint subcollection. On the bottom: the subcollection with three times the radius covers all the balls.
Some systems also support subtyping of labeled disjoint union types (such as algebraic data types). The rule for width subtyping is reversed: every tag appearing in the width subtype must appear in the width supertype.
In inversive geometry, the inversive distance is a way of measuring the "distance" between two circles, regardless of whether the circles cross each other, are tangent to each other, or are disjoint from each other.
In particular: # If the common intersection of all sets is not empty ( \bigcap X_i eq \emptyset), then obviously they cannot be partitioned to collections with disjoint unions. Hence the union of connected sets with non-empty intersection is connected. # If the intersection of each pair of sets is not empty (\forall i,j: X_i \cap X_j eq \emptyset) then again they cannot be partitioned to collections with disjoint unions, so their union must be connected. # If the sets can be ordered as a "linked chain", i.e.
It provides operations for adding new sets, merging sets (replacing them by their union), and finding a representative member of a set. The last operation allows to find out efficiently if any two elements are in the same or different sets. While there are several ways of implementing disjoint-set data structures, in practice they are often identified with a particular implementation called a disjoint- set forest. This is a specialized type of forest which performs unions and finds in near constant amortized time.
This data structure is used by the Boost Graph Library to implement its Incremental Connected Components functionality. It is also a key component in implementing Kruskal's algorithm to find the minimum spanning tree of a graph. Note that the implementation as disjoint-set forests doesn't allow the deletion of edges, even without path compression or the rank heuristic. Sharir and Agarwal report connections between the worst-case behavior of disjoint-sets and the length of Davenport–Schinzel sequences, a combinatorial structure from computational geometry.
Fix some point p on the Riemann sphere. Each Jordan curve not passing through p divides the Riemann sphere into two pieces, and we call the piece containing p the "exterior" of the curve, and the other piece its "interior". Suppose there are 2g disjoint Jordan curves A1, B1,..., Ag, Bg in the Riemann sphere with disjoint interiors. If there are Möbius transformations Ti taking the outside of Ai onto the inside of Bi, then the group generated by these transformations is a Kleinian group.
Addition of species is defined by the disjoint union of sets, and corresponds to a choice between structures. For species F and G, define (F + G)[A] to be the disjoint union (also written "+") of F[A] and G[A]. It follows that (F + G)(x) = F(x) + G(x). As a demonstration, take E+ to be the species of non-empty sets, whose generating function is E+(x) = ex − 1, and 1 the species of the empty set, whose generating function is 1(x) = 1.
For instance, they may be restricted to being the closures of disjoint open sets. The Bolyai–Gerwien theorem states that any polygon may be dissected into any other polygon of the same area, using interior-disjoint polygonal pieces. It is not true, however, that any polyhedron has a dissection into any other polyhedron of the same volume using polyhedral pieces. This process is possible, however, for any two honeycombs (such as cube) in three dimension and any two zonohedra of equal volume (in any dimension).
A diplexer is a passive device that implements frequency-domain multiplexing. Two ports (e.g., L and H) are multiplexed onto a third port (e.g., S). The signals on ports L and H occupy disjoint frequency bands.
In topology, a branch of mathematics, an extension topology is a topology placed on the disjoint union of a topological space and another set. There are various types of extension topology, described in the sections below.
When A is a Behrend sequence, one may derive another Behrend sequence by omitting from A any finite number of elements. Every Behrend sequence may be decomposed into the disjoint union of infinitely many Behrend sequences.
The disjoint union space X, together with the canonical injections, can be characterized by the following universal property: If Y is a topological space, and fi : Xi → Y is a continuous map for each i ∈ I, then there exists precisely one continuous map f : X → Y such that the following set of diagrams commute: Characteristic property of disjoint unions This shows that the disjoint union is the coproduct in the category of topological spaces. It follows from the above universal property that a map f : X → Y is continuous iff fi = f o φi is continuous for all i in I. In addition to being continuous, the canonical injections φi : Xi → X are open and closed maps. It follows that the injections are topological embeddings so that each Xi may be canonically thought of as a subspace of X.
The northernmost portion of the highway was constructed in the late 1930s as MD 630, which became a disjoint part of MD 84 in 1951. The gap in MD 84 north of Uniontown was filled in 1956.
The space of such functions consists of two disjoint convex sets: the increasing ones and the decreasing ones, and has the homotopy type of two points. A non-holonomic solution to this relation would consist in the data of two functions, a differentiable function f(x), and a continuous function g(x), with g(x) nowhere vanishing. A holonomic solution gives rise to a non-holonomic solution by taking g(x) = f'(x). The space of non-holonomic solutions again consists of two disjoint convex sets, according as g(x) is positive or negative.
This is in contrast to a packing problem, in which the units must be disjoint and their union may be smaller than the target polygon, and to a polygon partition problem, in which the units must be disjoint and their union must be equal to the target polygon. A polygon covering problem is a special case of the set cover problem. In general, the problem of finding a smallest set covering is NP- complete, but for special classes of polygons, a smallest polygon covering can be found in polynomial time.
The ESBT-broadcast (Edge-disjoint Spanning Binomial Tree) algorithm is a pipelined broadcast algorithm with optimal runtime for clusters with hypercube network topology. The algorithm embeds d edge-disjoint binomial trees in the hypercube, such that each neighbor of processing element 0 is the root of a spanning binomial tree on 2^d - 1 nodes. To broadcast a message, the source node splits its message into k chunks of equal size and cyclically sends them to the roots of the binomial trees. Upon receiving a chunk, the binomial trees broadcast it.
In 1972, Ahuja entered the Indian Institute of Technology, Kanpur (IIT Kanpur), to study mechanical engineering. He earned a Bachelor of Science in mechanical engineering in 1977 and continued his education as a graduate student at his alma mater. Between 1977 and 1979, he studied maximal arc-disjoint and node-disjoint flow in multicommodity networks and earned a Master of Science in Industrial & Management Engineering. In 1982, he earned a Ph.D. in Industrial & Management Engineering (BB) for his work on the role of parametric programming in network flow problems.
Analytic subsets of Polish spaces are closed under countable unions and intersections, continuous images, and inverse images. The complement of an analytic set need not be analytic. Suslin proved that if the complement of an analytic set is analytic then the set is Borel. (Conversely any Borel set is analytic and Borel sets are closed under complements.) Luzin proved more generally that any two disjoint analytic sets are separated by a Borel set: in other words there is a Borel set containing one and disjoint from the other.
Cistus heterophyllus has a disjoint distribution. C. h. subsp. heterophyllus is native to western North Africa, along the coastal Mediterranean region from the Spanish island of Peñón de Alhucemas and Targuist in Morocco to Algiers. C. h. subsp.
The solution need not be unique. The interval scheduling problem is 1-dimensional – only the time dimension is relevant. The Maximum disjoint set problem is a generalization to 2 or more dimensions. This generalization, too, is NP-complete.
In fact, each has two families of generating lines, the members of each family are disjoint and each member one family intersects, with just one exception, every member of the other family. Each family is called a regulus.
Beginning in 2021, its official name will be Judbarra National Park.NRETAS Judburra/Gregory National Park , Northern Territory Government The park consists of two geographically disjoint sections. The larger section lies to the southwest of the smaller northeastern section.
Disjoint circles. Intersecting circles. Congruent circles. In inversive geometry, the circle of antisimilitude (also known as mid-circle) of two circles, α and β, is a reference circle for which α and β are inverses of each other.
The second and third semiorder axioms forbid partial orders of four items forming two disjoint chains: the second axiom forbids two chains of two items each, while the third item forbids a three-item chain with one unrelated item.
A Büchi automaton (Q,Σ,∆,Q0,F) is called semi-deterministic if Q is the union of two disjoint subsets N and D such that F ⊆ D and, for every d ∈ D, automaton (D,Σ,∆,{d},F) is deterministic.
A pair of connected distance-regular graphs are cospectral if and only if they have the same intersection array. A distance-regular graph is disconnected if and only if it is a disjoint union of cospectral distance-regular graphs.
Alspach's conjecture is a mathematical theorem that characterizes the disjoint cycle covers of complete graphs with prescribed cycle lengths. It is named after Brian Alspach, who posed it as a research problem in 1981. A proof was published by .
A disjoint union of paths is called a linear forest. Paths are fundamental concepts of graph theory, described in the introductory sections of most graph theory texts. See, for example, Bondy and Murty (1976), Gibbons (1985), or Diestel (2005).
Otherwise, they are incomparable. Thus, two trapezoids are disjoint exactly if their corresponding boxes are comparable. The box representation is useful because the associated dominance order allows sweep line algorithms to be used.Stefan Felsner, Rudolf Muller, and Lorenz Wernisch.
Saussure's shrew (Sorex saussurei) is a species of mammal in the family Soricidae. It is found in Mexico. There is also a disjoint population of shrews in Guatemala that is provisionally assigned to this species, but may represent a distinct species.
Another segment of MD 216 was relocated when I-95 was built in the early 1970s, resulting in a disjoint route. The route was unified when the highway was relocated west through its interchange with US 29 in the mid-2000s.
The lexicographic product is in general noncommutative: . However it satisfies a distributive law with respect to disjoint union: . In addition it satisfies an identity with respect to complementation: . In particular, the lexicographic product of two self-complementary graphs is self-complementary.
Two vertices of the Gosset graph that come from the same copy are adjacent if they correspond to disjoint edges of K8; two vertices that come from different copies are adjacent if they correspond to edges that share a single vertex..
In other words, \textstyle N(a,b] is a Poisson random variable with mean \textstyle \lambda(b-a), where \textstyle a\le b. Furthermore, the number of points in any two disjoint intervals, say, \textstyle (a_1,b_1] and \textstyle (a_2,b_2] are independent of each other, and this extends to any finite number of disjoint intervals. In the queueing theory context, one can consider a point existing (in an interval) as an event, but this is different to the word event in the probability theory sense. It follows that \textstyle \lambda is the expected number of arrivals that occur per unit of time.
The vertical space is therefore a vector subspace of TeE. A horizontal space HeE is then a choice of a subspace of TeE such that TeE is the direct sum of VeE and HeE. The disjoint union of the vertical spaces VeE for each e in E is the subbundle VE of TE: this is the vertical bundle of E. Likewise, a horizontal bundle is the disjoint union of the horizontal subspaces HeE. The use of the words "the" and "a" in this definition is crucial: the vertical subspace is unique, it is determined solely by the fibration.
There is a cake C, which is usually assumed to be either a finite 1-dimensional segment, a 2-dimensional polygon or a finite subset of the multidimensional Euclidean plane Rd. There are n people with equal rights to C.I.e. there is no dispute over the rights of the people – the only problem is how to divide the cake such that each person receives a fair share. C has to be divided to n disjoint subsets, such that each person receives a disjoint subset. The piece allocated to person i is called X_i, and C = X_1 \sqcup \cdots \sqcup X_n.
The Fenchel–Nielsen coordinates for a point of the Teichmüller space of S consist of 3g − 3 positive real numbers called the lengths and 3g − 3 real numbers called the twists. A point of Teichmüller space is represented by a hyperbolic metric on S. The lengths of the Fenchel–Nielsen coordinates are the lengths of geodesics homotopic to the 3g − 3 disjoint simple closed curves. The twists of the Fenchel–Nielsen coordinates are given as follows. There is one twist for each of the 3g − 3 curves crossing one of the 3g − 3 disjoint simple closed curves γ.
The Angenent torus can be used to prove the existence of certain other kinds of singularities of the mean curvature flow. For instance, if a dumbbell shaped surface, consisting of a thin cylindrical "neck" connecting two large volumes, can have its neck surrounded by a disjoint Angenent torus, then the two surfaces of revolution will remain disjoint under the mean curvature flow until one of them reaches a singularity; if the ends of the dumbbell are large enough, this implies that the neck must pinch off, separating the two spheres from each other, before the torus surrounding the neck collapses..
For example: if X is Hausdorff and Y is locally compact and Hausdorff and is a proper local homeomorphism, then p is a covering map. There are local homeomorphisms where Y is a Hausdorff space and X is not. Consider for instance the quotient space , where the equivalence relation ~ on the disjoint union of two copies of the reals identifies every negative real of the first copy with the corresponding negative real of the second copy. The two copies of 0 are not identified and they do not have any disjoint neighborhoods, so X is not Hausdorff.
His cousin and her husband Spink continue to deal with events in Gettys. Nevare's disjoint personalities are united to realize the initial goal of the magic, in spite of the serious consequences it has on Nevare's life. But balance is finally achieved.
This disjoint between transcription and spoken value explains the romanisation for Sanskrit names in Thailand that many foreigners find confusing. For example, สุวรรณภูมิ is romanised as Suvarnabhumi, but pronounced su-wan-na-phum. ศรีนครินทร์ is romanised as Srinagarindra but pronounced si-nakha-rin.
An orientation on A is a partition of E into the union of two disjoint subsets E+ and E− so that for every edge e exactly one of the edges from the pair e, belongs to E+ and the other belongs to E−.
Every point in the plane, except for the shared center, belongs to exactly one of the circles in the pencil. Every two disjoint circles, and every hyperbolic pencil of circles, may be transformed into a set of concentric circles by a Möbius transformation...
Figure 1. Descriptively, very near sets In mathematics, near sets are either spatially close or descriptively close. Spatially close sets have nonempty intersection. In other words, spatially close sets are not disjoint sets, since they always have at least one element in common.
A random measure \xi has got independent increments if and only if the random variables \xi(B_1), \xi(B_2), \dots, \xi(B_m) are stochastically independent for every selection of pairwise disjoint measurable sets B_1, B_2, \dots, B_m and every m \in \N .
Note that the existence of a hyperplane that only "separates" two convex sets in the weak sense of both inequalities being non-strict obviously does not imply that the two sets are disjoint. Both sets could have points located on the hyperplane.
A 3-dimensional matching is a special case of a set packing: we can interpret each element (x, y, z) of T as a subset {x, y, z} of X ∪ Y ∪ Z; then a 3-dimensional matching M consists of pairwise disjoint subsets.
A thick end of a graph G is an end that contains infinitely many pairwise-disjoint rays. Halin's grid theorem characterizes the graphs that contain thick ends: they are exactly the graphs that have a subdivision of the hexagonal tiling as a subgraph.; .
The points and do not have disjoint neighborhoods in X. Any covering space of a differentiable manifold may be equipped with a (natural) differentiable structure that turns p (the covering map in question) into a local diffeomorphism – a map with constant rank n.
In particular, all T1 spaces, i.e., all spaces in which for every pair of distinct points, each has a neighborhood not containing the other, are T0 spaces. This includes all T2 (or Hausdorff) spaces, i.e., all topological spaces in which distinct points have disjoint neighbourhoods.
A related, but weaker, notion is that of a preregular space. X is a preregular space if any two topologically distinguishable points can be separated by disjoint neighbourhoods. Preregular spaces are also called R_1 spaces. The relationship between these two conditions is as follows.
The simplest construction of a Coxeter graph is from a Fano plane. Take the 7C3 = 35 possible 3-combinations on 7 objects. Discard the 7 triplets that correspond to the lines of the Fano plane, leaving 28 triplets. Link two triplets if they are disjoint.
The weeper capuchin is found over much of Venezuela and over The Guianas, as well as part of northern Brazil. The Kaapori capuchin has a range that is disjoint from the other gracile capuchins, living in northern Brazil within the states of Pará and Maranhão.
A data-parallel model focuses on performing operations on a data set, typically a regularly structured array. A set of tasks will operate on this data, but independently on disjoint partitions. In Flynn's taxonomy, data parallelism is usually classified as MIMD/SPMD or SIMD.
For a rule F there isn't the second rule F to produce a disjoint sum F+F. In this approach the definition of summing is actually a definition by example. The advantage is the natural insertion of the cycle index as the power tool.
He proved that the maximum number of edges that a geometric graph with n > 2 vertices can have without containing two disjoint edges (that cannot even share an endpoint) is n. John Conway conjectured that this statement can be generalized to simple topological graphs. A topological graph is called "simple" if any pair of its edges share at most one point, which is either an endpoint or a common interior point at which the two edges properly cross. Conway's thrackle conjecture can now be reformulated as follows: A simple topological graph with n > 2 vertices and no two disjoint edges has at most n edges.
Given a finite group G, the generators of its Burnside ring Ω(G) are the formal differences of isomorphism classes of finite G-sets. For the ring structure, addition is given by disjoint union of G-sets and multiplication by their Cartesian product. The Burnside ring is a free Z-module, whose generators are the (isomorphism classes of) orbit types of G. If G acts on a finite set X, then one can write X = \bigcup_i X_i (disjoint union), where each Xi is a single G-orbit. Choosing any element xi in Xi creates an isomorphism G/Gi → Xi, where Gi is the stabilizer (isotropy) subgroup of G at xi.
Let G be a directed graph, S be a set of starting vertices, and T be a set of destination vertices (not necessarily disjoint from S). The gammoid \Gamma derived from this data has T as its set of elements. A subset I of T is independent in \Gamma if there exists a set of vertex-disjoint paths whose starting points all belong to S and whose ending points are exactly I.. A strict gammoid is a gammoid in which the set T of destination vertices consists of every vertex in G. Thus, a gammoid is a restriction of a strict gammoid, to a subset of its elements.
A geometric intersection graph is a graph in which the nodes are geometric shapes and there is an edge between two shapes iff they intersect. An independent set in a geometric intersection graph is just a set of disjoint (non-overlapping) shapes. The problem of finding maximum independent sets in geometric intersection graphs has been studied, for example, in the context of Automatic label placement: given a set of locations in a map, find a maximum set of disjoint rectangular labels near these locations. Finding a maximum independent set in intersection graphs is still NP-complete, but it is easier to approximate than the general maximum independent set problem.
Left: Left-turn solver trapped Right: Pledge algorithm solution Disjoint mazes can be solved with the wall follower method, so long as the entrance and exit to the maze are on the outer walls of the maze. If however, the solver starts inside the maze, it might be on a section disjoint from the exit, and wall followers will continually go around their ring. The Pledge algorithm (named after Jon Pledge of Exeter) can solve this problem.Seymour Papert, "Uses of Technology to Enhance Education", MIT Artificial Intelligence Memo No. 298, June 1973 The Pledge algorithm, designed to circumvent obstacles, requires an arbitrarily chosen direction to go toward, which will be preferential.
Additionally, consider for instance the unit circle S, and the action on S by a group G consisting of all rational rotations. Namely, these are rotations by angles which are rational multiples of π. Here G is countable while S is uncountable. Hence S breaks up into uncountably many orbits under G. Using the axiom of choice, we could pick a single point from each orbit, obtaining an uncountable subset X of S with the property that all of its translates by G are disjoint from X. The set of those translates partitions the circle into a countable collection of disjoint sets, which are all pairwise congruent.
Many authors require that neighbourhoods be open; be careful to note conventions.) ;Neighbourhood base/basis: See Local base. ;Neighbourhood system for a point x: A neighbourhood system at a point x in a space is the collection of all neighbourhoods of x. ;Net: A net in a space X is a map from a directed set A to X. A net from A to X is usually denoted (xα), where α is an index variable ranging over A. Every sequence is a net, taking A to be the directed set of natural numbers with the usual ordering. ;Normal: A space is normal if any two disjoint closed sets have disjoint neighbourhoods.
An end E of a graph G is defined to be a free end if there is a finite set X of vertices with the property that X separates E from all other ends of the graph. (That is, in terms of havens, βE(X) is disjoint from βD(X) for every other end D.) In a graph with finitely many ends, every end must be free. proves that, if G has infinitely many ends, then either there exists an end that is not free, or there exists an infinite family of rays that share a common starting vertex and are otherwise disjoint from each other.
This configuration, like Möbius, can also be represented as two tetrahedra, mutually inscribed and circumscribed: in the integer representation the tetrahedra can be 0347 and 1256. However, these two 8_4 configurations are non-isomorphic, since Möbius has four pairs of disjoint planes, while the latter one has no disjoint planes. For a similar reason (and because pairs of planes are degenerate quadratic surfaces), the Möbius configuration is on more quadratic surfaces of three-dimensional space than the latter configuration. The Levi graph of the Möbius configuration has 16 vertices, one for each point or plane of the configuration, with an edge for every incident point-plane pair.
SuperPascal enforces certain restrictions on the use of variables and communication to minimise or eliminate time-dependent errors. With variables, a simple rule is required: parallel processes can only update disjoint sets of variables. For example, in a `parallel` statement a target variable cannot be updated by more than a single process, but an expression variable (which can't be updated) may be used by multiple processes. In some circumstances, when a variable such as an array is the target of multiple parallel processes, and the programmer knows its element-wise usage is disjoint, then the disjointness restriction may be overridden with a preceding `[sic]` statement.
An irreducible component in a topological space is a maximal irreducible subset (i.e. an irreducible set that is not contained in any larger irreducible set). The irreducible components are always closed. Unlike the connected components of a space, the irreducible components need not be disjoint (i.e.
Its edge figure is a square. There are 4 tetrahedra and 4 triangles meeting at every edge. The 16-cell can be decomposed into two similar disjoint circular chains of eight tetrahedrons each, four edges long. Each chain, when stretched out straight, forms a Boerdijk–Coxeter helix.
In point-set topology, the composant of a point p in a continuum A is the union of all proper subcontinua of A that contain p. If a continuum is indecomposable, then its composants are pairwise disjoint. The composants of a continuum are dense in that continuum.
More generally, one can partition any configuration of Rule 90 into two subsets with disjoint nonzero cells, evolve the two subsets separately, and compute each successive configuration of the original automaton as the exclusive or of the configurations on the same time steps of the two subsets.
In convex geometry, two disjoint convex sets in n-dimensional Euclidean space are separated by a hyperplane, a result called the hyperplane separation theorem. In machine learning, hyperplanes are a key tool to create support vector machines for such tasks as computer vision and natural language processing.
It is on the east coast of the Cape York Peninsula excised from the Shire of Douglas and consists of a single locality, Wujal Wujal which is split into two disjoint areas separated by the Bloomfield River (the river itself remaining part of Shire of Douglas).
For example, the even and odd natural numbers form disjoint cofinal subsets of the set of all natural numbers. If a partially ordered set A admits a totally ordered cofinal subset, then we can find a subset B that is well-ordered and cofinal in A.
The Gewirtz graph can be constructed as follows. Consider the unique S(3, 6, 22) Steiner system, with 22 elements and 77 blocks. Choose a random element, and let the vertices be the 56 blocks not containing it. Two blocks are adjacent when they are disjoint.
A group-interval scheduling problem, i.e. GISMPk, can be described by a similar interval-intersection graph, with additional edges between each two intervals of the same group, i.e., this is the edge union of an interval graph and a graph consisting of n disjoint cliques of size k.
Every threshold graph is also a cograph. A threshold graph may be formed by repeatedly adding one vertex, either connected to all previous vertices or to none of them; each such operation is one of the disjoint union or join operations by which a cotree may be formed.
The following summarises the features and operations on polygons supported by GPC: GPC can compute the following clip operations: difference, intersection, exclusive-or and union. Polygons may comprise multiple disjoint contours. Contour vertices may be specified as clockwise or anticlockwise. Contours may be convex, concave or self- intersecting.
Characteristic functions are often assumed to be superadditive . This means that the value of a union of disjoint coalitions is no less than the sum of the coalitions' separate values: v ( S \cup T ) \geq v (S) + v (T) whenever S, T \subseteq N satisfy S \cap T = \emptyset .
It is generated by the following basis: for every n-tuple U1, ..., Un of open sets in X and for every compact set K, the set of all subsets of X that are disjoint from K and have nonempty intersections with each Ui is a member of the basis.
W is not primitive if it is periodic, where the population can perpetually cycle through different disjoint sets of compositions, or if it is reducible, where the dominant species (or quasispecies) that develops can depend on the initial population, as is the case in the simple example given below.
300px Bugac is a village in Bács-Kiskun county, in the Southern Great Plain region of southern Hungary. It covers an area of and has a population of 2889 people (2010). The surrounding area, Bugac puszta is the largest of seven disjoint units making up the Kiskunság National Park.
The following conditions are equivalent for a poset P: #P is a disjoint union of zigzag posets. #If a ≤ b ≤ c in P, either a = b or b = c. #< \circ < = \emptyset, i.e. it is never the case that a < b and b < c, so that < is vacuously transitive.
Another useful form of the formula is: :\chi(S')- r = N \cdot (\chi(S) - b) where r is the number points in S' at which the cover has nontrivial ramification (ramification points) and b is the number of points in S that are images of such points (branch points). Indeed, to obtain this formula, remove disjoint disc neighborhoods of the branch points from S and disjoint disc neighborhoods of the ramification points in S' so that the restriction of \pi is a covering. Then apply the general degree formula to the restriction, use the fact that the Euler characteristic of the disc equals 1, and use the additivity of the Euler characteristic under connected sums.
Informally, the tangent bundle of a manifold (in this case a circle) is obtained by considering all the tangent spaces (top), and joining them together in a smooth and non-overlapping manner (bottom). In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector. This is graphically illustrated in the accompanying picture for tangent bundle of circle S1, see Examples section: all tangents to a circle lie in the plane of the circle.
In a Hanner polytope, every two opposite facets are disjoint, and together include all of the vertices of the polytope, so that the convex hull of the two facets is the whole polytope.. As a simple consequence of this fact, all facets of a Hanner polytope have the same number of vertices as each other (half the number of vertices of the whole polytope). However, the facets may not all be isomorphic to each other. For instance, in the octahedral prism, two of the facets are octahedra, and the other eight facets are triangular prisms. Dually, in every Hanner polytope, every two opposite vertices touch disjoint sets of facets, and together touch all of the facets of the polytope.
In a letter to Wacław Sierpiński, motivated by some results of Giuseppe Vitali, Tibor Radó observed that for every covering of a unit interval, one can select a subcovering consisting of pairwise disjoint intervals with total length at least 1/2 and that this number cannot be improved. He then asked for an analogous statement in the plane. : If the area of the union of a finite set of squares in the plane with parallel sides is one, what is the guaranteed maximum total area of a pairwise disjoint subset? Radó proved that this number is at least 1/9 and conjectured that it is at least 1/4 a constant which cannot be further improved.
In fact, every vertex order of a cograph is a perfect order which further implies that max clique finding and min colouring can be found in linear time with any greedy colouring and without the need for a cotree decomposition. Every cograph is a distance-hereditary graph, meaning that every induced path in a cograph is a shortest path. The cographs may be characterized among the distance-hereditary graphs as having diameter two in each connected component. Every cograph is also a comparability graph of a series-parallel partial order, obtained by replacing the disjoint union and join operations by which the cograph was constructed by disjoint union and ordinal sum operations on partial orders.
In the mathematical discipline of graph theory, the Erdős–Pósa theorem, named after Paul Erdős and Lajos Pósa, states that there is a function such that for each positive integer , every graph either contains at least vertex-disjoint cycles or it has a feedback vertex set of at most vertices that intersects every cycle. Furthermore, in the sense of Big O notation. Because of this theorem, cycles are said to have the Erdős–Pósa property. The theorem claims that for any finite number there is an appropriate (least) value , with the property that in every graph without a set of vertex-disjoint cycles, all cycles can be covered by no more than vertices.
There also exist infinite sets of rays that are all disjoint from each other, for instance the sets of rays that use only two of the six directions that a path can follow within the tiling. Because it has infinitely many pairwise disjoint rays, all equivalent to each other, this graph has a thick end. Halin's theorem states that this example is universal: every graph with a thick end contains as a subgraph either this graph itself, or a graph formed from it by modifying it in simple ways, by subdividing some of its edges into finite paths. The subgraph of this form can be chosen so that its rays belong to the given thick end.
In linear algebra and matroid theory, Rota's basis conjecture is an unproven conjecture concerning rearrangements of bases, named after Gian-Carlo Rota. It states that, if X is either a vector space of dimension n or more generally a matroid of rank n, with n disjoint bases Bi, then it is possible to arrange the elements of these bases into an n × n matrix in such a way that the rows of the matrix are exactly the given bases and the columns of the matrix are also bases. That is, it should be possible to find a second set of n disjoint bases Ci, each of which consists of one element from each of the bases Bi.
The only finite homogeneous graphs are the cluster graphs mKn formed from the disjoint unions of isomorphic complete graphs, the Turán graphs formed as the complement graphs of mKn, the 3 × 3 rook's graph, and the 5-cycle. The only countably infinite homogeneous graphs are the disjoint unions of isomorphic complete graphs (with the size of each complete graph, the number of complete graphs, or both numbers countably infinite), their complement graphs, the Henson graphs together with their complement graphs, and the Rado graph. If a graph is 5-ultrahomogeneous, then it is ultrahomogeneous for every k. There are only two connected graphs that are 4-ultrahomogeneous but not 5-ultrahomogeneous: the Schläfli graph and its complement.
Two linked curves forming a Hopf link. When the circle is mapped to three- dimensional Euclidean space by an injective function (a continuous function that does not map two different points of the circle to the same point of space), its image is a closed curve. Two disjoint closed curves that both lie on the same plane are unlinked, and more generally a pair of disjoint closed curves is said to be unlinked when there is a continuous deformation of space that moves them both onto the same plane, without either curve passing through the other or through itself. If there is no such continuous motion, the two curves are said to be linked.
For example: If , then If , then # Create a residual graph formed from by removing the edges of on path that are directed into and then reverse the direction of the zero length edges along path (figure D). # Find the shortest path in the residual graph by running Dijkstra's algorithm (figure E). # Discard the reversed edges of from both paths. The remaining edges of and form a subgraph with two outgoing edges at , two incoming edges at , and one incoming and one outgoing edge at each remaining vertex. Therefore, this subgraph consists of two edge-disjoint paths from to and possibly some additional (zero-length) cycles. Return the two disjoint paths from the subgraph.
In graph theory, the Kneser graph (alternatively ) is the graph whose vertices correspond to the -element subsets of a set of elements, and where two vertices are adjacent if and only if the two corresponding sets are disjoint. Kneser graphs are named after Martin Kneser, who first investigated them in 1955.
Lu Jiaxi (; June 10, 1935 – October 31, 1983) was a self-taught Chinese mathematician who made important contributions in combinatorial design theory. He was a high school physics teacher in a remote city and worked in his spare time on the problem of large sets of disjoint Steiner triple systems.
There was no migration path for existing NonStop system software coded in TAL. The OS and database and Cobol compilers were entirely redesigned. Customers would see it as a totally disjoint product line requiring all-new software from them. The software side of this ambitious project took much longer than planned.
A partition of X is a set P of nonempty subsets of X, such that every element of X is an element of a single element of P. Each element of P is a cell of the partition. Moreover, the elements of P are pairwise disjoint and their union is X.
Main campus Lecture wing The main campus is located in Daedeok Science Town at the city of Daejeon. Most lecture, administrative, and faculty facilities are located in adjoined buildings. Research facilities are dispersed throughout several disjoint buildings. Other facilities on the campus include a cafeteria, a dormitory, and a playing field.
B. sessilis var. cordata occurs only in the Warren region of the Southwest Botanical Province of Western Australia. It occurs in two disjoint areas: along the west coast between Cape Naturaliste and Cape Leeuwin; and along the south coast between Point D'Entrecasteaux and Denmark. It grows in sand over limestone, amongst coastal heath.
The first piece of MD 563 was constructed as a gravel road from MD 224 south to Sandy Point Road in 1934. The next portion of Riverside Road was a disjoint segment from Liverpool Point Road south to Smith Point Road in 1935. The two sections of MD 563 were united in 1936.
But if a and b are in X such that p, a, and b are three distinct points, then {a} and {b} are disjoint closed sets and thus X is not ultraconnected. Note that if X is the Sierpiński space then no such a and b exist and X is in fact ultraconnected.
In computer science, specifically formal languages, convolution (sometimes referred to as zip) is a function which maps a tuple of sequences into a sequence of tuples. This name zip derives from the action of a zipper in that it interleaves two formerly disjoint sequences. The reverse function is unzip which performs a deconvolution.
In computability theory, two disjoint sets of natural numbers are called recursively inseparable if they cannot be "separated" with a recursive set.Monk 1976, p. 100 These sets arise in the study of computability theory itself, particularly in relation to Π classes. Recursively inseparable sets also arise in the study of Gödel's incompleteness theorem.
If u is empty, then x is ranked and we are done. Otherwise, apply the axiom of regularity to u to get an element w of u which is disjoint from u. Since w is in u, w is unranked. w is a subset of t by the definition of transitive closure.
Other structures will have F and G splitting the set in a different way. The set (F · G)[A], where A is the base set, is the disjoint union of all such structures. center The addition and multiplication of species are the most comprehensive expression of the sum and product rules of counting.
This branch cut separates the principal branch from the two branches and . In all branches with , there is a branch point at and a branch cut along the entire negative real axis. The functions are all injective and their ranges are disjoint. The range of the entire multivalued function is the complex plane.
The Cambridge Companion to Augustine refers to two volumes of essays about Augustine of Hippo and Augustinianism published in 2001 and 2014 by Cambridge University Press, with largely disjoint contents. The editors of the first version were Eleonore Stump and Norman Kretzmann, and for the second version Stump and David Vincent Meconi.
Kentucky Route 379 (KY 379) is a state highway in Kentucky with two disjoint sections. The southern segment runs from KY 1880 near Claywell to the Cumberland River in Cumberland County. The northern segment begins at KY 771 near Ribbon and extends north to (US 127) in Russell Springs in Russell County.
A graph with n nodes can contain at most n-1 bridges, since adding additional edges must create a cycle. The graphs with exactly n-1 bridges are exactly the trees, and the graphs in which every edge is a bridge are exactly the forests. In every undirected graph, there is an equivalence relation on the vertices according to which two vertices are related to each other whenever there are two edge-disjoint paths connecting them. (Every vertex is related to itself via two length-zero paths, which are identical but nevertheless edge-disjoint.) The equivalence classes of this relation are called 2-edge-connected components, and the bridges of the graph are exactly the edges whose endpoints belong to different components.
If two bounded connected planar shapes have disjoint convex hulls that are separated by a positive distance, then they necessarily have exactly four common lines of support, the bitangents of the two convex hulls. Two of these lines of support separate the two shapes, and are called critical support lines. Otherwise there may be more or fewer than four lines of support, even if the shapes themselves are disjoint. For instance, if one shape is an annulus that contains the other, then there are no common lines of support, while if each of two shapes consists of a pair of small disks at opposite corners of a square then there may be as many as 16 common lines of support.
The Oberwolfach problem on decompositions of complete graphs into copies of a given 2-regular graph is related, but neither is a special case of the other. If G is a 2-regular graph, with n vertices, formed from a disjoint union of cycles of certain lengths, then a solution to the Oberwolfach problem for G would also provide a decomposition of the complete graph into (n-1)/2 copies of each of the cycles of G. However, not every decomposition of K_n into this many cycles of each size can be grouped into disjoint cycles that form copies of G, and on the other hand not every instance of Alspach's conjecture involves sets of cycles that have (n-1)/2 copies of each cycle.
A domatic partitionIn graph theory, a domatic partition of a graph G = (V,E) is a partition of V into disjoint sets V_1, V_2,...,V_K such that each Vi is a dominating set for G. The figure on the right shows a domatic partition of a graph; here the dominating set V_1 consists of the yellow vertices, V_2 consists of the green vertices, and V_3 consists of the blue vertices. The domatic number is the maximum size of a domatic partition, that is, the maximum number of disjoint dominating sets. The graph in the figure has domatic number 3. It is easy to see that the domatic number is at least 3 because we have presented a domatic partition of size 3.
Suppose, to the contrary, that there is a function, f, on the natural numbers with f(n+1) an element of f(n) for each n. Define S = {f(n): n a natural number}, the range of f, which can be seen to be a set from the axiom schema of replacement. Applying the axiom of regularity to S, let B be an element of S which is disjoint from S. By the definition of S, B must be f(k) for some natural number k. However, we are given that f(k) contains f(k+1) which is also an element of S. So f(k+1) is in the intersection of f(k) and S. This contradicts the fact that they are disjoint sets.
A measure on X is a function that assigns a non-negative real number to subsets of X; this can be thought of as making precise a notion of "size" or "volume" for sets. We want the size of the union of disjoint sets to be the sum of their individual sizes, even for an infinite sequence of disjoint sets. One would like to assign a size to every subset of X, but in many natural settings, this is not possible. For example, the axiom of choice implies that, when the size under consideration is the ordinary notion of length for subsets of the real line, then there exist sets for which no size exists, for example, the Vitali sets.
There are different kinds of such windows like sliding windows that are similar to FIFO lists or tumbling windows that cut out disjoint parts. Furthermore, the windows can also be differentiated into element-based windows, e.g., to consider the last ten elements, or time-based windows, e.g., to consider the last ten seconds of data.
Fischer has done multifaceted work in theoretical computer science in general. Fischer's early work, including his PhD thesis, focused on parsing and formal grammars. Slides from PODC 2003. One of Fischer's most- cited works deals with string matching.. Already during his years at Michigan, Fischer studied disjoint-set data structures together with Bernard Galler.
The highway's northern end was moved to its present terminus just north of I-270 when Stringtown Road was constructed in the mid-2000s. In addition to the Boyds- Clarksburg route, MD 121 has also included three disjoint segments in Dawsonville and Germantown. All three of these routes were segments of the original MD 119.
Source basis for Haakon Jarl are considerable. He was given coverage in several sagas, including by Snorri Sturluson in Heimskringla, Ágrip af Nóregskonungasögum and more. According to Hallfreðar saga the poet Hallfreðr composed a drápa on the earl. Several disjoint stanzas by Hallfreðr in Skáldskaparmál are often thought to belong to this otherwise lost poem.
A tree is an undirected graph in which any two vertices are connected by exactly one path, or equivalently a connected acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by at most one path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees.
The index is 4. In mathematics, specifically group theory, a subgroup of a group may be used to decompose the underlying set of into disjoint equal-size pieces called cosets. There are two types of cosets: left cosets and right cosets. Cosets (of either type) have the same number of elements (cardinality) as does .
Then, where , must be in the form where are distinct elements of . The other four elements in are cycles of length 3. Note that the cosets generated by a subgroup of a group is a partition of the group. The cosets generated by a specific subgroup are either identical to each other or disjoint.
The connected sum of two n-manifolds is defined by removing an open ball from each manifold and taking the quotient of the disjoint union of the resulting manifolds with boundary, with the quotient taken with regards to a homeomorphism between the boundary spheres of the removed balls. This results in another n-manifold.
The permanent of a (0,1)-matrix is equal to the number of vertex-disjoint cycle covers of a directed graph with this adjacency matrix. This fact is used in a simplified proof showing that computing the permanent is #P-complete.Ben-Dor, Amir and Halevi, Shai. (1993). "Zero-one permanent is #P-complete, a simpler proof".
If y ≤ d, then g would not reach its maximum on [c,d] at z. Thus, y ∈ (d,b], and g(d) ≤ g(z) < g(y). This means that d ∈ S, which is a contradiction, thus establishing the lemma. The set E is open, so it is composed of a countable union of disjoint intervals (ak,bk).
The shapes employed in a setiset need not be connected regions. Disjoint pieces composed of two or more separated islands are also permitted. Such pieces are described as disconnected, or weakly-connected (when islands join only at a point), as seen in the setiset shown in Figure 3. The fewest pieces in a setiset is two.
The rank of a set X\subset T in a gammoid defined from a graph G and vertex subsets S and T is, by definition, the maximum number of vertex-disjoint paths from S to X. By Menger's theorem, it also equals the minimum cardinality of a set Y that intersects every path from S to X.
Part of the road in east. Tsat Tsz Mui Road Tsat Tsz Mui Road () is a road in Tsat Tsz Mui in Hong Kong. The road runs in the area of Tsat Tsz Mui and eastern North Point from west to east, parallel to King's Road, except disjoint by a residential-commercial complex of Island Place.
Consider graphs G and H. If both G and H admit perfect state transfer at time t, then their Cartesian product G \, \square \, H admits perfect state transfer at time t. If either G or H admits perfect state transfer at time t, then their disjoint union G \sqcup H admits perfect state transfer at time t.
Spiringen is a village and a municipality in the canton of Uri in Switzerland. The municipality comprises two disjoint areas, separated by the municipality of Unterschächen and the Klausen Pass. The western area includes the village of Spiringen in the Schächen Valley, whilst the eastern area includes the Urner Boden alp above Linthal and the canton of Glarus.
In particular, initializing a disjoint-set forest with nodes requires time. In practice, MakeSet must be preceded by an operation that allocates memory to hold . As long as memory allocation is an amortized constant-time operation, as it is for a good dynamic array implementation, it does not change the asymptotic performance of the random- set forest.
Although Teirlinck's proof did not follow the outline in the manuscript, it nevertheless made use of the combinatorial structures that Lu had constructed. Lu Jiaxi was awarded posthumously in 1987 the First Class Award of the State Natural Science Award, then the highest honor in science in China, for his work on large sets of disjoint Steiner triple systems.
In mathematics, a topological space X is said to be ultraconnected if no pair of nonempty closed sets of X is disjoint. Equivalently, a space is ultraconnected if and only if the closures of two distinct points always have non trivial intersection. Hence, no T_1 space with more than 1 point is ultraconnected.Steen and Seeback, Sect.
Kelsey is a rural municipality in the province of Manitoba in Western Canada. It consists of several disjoint parts. The largest part is Carrot Valley, located around and southwest of The Pas along the Carrot River, but the communities of Wanless () and Cranberry Portage (), located further north, are also part of the municipality. It is 867.64 km² large.
The rules may be varied by requiring lines of 4 marked points in a row rather than 5, with a reduced starting configuration. Also, the "disjoint" variation of the game does not allow two parallel lines to share an endpoint, whereas the standard "touching" version does allow this.Christian Boyer, "Morpion Solitaire – Rules of the Game", accessed August 8, 2010.
For a connected manifold, "open" is equivalent to "without boundary and non-compact", but for a disconnected manifold, open is stronger. For instance, the disjoint union of a circle and a line is non-compact since a line is non-compact, but this is not an open manifold since the circle (one of its components) is compact.
Given a directed graph G = (V, E), a path cover is a set of directed paths such that every vertex v ∈ V belongs to at least one path. Note that a path cover may include paths of length 0 (a single vertex)., Section 2.5. A path cover may also refer to a vertex-disjoint path cover, i.e.
Given two subgroups, and (which need not be distinct) of a group , the double cosets of and in are the sets of the form }. These are the left cosets of and right cosets of when and respectively. Two double cosets and are either disjoint or identical. The set of all double cosets for fixed and form a partition of .
Decomposition methods translate a problem into a new one that is easier to solve. The new problem only contains binary constraints; their scopes form a directed acyclic graph. The variables of the new problem represent each a set of variables of the original one. These sets are not necessarily disjoint, but they cover the set of the original variables.
In mathematics, a Ford circle is a circle with center at (p/q,1/(2q^2)) and radius 1/(2q^2), where p/q is an irreducible fraction, i.e. p and q are coprime integers. Each Ford circle is tangent to the horizontal axis y=0, and any two Ford circles are either tangent or disjoint from each other.
This would generate a daughter cell lacking a copy and a daughter cell with an extra copy. Completely inactive mitotic checkpoints may cause nondisjunction at multiple chromosomes, possibly all. Such a scenario could result in each daughter cell possessing a disjoint set of genetic material. Merotelic attachment occurs when one kinetochore is attached to both mitotic spindle poles.
A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points.
The mapping class groups satisfy the Tits alternative: that is, any subgroup of it either contains a non-abelian free subgroup or it is virtually solvable (in fact abelian). Any subgroup which is not reducible (that is it does not preserve a set of isotopy class of disjoint simple closed curves) must contain a pseudo-Anosov element.
41 ;Discrete topology: See discrete space. ;Disjoint union topology: See Coproduct topology. ;Dispersion point: If X is a connected space with more than one point, then a point x of X is a dispersion point if the subspace X − {x} is hereditarily disconnected (its only connected components are the one- point sets). ;Distance: See metric space.
Maryland Route 680 was the designation for two disjoint segments of Gillis Falls Road near Woodbine in southwestern Carroll County. The first segment was built west from MD 94 in 1939. The second segment was built between the first segment and MD 27 by 1946. Both portions of MD 680 were removed from the state highway system in 1954.
In geometry, space partitioning is the process of dividing a space (usually a Euclidean space) into two or more disjoint subsets (see also partition of a set). In other words, space partitioning divides a space into non-overlapping regions. Any point in the space can then be identified to lie in exactly one of the regions.
A graph is a 2-leaf power if and only if it is the disjoint union of cliques (i.e., a cluster graph). A graph is a 3-leaf power if and only if it is a (bull, dart, gem)-free chordal graph.; Based on this characterization and similar ones, 3-leaf powers can be recognized in linear time.
Let be the fundamental group corresponding to the spanning tree . For every vertex and edge , and can be identified with their images in . It is possible to define a graph with vertices and edges the disjoint union of all coset spaces and respectively. This graph is a tree, called the universal covering tree, on which acts.
Both disjoint segments of County 49 were decommissioned in 2006. County Road 50 begins at County 10 in Corcoran and runs through Greenfield to Rockford, where it then runs along the east bank of the Crow River through the Lake Rebecca Park Reserve to the Wright County line, where the road continues as Wright County Road 17 into Delano.
They can be constructed as induced subgraphs of the Rado graph. The Rado graph, the Henson graphs and their complements, disjoint unions of countably infinite cliques and their complements, and infinite disjoint unions of isomorphic finite cliques and their complements are the only possible countably infinite homogeneous graphs. The universality property of the Rado graph can be extended to edge-colored graphs; that is, graphs in which the edges have been assigned to different color classes, but without the usual edge coloring requirement that each color class form a matching. For any finite or countably infinite number of colors χ, there exists a unique countably-infinite χ-edge-colored graph Gχ such that every partial isomorphism of a χ-edge-colored finite graph can be extended to a full isomorphism.
One of Baumgartner's results is the consistency of the statement that any two \aleph_1-dense sets of reals are order isomorphic (a set of reals is \aleph_1-dense if it has exactly \aleph_1 points in every open interval). With András Hajnal he proved the Baumgartner–Hajnal theorem, which states that the partition relation \omega_1\to(\alpha)^2_n holds for \alpha<\omega_1 and n<\omega. He died in 2011 of a heart attack at his home in Hanover, New Hampshire. The mathematical context in which Baumgartner worked spans Suslin's problem, Ramsey theory, uncountable order types, disjoint refinements, almost disjoint families, cardinal arithmetics, filters, ideals, and partition relations, iterated forcing and Axiom A, proper forcing and the proper forcing axiom, chromatic number of graphs, a thin very-tall superatomic Boolean algebra, closed unbounded sets, and partition relations.
A topological generalization of Radon's theorem states that, if ƒ is any continuous function from a (d + 1)-dimensional simplex to d-dimensional space, then the simplex has two disjoint faces whose images under ƒ are not disjoint.; . Radon's theorem itself can be interpreted as the special case in which ƒ is the unique affine map that takes the vertices of the simplex to a given set of d + 2 points in d-dimensional space. More generally, if K is any (d + 1)-dimensional compact convex set, and ƒ is any continuous function from K to d-dimensional space, then there exists a linear function g such that some point where g achieves its maximum value and some other point where g achieves its minimum value are mapped by ƒ to the same point.
The point x, represented by a dot to the left of the picture, and the closed set F, represented by a closed disk to the right of the picture, are separated by their neighbourhoods U and V, represented by larger open disks. The dot x has plenty of room to wiggle around the open disk U, and the closed disk F has plenty of room to wiggle around the open disk V, yet U and V do not touch each other. A topological space X is a regular space if, given any closed set F and any point x that does not belong to F, there exists a neighbourhood U of x and a neighbourhood V of F that are disjoint. Concisely put, it must be possible to separate x and F with disjoint neighborhoods.
In some cases, a graph may be embedded in space in such a way that, for each cycle in the graph, one can find a disk bounded by that cycle that does not cross any other feature of the graph. In this case, the cycle must be unlinked from all the other cycles disjoint from it in the graph. The embedding is said to be flat if every cycle bounds a disk in this way.. A similar definition of a "good embedding" appears in ; see also and . A flat embedding is necessarily linkless, but there may exist linkless embeddings that are not flat: for instance, if G is a graph formed by two disjoint cycles, and it is embedded to form the Whitehead link, then the embedding is linkless but not flat.
There is a direct transformation for an instance of the set TSP to an instance of the standard asymmetric TSP. The idea is to first create disjoint sets and then assign a directed cycle to each set. The salesman, when visiting a vertex in some set, then walks around the cycle for free. To not use the cycle would ultimately be very costly.
In mathematics, the 2 theorem of Gromov and Thurston states a sufficient condition for Dehn filling on a cusped hyperbolic 3-manifold to result in a negatively curved 3-manifold. Let M be a cusped hyperbolic 3-manifold. Disjoint horoball neighborhoods of each cusp can be selected. The boundaries of these neighborhoods are quotients of horospheres and thus have Euclidean metrics.
In type theory, a union has a sum type; this corresponds to disjoint union in mathematics. Depending on the language and type, a union value may be used in some operations, such as assignment and comparison for equality, without knowing its specific type. Other operations may require that knowledge, either by some external information, or by the use of a tagged union.
Music supports an advertisement's structure and continuity by mediating between disjoint images. Accompanying a TV commercial, music either structures the narrative or tells a narrative itself. It can also create an antagonist and protagonist within this narrative by giving them typical musical figures, harmonies or melodies. Moreover, music can emphasize dramatic moments within the advertisement, and therefore creates both structure and continuity.
With isorecursive types, the recursive type \mu \alpha . T and its expansion (or unrolling) T[\mu \alpha.T / \alpha] (Where the notationX[Y/Z] indicates that all instances of Z are replaced with Y in X) are distinct (and disjoint) types with special term constructs, usually called roll and unroll, that form an isomorphism between them. To be precise: roll : T[\mu\alpha.
The underpinning fact from which the theorem is established is the expandability of a holomorphic function into its Taylor series. The connectedness assumption on the domain D is necessary. For example, if D consists of two disjoint open set, f can be 0 on one open set, and 1 on another, while g is 0 on one, and 2 on another.
One can consider the complement of each interval, written as (-\infty,a_n) \cup (b_n, \infty). By De Morgan's laws, the complement of the intersection is a union of two disjoint open sets. By the connectedness of the real line there must be something between them. This shows that the intersection of (even an uncountable number of) nested, closed, and bounded intervals is nonempty.
Theorem: A real Banach space is reflexive if and only if every pair of non- empty disjoint closed convex subsets, one of which is bounded, can be strictly separated by a hyperplane. James' theorem: A Banach space B is reflexive if and only if every continuous linear functional on B attains its supremum on the closed unit ball in B.
In turn, `TextPattern` requires the support of the `TextPatternRange` class to expose format and style attributes. `TextPatternRange` supports `TextPattern` by representing a contiguous text span in a text container with the `Start` and `End` endpoints. Multiple or disjoint text spans can be represented by more than one `TextPatternRange` objects. `TextPatternRange` supports functionality such as clone, selection, comparison, retrieval and traversal.
In 2012, he was awarded the Fulkerson Prize (jointly with Anders Johansson and Van H. Vu) for determining the threshold of edge density above which a random graph can be covered by disjoint copies of a given smaller graph. Also in 2012, he became a fellow of the American Mathematical Society.List of Fellows of the American Mathematical Society, retrieved 2013-01-27.
This assertion was proved for the case of equal squares independently by A. Sokolin, R. Rado, and V. A. Zalgaller. However, in 1973, Miklós Ajtai disproved Radó's conjecture, by constructing a system of squares of two different sizes for which any subsystem consisting of disjoint squares covers the area at most 1/4 − 1/1728 of the total area covered by the system.
In this section, another broadcasting algorithm with an underlying telephone communication model will be introduced. A Hypercube creates network system with p = 2^d (d = 0,1,2,3,...) . Every node is represented by binary {0,1} depending on the number of dimensions. Fundamentally ESBT(Edge-disjoint Spanning Binomial Trees) is based on hypercube graphs, pipelining( m messages are divided by k packets) and binomial trees.
A small category is connected if and only if its underlying graph is weakly connected, meaning that it is connected if one disregard the direction of the arrows. Each category J can be written as a disjoint union (or coproduct) of a collection of connected categories, which are called the connected components of J. Each connected component is a full subcategory of J.
It will be shown that , and hence a one-to-one correspondence between and the group exists. For , let be the subset of consisting of all subsets of cardinality exactly . Then is the disjoint union of the . The number of subsets of of cardinality is at most because every subset with elements is an element of the -fold cartesian product of .
Carcharodus orientalis, the Oriental skipper, or Oriental marbled skipper, is a butterfly of the family Hesperiidae. It is found in Montenegro, Albania, North Macedonia, Romania, Bulgaria and Greece, east to Asia Minor, northern Iran, Ukraine, the Caucasus to Kazachstan and Turkmenistan. There is a disjoint population in northern Hungary. In the south it is also found in Wadi Al Hisha (Jordan) and Israel.
The parallel paths in sub-graphs (circled in blue) belong to the same SRLG. Finding an SRG diverse path is the same as finding two disjoint subsets, such that each subset contains at least one common element. This is equivalent to the set-splitting problem, which has been proven NP-complete. Therefore, the SRG diverse routing problem is also NP-complete.
The odd graph On has one vertex for each of the (n − 1)-element subsets of a (2n − 1)-element set. Two vertices are connected by an edge if and only if the corresponding subsets are disjoint. That is, On is the Kneser graph KG(2n − 1,n − 1). O2 is a triangle, while O3 is the familiar Petersen graph.
A packing of PG(3,2) is a partition of the 35 lines into 7 disjoint spreads of 5 lines each, and corresponds to a solution for all seven days. There are 240 packings of PG(3,2), that fall into two conjugacy classes of 120 under the action of PGL(4,2) (the collineation group of the space); a correlation interchanges these two classes.
In topology, Urysohn's lemma is a lemma that states that a topological space is normal if and only if any two disjoint closed subsets can be separated by a continuous function. Section 15. Urysohn's lemma is commonly used to construct continuous functions with various properties on normal spaces. It is widely applicable since all metric spaces and all compact Hausdorff spaces are normal.
Then the children of each nodes are obtained by negating single leading zeroes. This results in a single binomial spanning tree. To obtain d edge-disjoint copies of the tree, translate and rotate the nodes: for the k-th copy of the tree, apply a XOR operation with 2^k to each node. Subsequently, right-rotate all nodes by k digits.
Mountain is a rural municipality in the province of Manitoba in Western Canada. It consists of two disjoint parts, Mountain (North) at , and Mountain (South) at , separated by the Municipality of Minitonas – Bowsman. The two parts are separated by approximately 10 km (6.2 mi) at the northeast corner of Minitonas – Bowsman, along its border with unorganized territory of Division No. 19.
Chris Rosin, "A new Morpion Solitaire record via Monte-Carlo search", accessed January 28, 2011. For the "disjoint" version of the game with lines consisting of 5 marked points, the record of 80 linesChristian Boyer, "Morpion Solitaire – Records Grids (5D game)", accessed August 8, 2010. has been obtained by computer search.T. Cazenave, "Nested Monte-Carlo Search", IJCAI 2009, pp. 456–461.
For, the product is given by the Cartesian product, whereas the coproduct is given by the disjoint union. This category does not have a zero object. Block matrix algebra relies upon biproducts in categories of matrices.H.D. Macedo, J.N. Oliveira, Typing linear algebra: A biproduct-oriented approach, Science of Computer Programming, Volume 78, Issue 11, 1 November 2013, Pages 2160-2191, , .
In mathematics, a superadditive set function is a set function whose value when applied to the union of two disjoint sets is greater than or equal to the sum of values of the function applied to each of the sets separately. This definition is analogous to the notion of superadditivity for real-valued functions. It is contrasted to subadditive set function.
There have been several changes in the existing segments of MD 850 since they were designated. MD 850D was once a single segment that spanned two counties. The section of Old Liberty Road between the now-disjoint segments of MD 850D was transferred from state to private maintenance in 2001. In 2004, MD 850I was shortened from to its current length.
The second axiom means that elements with distinct indices behave as disjoint variables, so that storing a value in one element does not affect the value of any other element. These axioms do not place any constraints on the set of valid index tuples I, therefore this abstract model can be used for triangular matrices and other oddly-shaped arrays.
Nearly all studied TPMS are free of self-intersections (i.e. embedded in ℝ3): from a mathematical standpoint they are the most interesting (since self-intersecting surfaces are trivially abundant). All connected TPMS have genus ≥ 3, and in every lattice there exist orientable embedded TPMS of every genus ≥3. Embedded TPMS are orientable and divide space into two disjoint sub-volumes (labyrinths).
A second disjoint section of the state highway was built from Creagerstown to Thurmont in the late 1930s. The two segments of MD 550 were united when the designation was assigned to the highway between Woodsboro and Creagerstown in the mid-1950s. MD 550 was extended northwest from Thurmont to Fort Ritchie, assuming all of MD 81, in the late 1970s.
The Mirsky–Newman theorem, a special case of the Herzog–Schönheim conjecture, states that there is no disjoint distinct covering system. This result was conjectured in 1950 by Paul Erdős and proved soon thereafter by Leon Mirsky and Donald J. Newman. However, Mirsky and Newman never published their proof. The same proof was also found independently by Harold Davenport and Richard Rado.
That is, G1 and G2 can be represented on the same set of n vertices with no edges in common. The Hajnal–Szemerédi theorem is the special case of this conjecture in which G2 is a disjoint union of cliques. provides a similar but stronger condition on Δ1 and Δ2 under which such a packing can be guaranteed to exist.
The tensor algebra of a Lie algebra has a Poisson algebra structure. A very explicit construction of this is given in the article on universal enveloping algebras. The construction proceeds by first building the tensor algebra of the underlying vector space of the Lie algebra. The tensor algebra is simply the disjoint union (direct sum ⊕) of all tensor products of this vector space.
On the surface COBOL ReSource appeals to VS users and software developers because it is faithful to the VS look and feel with 32 PFKeys, foreground suspension via Help, VS Field Attribute Characters, underlining, etc. Under the covers, however, are more significant compatibility features such as VS-style argument passing and return by reference between disjoint processes, and full PUTPARM/GETPARM functionality.
When , the graph must itself be a matching, with no two edges adjacent, and its edge chromatic number is one. That is, all graphs with are of class one. When , the graph must be a disjoint union of paths and cycles. If all cycles are even, they can be 2-edge-colored by alternating the two colors around each cycle.
An unordered tree is well-founded if the strict partial order is a well-founded relation. In particular, every finite tree is well-founded. Assuming the axiom of dependent choice a tree is well-founded if and only if it has no infinite branch. Well-founded trees can be defined recursively - by forming trees from a disjoint union of smaller trees.
A topological space X is said to be equidimensional if for all points p in X the dimension at p that is, dim p(X) is constant. The Euclidean space is an example of an equidimensional space. The disjoint union of two spaces X and Y (as topological spaces) of different dimension is an example of a non- equidimensional space.
Let Ci be the boundary circles of a finite collection of disjoint closed disks. The group generated by inversion in each circle has limit set a Cantor set, and the quotient H3/G is a mirror orbifold with underlying space a ball. It is double covered by a handlebody; the corresponding index 2 subgroup is a Kleinian group called a Schottky group.
A Seifert fiber space is a 3-manifold together with a decomposition as a disjoint union of circles. In other words, it is a S^1-bundle (circle bundle) over a 2-dimensional orbifold. Many 3-manifolds are Seifert fiber spaces, and they account for all compact oriented manifolds in 6 of the 8 Thurston geometries of the geometrization conjecture.
The matroid partitioning problem is to partition the elements of a matroid into as few independent sets as possible, and the matroid packing problem is to find as many disjoint spanning sets as possible. Both can be solved in polynomial time, and can be generalized to the problem of computing the rank or finding an independent set in a matroid sum.
If is a power of two, there is an optimal broadcasting algorithm based on edge disjoint spanning binomial trees (ESBT) in a hypercube. The hypercube, excluding the root , is split into ESBTs. The algorithm uses pipelining by splitting the broadcast data into blocks. Processor cyclically distributes blocks to the roots of the ESBTs and each ESBT performs a pipelined binary tree broadcast.
1\. Locally discrete collections are always locally finite. See the page on local finiteness. 2\. If a collection of subsets of a topological space X is locally discrete, it must satisfy the property that each point of the space belongs to at most one element of the collection. This means that only collections of pairwise disjoint sets can be locally discrete. 3\.
In mathematics, an extensive category is a category C with finite coproducts that are disjoint and well-behaved with respect to pullbacks. Equivalently, C is extensive if the coproduct functor from the product of the slice categories C/X × C/Y to the slice category C/(X + Y) is an equivalence of categories for all objects X and Y of C.
The concept of duality applies as well to infinite graphs embedded in the plane as it does to finite graphs. However, care is needed to avoid topological complications such as points of the plane that are neither part of an open region disjoint from the graph nor part of an edge or vertex of the graph. When all faces are bounded regions surrounded by a cycle of the graph, an infinite planar graph embedding can also be viewed as a tessellation of the plane, a covering of the plane by closed disks (the tiles of the tessellation) whose interiors (the faces of the embedding) are disjoint open disks. Planar duality gives rise to the notion of a dual tessellation, a tessellation formed by placing a vertex at the center of each tile and connecting the centers of adjacent tiles.
In 1962, Paul Erdős and Lajos Pósa proved that for every positive integer k there exists a positive integer k' such that for every graph G, either (i) G has k vertex-disjoint (long and/or even) cycles or (ii) there exists a subset X of less than k' vertices of G such that G \ X has no (long and/or even) cycles. This result, known today as the Erdős–Pósa theorem, cannot be extended to odd cycles. In fact, in 1987 Dejter and Víctor Neumann-LaraDejter I. J.; Neumann-Lara V. "Unboundedness for odd cyclic transversality", Coll. Math. Soc. J. Bolyai, 52 (1987), 195–203 showed that given an integer k > 0, there exists a graph G not possessing disjoint odd cycles such that the number of vertices of G whose removal destroys all odd cycles of G is higher than k.
In the same way, one may consider an arbitrary sequence of enqueue and dequeue operations of a queue data structure, and form a graph that has these operations as its vertices, placed in order on the spine of a single page, with an edge between each enqueue operation and the corresponding dequeue. Then, in this graph, each two edges will either cross or cover disjoint intervals on the spine. By analogy, researchers have defined a queue embedding of a graph to be an embedding in a topological book such that each vertex lies on the spine, each edge lies in a single page, and each two edges in the same page either cross or cover disjoint intervals on the spine. The minimum number of pages needed for a queue embedding of a graph is called its queue number...
2.5D datasets can be conveniently represented on a framework of boxels, which are axis-aligned non- intersecting boxes that can be used to directly represent objects in the scene or as bounding volumes. Leonidas J. Guibas and Yuan Yao's work showed that axis- aligned disjoint rectangles in the plane can be ordered into four total orders so that any ray meets them in one of the four orders. This work has been proven to also be applicable to boxels in this context, and it is shown that there exist four different partitionings of the boxels into ordered sequences of disjoint sets, called antichains, so that boxels in one antichain can act as occluders of the boxels in subsequent antichains. The expected runtime for the antichain partitioning is O(n log n), where n is the number of boxels.
MathSciNet lists 73 publications for Heinrich, dated from 1976 to 2012. Several of Heinrich's research publications concern orthogonal Latin squares, analogous concepts in graph theory, and applications of these concepts in parallel computing. As well, she has published works on finding spanning subgraphs with constraints on the degree of each vertex, and on Alspach's conjecture on disjoint cycle covers of complete graphs, among other topics.
When MD 100 was under construction between MD 104 and I-95 in the late 1990s, a standard intersection with MD 104 served as the eastern terminus of the two-lane, disjoint section of MD 100 between US 29 and MD 104. MD 104's interchange with MD 100 was completed in 1998 concurrent with the portion of the freeway from MD 104 to I-95.
The set of reals R = T ∪ {an} = T0 ∪ {tn} ∪ {an} where an is the sequence of real algebraic numbers. So both T and R are the union of three pairwise disjoint sets: T0 and two countable sets. A one-to-one correspondence between T and R is given by the function: f(t) = t if t ∈ T0, f(t2n-1) = tn, and f(t2n) = an.
The incidence between the 35 triplets and 15 Fano planes induces PG(3,2), with 15 points and 35 lines. To make the Hoffman-Singleton graph, create a graph vertex for each of the 15 Fano planes and 35 triplets. Connect each Fano plane to its 7 triplets, like a Levi graph, and also connect disjoint triplets to each other like the odd graph O(4).
Fig.3 Generalization Fig. Example generalization Generalization is a way to express a relationship between a general concept and a more specific concept. Also, if necessary, one can indicate whether the groups of concepts that are identified are overlapping or disjoint, complete or incomplete. Generalization is visualized by a solid arrow with an open arrowhead, pointing to the parent, as is illustrated in Figure 3.
A binary matroid is graphic if and only if its minors do not include the dual of the graphic matroid of K_5 nor of K_{3,3}., Theorem 10.5.1, p. 176. If every circuit of a binary matroid has odd cardinality, then its circuits must all be disjoint from each other; in this case, it may be represented as the graphic matroid of a cactus graph.
In the case where K is a simplex, the two simplex faces formed by the maximum and minimum points of g must then be two disjoint faces whose images have a nonempty intersection. This same general statement, when applied to a hypersphere instead of a simplex, gives the Borsuk–Ulam theorem, that ƒ must map two opposite points of the sphere to the same point.
For example, Cartesian products and disjoint unions of sets are dual to each other in the sense that and for any set . This is a particular case of a more general duality phenomenon, under which limits in a category correspond to colimits in the opposite category ; further concrete examples of this are epimorphisms vs. monomorphism, in particular factor modules (or groups etc.) vs. submodules, direct products vs.
If edges can only be added, then the dynamic connectivity problem can be solved by a Disjoint-set data structure. Each set represents a connected component; there is a path between x and y if and only if they belong to the same set. The amortized time per operation is \Theta(\alpha(n)), where n is the number of vertices and α is the inverse Ackermann function.
Shape distortion may be diminished by using an interrupted version. A sinusoidal interrupted Mollweide projection discards the central meridian in favor of alternating half-meridians which terminate at right angles to the equator. This has the effect of dividing the globe into lobes. In contrast, a parallel interrupted Mollweide projection uses multiple disjoint central meridians, giving the effect of multiple ellipses joined at the equator.
The Casimir effect is an interaction between disjoint neutral bodies provoked by the fluctuations of the electrodynamical vacuum. Mathematically it can be explained by considering the normal modes of electromagnetic fields, which explicitly depend on the boundary (or matching) conditions on the interacting bodies' surfaces. Since graphene/electromagnetic field interaction is strong for a one-atom-thick material, the Casimir effect is of interest.
The set S is called the orbit of the cycle. Every permutation on finitely many elements can be decomposed into cycles on disjoint orbits. The cyclic parts of a permutation are cycles, thus the second example is composed of a 3-cycle and a 1-cycle (or fixed point) and the third is composed of two 2-cycles, and denoted (1, 3) (2, 4).
The disjointness graph of a hypergraph H, denoted D(H), is the graph whose vertex set is the set of the hyperedges of H, with two vertices adjacent in D(H) when their corresponding hyperedges are disjoint in H. In other words, D(H) is the complement graph of L(H). A clique in D(H) corresponds to an independent set in L(H), and vice versa.
Packing dimension is constructed in a very similar way to Hausdorff dimension, except that one "packs" E from inside with pairwise disjoint balls of diameter at most δ. Just as before, one can consider functions h : [0, +∞) → [0, +∞] more general than h(δ) = δs and call h an exact dimension function for E if the h-packing measure of E is finite and strictly positive.
Taking history preserving maps as morphisms in the category of prefix orders leads to a notion of product that is not the Cartesian product of the two orders since the Cartesian product is not always a prefix order. Instead, it leads to an arbitrary interleaving of the original prefix orders. The union of two prefix orders is the disjoint union, as it is with partial orders.
In some games, it is possible to partition the elements of X (or a subset of them) into a set of pairwise-disjoint pairs. Under certain conditions, a player can win using the following greedy strategy: "whenever your opponent picks an element of pair i, pick the other element of pair i". The "certain conditions" are different for Maker and for Breaker; see pairing strategy.
In mathematics, a covering number is the number of spherical balls of a given size needed to completely cover a given space, with possible overlaps. Two related concepts are the packing number, the number of disjoint balls that fit in a space, and the metric entropy, the number of points that fit in a space when constrained to lie at some fixed minimum distance apart.
Above:A 3:1-coloring of the cycle on 5 vertices, and the corresponding 6:2-coloring. Below: A 5:2 coloring of the same graph. A b-fold coloring of a graph G is an assignment of sets of size b to vertices of a graph such that adjacent vertices receive disjoint sets. An a:b-coloring is a b-fold coloring out of a available colors.
Then S can be partitioned into \kappa many disjoint stationary sets. This result is due to Solovay. If \kappa is a successor cardinal, this result is due to Ulam and is easily shown by means of what is called an Ulam matrix. H. Friedman has shown that for every countable successor ordinal \beta, every stationary subset of \omega_1 contains a closed subset of order type \beta.
In mathematical morphology, hit-or-miss transform is an operation that detects a given configuration (or pattern) in a binary image, using the morphological erosion operator and a pair of disjoint structuring elements. The result of the hit-or-miss transform is the set of positions where the first structuring element fits in the foreground of the input image, and the second structuring element misses it completely.
Image fragments are grouped together based on similarity, but unlike standard k-means clustering and such cluster analysis methods, the image fragments are not necessarily disjoint. This block-matching algorithm is less computationally demanding and is useful later-on in the aggregation step. Fragments do however have the same size. A fragment is grouped if its dissimilarity with a reference fragment falls below a specified threshold.
Ryan can certainly be funny, but it is rarely without a sting." Some of these disjoint qualities in her work are illustrated by her poem "Outsider Art", which Harold Bloom selected for the anthology The Best of the Best American Poetry 1988–1997. Ryan is also known for her extensive use of internal rhyme. She refers to her specific methods of using internal rhyme as "recombinant rhyme.
If A were a subspace of (R, T) containing K, K would have no limit point in A so that A can not be limit point compact. Therefore, A cannot be compact 8\. The quotient space of (R, T) obtained by collapsing K to a point is not Hausdorff. K is distinct from 0, but can't be separated from 0 by disjoint open sets.
It is not a single territory, but comprises seven disjoint units, scattered throughout the area. One of these is the Kiskunság's Puszta where annual events are held reviving the old pastoral life and cattle breeding customs. Another is Lake Kolon near the town of Izsák. It is famous for its marsh tortoises, herons, expanses of untouched reeds and nine species of orchids which grow in the vicinity.
MD 117 was the inspiration for the 1971 hit song "Take Me Home, Country Roads". MD 117 originally consisted of three disjoint segments. The segment from MD 28 to west of Boyds was built in the early 1910s and extended to Boyds in the late 1920s. The two other segments were built west from Germantown and west from MD 124 in Gaithersburg in the early 1930s.
This can be shown by the isomorphisms {(3,4), (3,4), (3,4)} and {(2,3), (2,3), (2,3)} respectively. Since isotopy classes are disjoint, the number of reduced Latin squares gives an upper bound on the number of isotopy classes. Also, the total number of Latin squares is times the number of reduced squares. Normalize a Cayley table of a quasigroup in the same manner as a reduced Latin square.
Hallar-Steinn was an Icelandic poet active around the year 1200. He is best known for the poem Rekstefja, preserved in Bergsbók and Óláfs saga Tryggvasonar en mesta. A few other disjoint verses by him are also known, quoted in Skáldskaparmál, Laufás-Edda and the Third Grammatical Treatise. Rekstefja traces the career of King Óláfr Tryggvason from his upbringing in Russia to his fall at Svöldr.
In computer science, terminal and nonterminal symbols are the lexical elements used in specifying the production rules constituting a formal grammar. Terminal symbols are the elementary symbols of the language defined by a formal grammar. Nonterminal symbols (or syntactic variables) are replaced by groups of terminal symbols according to the production rules. The terminals and nonterminals of a particular grammar are two disjoint sets.
As a special case, a topological space is zero-dimensional with respect to the covering dimension if every open cover of the space has a refinement consisting of disjoint open sets so that any point in the space is contained in exactly one open set of this refinement. It is often convenient to say that the covering dimension of the empty set is -1.
A cycle graph of length four or five is well-covered: in each case, every maximal independent set has size two. A cycle of length seven, and a path of length three, are also well- covered. Every complete graph is well-covered: every maximal independent set consists of a single vertex. Similarly, every cluster graph (a disjoint union of complete graphs) is well-covered.
Construction of a distance-hereditary graph of clique-width 3 by disjoint unions, relabelings, and label-joins. Vertex labels are shown as colors. In graph theory, the clique-width of a graph G is a parameter that describes the structural complexity of the graph; it is closely related to treewidth, but unlike treewidth it can be bounded even for dense graphs. It is defined as the minimum number of labels needed to construct G by means of the following 4 operations : #Creation of a new vertex v with label i ( noted i(v) ) #Disjoint union of two labeled graphs G and H ( denoted G \oplus H ) #Joining by an edge every vertex labeled i to every vertex labeled j (denoted η(i,j)), where i eq j #Renaming label i to label j ( denoted ρ(i,j) ) Graphs of bounded clique-width include the cographs and distance-hereditary graphs.
The Ugandan musk shrew has a disjoint distribution, having been found in Uganda, where the type locality is Kampala, in Tandala in the Democratic Republic of the Congo, and in the Dzanga-Sangha Special Reserve and Batouri in the Central African Republic. It is present in both primary and secondary forest, as well as in single species Gilbertiodendron dewevrei forest. This shrew seems to favour fairly open areas with little undergrowth.
In the mathematical discipline of graph theory, 2-factor theorem discovered by Julius Petersen, is one of the earliest works in graph theory and can be stated as follows: : 2-factor theorem. Let G be a regular graph whose degree is an even number, 2k. Then the edges of G can be partitioned into k edge- disjoint 2-factors.Lovász, László, and Plummer, M.D.. Matching Theory, American Mathematical Soc.
The connected sum is a local operation on manifolds, meaning that it alters the summands only in a neighborhood of V. This implies, for example, that the sum can be carried out on a single manifold M containing two disjoint copies of V, with the effect of gluing M to itself. For example, the connected sum of a two-sphere at two distinct points of the sphere produces the two-torus.
The Poisson random measure is independent on disjoint subspaces, whereas the other PT random measures (negative binomial and binomial) have positive and negative covariances. The PT random measures are discussedCaleb Bastian, Gregory Rempala. Throwing stones and collecting bones: Looking for Poisson-like random measures, Mathematical Methods in the Applied Sciences, 2020. doi:10.1002/mma.6224 and include the Poisson random measure, negative binomial random measure, and binomial random measure.
A graph and two of its cuts. The dotted line in red represents a cut with three crossing edges. The dashed line in green represents one of the minimum cuts of this graph, crossing only two edges. In graph theory, a minimum cut or min-cut of a graph is a cut (a partition of the vertices of a graph into two disjoint subsets) that is minimal in some sense.
In 1956, MD 97 was extended north from Howard County to Westminster, then assumed MD 32 northwest from there to Emmitsburg. Construction on the new alignment of MD 97 from east of Fountain Valley to east of Taneytown began by 1964 and was completed in 1965. The old alignment of MD 97 became a disjoint section of MD 32. Old Taneytown Road's designation was changed to MD 832 in 1978.
The Ruzzo–Tompa algorithm has been used in Bioinformatics tools to study biological data. The problem of finding disjoint maximal subsequences is of practical importance in the analysis of DNA. Maximal subsequences algorithms have been used in the identification of transmembrane segments and the evaluation of sequence homology. The algorithm is used in sequence alignment which is used as a method of identifying similar DNA, RNA, or protein sequences.
A topological space X is said to be disconnected if it is the union of two disjoint non-empty open sets. Otherwise, X is said to be connected. A subset of a topological space is said to be connected if it is connected under its subspace topology. Some authors exclude the empty set (with its unique topology) as a connected space, but this article does not follow that practice.
20 stacked pentagonal antiprisms occur in two disjoint rings of 10 antiprisms each. The antiprisms in each ring are joined to each other via their pentagonal faces. The two rings are mutually perpendicular, in a structure similar to a duoprism. The 300 tetrahedra join the two rings to each other, and are laid out in a 2-dimensional arrangement topologically equivalent to the 2-torus and the ridge of the duocylinder.
Any system under test can be described by a set of classifications, holding both input and output parameters. (Input parameters can also include environments states, pre- conditions and other, rather uncommon parameters). Each classification can have any number of disjoint classes, describing the occurrence of the parameter. The selection of classes typically follows the principle of equivalence partitioning for abstract test cases and boundary-value analysis for concrete test cases.
A box-making game (often called just a box game) is a biased positional game where two players alternately pick elements from a family of pairwise-disjoint sets ("boxes"). The first player - called BoxMaker - tries to pick all elements of a single box. The second player - called BoxBreaker - tries to pick at least one element of all boxes. The box game was first presented by Paul Erdős and Václav Chvátal.
The two disjoint segments of MD 128 were united when that segment returned to state control in 1987; MD 127 was again removed from Chatsworth Avenue. That same year, I-795 was completed north to MD 140 and the MD 795 connector was built between the I-795 - MD 140 interchange and the MD 30 - MD 128 intersection, resulting in a slight relocation of MD 128's western terminus.
The first indication that there might be a problem in defining length for an arbitrary set came from Vitali's theorem.Moore, Gregory H., Zermelo's Axiom of Choice, Springer-Verlag, 1982, pp. 100-101 When you form the union of two disjoint sets, one would expect the measure of the result to be the sum of the measure of the two sets. A measure with this natural property is called finitely additive.
The freeway was extended to its present terminus at Rolling Road and the ramps to UMBC Boulevard were constructed in 1975. Metropolitan Boulevard south of the I-95 interchange was marked as a second segment of MD 46--disjoint from the section between MD 295 and Baltimore/Washington International Airport--from when it opened. North of I-95, the freeway was marked as a relocation of MD 166.
The question of what the connection is between the apparently disjoint tales of Laban and Pharaoh is interpreted in several ways by rabbinical authorities. Rabbi Azriel Hildesheimer explains in his Hukkat HaPesach that Laban was, in fact, the primum mobile of the entire Exile and Exodus saga. Rachel was Jacob's divinely intended wife and could hypothetically have given birth to Joseph as Jacob's firstborn with rights of primogeniture.
Some languages allow both static and dynamic typing. For example, Java and some other ostensibly statically typed languages support downcasting types to their subtypes, querying an object to discover its dynamic type, and other type operations that depend on runtime type information. Another example is C++ RTTI. More generally, most programming languages include mechanisms for dispatching over different 'kinds' of data, such as disjoint unions, runtime polymorphism, and variant types.
The 12.6 km2 refuge was created in 1974 to provide habitat and nesting areas for waterfowl and other migratory birds. It includes a protected estuary, salt marshes and open mudflats, freshwater marshes, open grassland, and riparian woodland and brush. An additional is protected by the disjoint Black River Unit on a tributary of the Chehalis River. Local environmentalist Margaret McKenny is attributed for the preservation of this area.
Illustration of the hyperplane separation theorem. In geometry, the hyperplane separation theorem is a theorem about disjoint convex sets in n-dimensional Euclidean space. There are several rather similar versions. In one version of the theorem, if both these sets are closed and at least one of them is compact, then there is a hyperplane in between them and even two parallel hyperplanes in between them separated by a gap.
In Dürr et al., the authors describe an extension of de Broglie–Bohm theory for handling creation and annihilation operators, which they refer to as "Bell-type quantum field theories". The basic idea is that configuration space becomes the (disjoint) space of all possible configurations of any number of particles. For part of the time, the system evolves deterministically under the guiding equation with a fixed number of particles.
In mathematics, a pullback bundle or induced bundle is the fiber bundle that is induced by a map of its base-space. Given a fiber bundle and a continuous map one can define a "pullback" of by as a bundle over . The fiber of over a point in is just the fiber of over . Thus is the disjoint union of all these fibers equipped with a suitable topology.
To meaningfully bring together the still disjoint parts (syntax expressions and types) a third part is needed: context. Syntactically, a context is a list of pairs x:\sigma, called assignments, assumptions or bindings, each pair stating that value variable x_ihas type \sigma_i. All three parts combined give a typing judgment of the form \Gamma\ \vdash\ e:\sigma, stating that under assumptions \Gamma, the expression e has type \sigma.
Alternative Technologies. Most business process technology focused on analyzing and documenting existing business processes, then manually "reengineering" the processes to eliminate waste, remove bottlenecks, and improve cycle times. These efforts were largely disjoint from process automation systems and distributed control systems (which focused on highly repetitive, often continuous processes), and workflow technologies (which focused on highly repetitive sequential processes like document processing). McGoveran postulatedPascal, F. (April 11, 2015).
Multiple description coding (MDC) is a coding technique that fragments a single media stream into n substreams (n ≥ 2) referred to as descriptions. The packets of each description are routed over multiple, (partially) disjoint paths. In order to decode the media stream, any description can be used, however, the quality improves with the number of descriptions received in parallel. The idea of MDC is to provide error resilience to media streams.
The R27 is a provincial route in South Africa that consists of two disjoint segments. The first segment, also known as the West Coast Highway, connects Cape Town with Velddrif along the West Coast. The second runs from Vredendal via Vanrhynsdorp, Calvinia, Brandvlei and Kenhardt to Keimoes on the N14 near Upington. The connection between Velddrif and Vredendal has never been built, although it can be driven on various gravel roads.
In geometric topology, a cellular decomposition G of a manifold M is a decomposition of M as the disjoint union of cells (spaces homeomorphic to n-balls Bn). The quotient space M/G has points that correspond to the cells of the decomposition. There is a natural map from M to M/G, which is given the quotient topology. A fundamental question is whether M is homeomorphic to M/G.
However, the connected components of a locally connected space are also open, and thus are clopen sets.Willard, Corollary 27.10, p. 200 It follows that a locally connected space X is a topological disjoint union \coprod C_x of its distinct connected components. Conversely, if for every open subset U of X, the connected components of U are open, then X admits a base of connected sets and is therefore locally connected.
However, polynomially convex sets do not behave as nicely as convex sets. Kallin studied conditions under which unions of convex balls are polynomially convex, and found an example of three disjoint cubical cylinders whose union is not polynomially convex.. As part of her work on polynomial convexity, she proved a result now known as Kallin's lemma, giving conditions under which the union of two polynomially convex sets remains itself polynomially convex...
Equilateral dimension has also been considered for normed vector spaces with norms other than the Lp norms. The problem of determining the equilateral dimension for a given norm is closely related to the kissing number problem: the kissing number in a normed space is the maximum number of disjoint translates of a unit ball that can all touch a single central ball, whereas the equilateral dimension is the maximum number of disjoint translates that can all touch each other. For a normed vector space of dimension d, the equilateral dimension is at most 2d; that is, the L∞ norm has the highest equilateral dimension among all normed spaces.. asked whether every normed vector space of dimension d has equilateral dimension at least , but this remains unknown. There exist normed spaces in any dimension for which certain sets of four equilateral points cannot be extended to any larger equilateral set but these spaces may have larger equilateral sets that do not include these four points.
Maryland has a unitary system of numbered state highways with numbers between 2 and 999. The longest Maryland state highway is Maryland Route 2, while several state highways are less than in length. Most of the shortest highways are unsigned. Several state highways have multiple disjoint segments that are denoted internally by suffixes, encompassing either old alignments of a major highway or a collection of service roads related to a particular highway.
A matroid is said to be connected if it is not the direct sum of two smaller matroids; that is, it is connected if and only if there do not exist two disjoint subsets of elements such that the rank function of the matroid equals the sum of the ranks in these separate subsets. Graphic matroids are connected if and only if the underlying graph is both connected and 2-vertex-connected.
The maximal connected subsets (ordered by inclusion) of a non-empty topological space are called the connected components of the space. The components of any topological space X form a partition of X: they are disjoint, non-empty, and their union is the whole space. Every component is a closed subset of the original space. It follows that, in the case where their number is finite, each component is also an open subset.
Two intertwining rings of the 120-cell. Two orthogonal rings in a cell-centered projection The 120-cell can be partitioned into 12 disjoint 10-cell great circle rings, forming a discrete/quantized Hopf fibration. Starting with one 10-cell ring, one can place another ring alongside it that spirals around the original ring one complete revolution in ten cells. Five such 10-cell rings can be placed adjacent to the original 10-cell ring.
In addition the 300 tetrahedra can be partitioned into 10 disjoint Boerdijk–Coxeter helices of 30 cells each that close back on each other. The two pentagonal antiprism tubes, plus the 10 BC helices, form an irregular discrete Hopf fibration of the grand antiprism that Hopf maps to the faces of a pentagonal antiprism. The two tubes map to the two pentagonal faces and the 10 BC helices map to the 10 triangular faces.
A projective cone (or just cone) in projective geometry is the union of all lines that intersect a projective subspace R (the apex of the cone) and an arbitrary subset A (the basis) of some other subspace S, disjoint from R. In the special case that R is a single point, S is a plane, and A is a conic section on S, the projective cone is a conical surface; hence the name.
There are also second, third, etc., Brillouin zones, corresponding to a sequence of disjoint regions (all with the same volume) at increasing distances from the origin, but these are used less frequently. As a result, the first Brillouin zone is often called simply the Brillouin zone. In general, the n-th Brillouin zone consists of the set of points that can be reached from the origin by crossing exactly n − 1 distinct Bragg planes.
The first stage of the algorithm decomposes the tree into a number of disjoint paths. In the second stage of the algorithm, each path is extended and therefore the resulting paths will not be mutually exclusive. In the first stage of the algorithm, each path is associated with an array of size h' . We extend this path by adding the h' immediate ancestors at the top of the path in the same array.
The northern shrew tenrec (Microgale jobihely) is a species of mammal in the family Tenrecidae. It is endemic to Madagascar, where it has a restricted disjoint range in two locations 485 km apart in the north and east of the island. In the north, it is found on the southwestern slopes of the Tsaratanana Massif at elevations from 1420 to 1680 m. In central eastern Madagascar, it is found in the Ambatovy Forest.
But it makes future Find operations faster, not only for the nodes between the query node and the root, but also for their descendants. This updating is an important part of the disjoint-set forest's amortized performance guarantee. There are several algorithms for Find that achieve the asymptotically optimal time complexity. One family of algorithms, known as path compression, makes every node between the query node and the root point to the root.
Standard measure theory takes the third option. One defines a family of measurable sets, which is very rich, and almost any set explicitly defined in most branches of mathematics will be among this family. It is usually very easy to prove that a given specific subset of the geometric plane is measurable. The fundamental assumption is that a countably infinite sequence of disjoint sets satisfies the sum formula, a property called σ-additivity.
The domination problem was studied from the 1950s onwards, but the rate of research on domination significantly increased in the mid-1970s. In 1972, Richard Karp proved the set cover problem to be NP-complete. This had immediate implications for the dominating set problem, as there are straightforward vertex to set and edge to non-disjoint-intersection bijections between the two problems. This proved the dominating set problem to be NP- complete as well.
Due to the nontrivial topology, a Weyl semimetal is expected to demonstrate Fermi arc electron states on its surface. These arcs are discontinuous or disjoint segments of a two dimensional Fermi contour, which are terminated onto the projections of the Weyl fermion nodes on the surface. A 2012 theoretical investigation of superfluid Helium-3 suggested Fermi arcs in neutral superfluids. A detector image (top) signals the existence of Weyl fermion nodes and the Fermi arcs.
Then the algorithm shrinks the edge between s and t to search for non s-t cuts. The minimum cut found in all phases will be the minimum weighted cut of the graph. A cut is a partition of the vertices of a graph into two non- empty, disjoint subsets. A minimum cut is a cut for which the size or weight of the cut is not larger than the size of any other cut.
Edges are considered in increasing order of weight; their endpoint pixels are merged into a region if this doesn't cause a cycle in the graph, and if the pixels are 'similar' to the existing regions' pixels. Detecting cycles is possible in near-constant time with the aid of a disjoint- set data structure.G. Harfst, E. Reingold: A Potential-Based Amortized Analysis of the Union-Find Data Structure. SIGACT 31 (September 2000) pp.
The Rural Municipality of Park is a former rural municipality (RM) in the Canadian province of Manitoba. It was originally incorporated as a rural municipality on January 1, 1997. It ceased on January 1, 2015 as a result of its provincially mandated amalgamation with the RM of Harrison to form the Municipality of Harrison Park. Prior to 2006, the former RM of Park comprised two disjoint parts, called Park (South) and Park (North).
Example of Exact Coloring with 7 colors and 14 vertices In graph theory, an exact coloring is a (proper) vertex coloring in which every pair of colors appears on exactly one pair of adjacent vertices. That is, it is a partition of the vertices of the graph into disjoint independent sets such that, for each pair of distinct independent sets in the partition, there is exactly one edge with endpoints in each set...
In abstract algebra and mathematical logic, if U is an ultrafilter on a set X, "almost all elements of X" sometimes means "the elements of some element of U". For any partition of X into two disjoint sets, one of them will necessarily contain almost all elements of X. It is possible to think of the elements of a filter on X as containing almost all elements of X, even if it isn't an ultrafilter.
In graph theory, a cut is a partition of the vertices of a graph into two disjoint subsets. Any cut determines a cut-set, the set of edges that have one endpoint in each subset of the partition. These edges are said to cross the cut. In a connected graph, each cut-set determines a unique cut, and in some cases cuts are identified with their cut-sets rather than with their vertex partitions.
Dual to the notion of a fundamental cycle is the notion of a fundamental cutset. By deleting just one edge of the spanning tree, the vertices are partitioned into two disjoint sets. The fundamental cutset is defined as the set of edges that must be removed from the graph G to accomplish the same partition. Thus, each spanning tree defines a set of V − 1 fundamental cutsets, one for each edge of the spanning tree.
In 2016, Chomsky and Berwick defined the minimalist program under the Strong Minimalist Thesis in their book Why Only Us by saying that language is mandated by efficient computations and, thus, keeps to the simplest recursive operations. The main basic operation in the minimalist program is merge. Under merge there are two ways in which larger expressions can be constructed: externally and internally. Lexical items that are merged externally build argument representations with disjoint constituents.
Usually, the easiest part of model evaluation is checking whether a model fits experimental measurements or other empirical data. In models with parameters, a common approach to test this fit is to split the data into two disjoint subsets: training data and verification data. The training data are used to estimate the model parameters. An accurate model will closely match the verification data even though these data were not used to set the model's parameters.
Indy Parks established the Pogue's Run Trail alongside the creek bed on the section northeast of downtown. New sections of trail are being planned for construction to connect the Pogue's Run Trail to downtown. , approximately of disjoint sections of the planned trail have been completed. The trail will run from the Monon Trail at 10th Street along the creek to the Pogues Run Art and Nature Park a few blocks west of Emerson Avenue.
Since the function considered above grows very rapidly, its inverse function, f, grows very slowly. This inverse Ackermann function f−1 is usually denoted by α. In fact, α(n) is less than 5 for any practical input size n, since is on the order of 2^{2^{2^{2^{16}}}}. This inverse appears in the time complexity of some algorithms, such as the disjoint-set data structure and Chazelle's algorithm for minimum spanning trees.
The "detection efficiency", or "fair sampling" problem is the most prevalent loophole in optical experiments. Another loophole that has more often been addressed is that of communication, i.e. locality. There is also the "disjoint measurement" loophole which entails multiple samples used to obtain correlations as compared to "joint measurement" where a single sample is used to obtain all correlations used in an inequality. To date, no test has simultaneously closed all loopholes.
Moreover, they compute all complex roots when only few are real. It follows that the standard way of computing real roots is to compute first disjoint intervals, called isolating intervals, such that each one contains exactly one real root, and together they contain all the roots. This computation is called real-root isolation. Having isolating interval, one may use fast numerical methods, such as Newton's method for improving the precision of the result.
Only tame games can be played using the same strategy as misère Nim. Nim is a special case of a poset game where the poset consists of disjoint chains (the heaps). The evolution graph of the game of Nim with three heaps is the same as three branches of the evolution graph of the Ulam- Warburton automaton. At the 1940 New York World's Fair Westinghouse displayed a machine, the Nimatron, that played Nim.
Signed sets may be represented mathematically as an ordered pair of disjoint sets, one set for their positive elements and another for their negative elements. Alternatively, they may be represented as a Boolean function, a function whose domain is the underlying unsigned set (possibly specified explicitly as a separate part of the representation) and whose range is a two-element set representing the signs. Signed sets may also be called \Z_2-graded sets.
Let X, Y, and Z be finite, disjoint sets, and let T be a subset of X × Y × Z. That is, T consists of triples (x, y, z) such that x ∈ X, y ∈ Y, and z ∈ Z. Now M ⊆ T is a 3-dimensional matching if the following holds: for any two distinct triples (x1, y1, z1) ∈ M and (x2, y2, z2) ∈ M, we have x1 ≠ x2, y1 ≠ y2, and z1 ≠ z2.
In fact they are a base for the standard topology on the real numbers. However, a base is not unique. Many different bases, even of different sizes, may generate the same topology. For example, the open intervals with rational endpoints are also a base for the standard real topology, as are the open intervals with irrational endpoints, but these two sets are completely disjoint and both properly contained in the base of all open intervals.
474,475 Originally the little n-cubes operad or the little intervals operad (initially called little n-cubes PROPs) was defined by Michael Boardman and Rainer Vogt in a similar way, in terms of configurations of disjoint axis-aligned n-dimensional hypercubes (n-dimensional intervals) inside the unit hypercube. Later it was generalized by May to little convex bodies operad, and "little disks" is a case of "folklore" derived from the "little convex bodies".
The nested osculating circles of an Archimedean spiral. The spiral itself is not shown, but is visible where the circles are more dense. In differential geometry, the Tait–Kneser theorem states that, if a smooth plane curve has monotonic curvature, then the osculating circles of the curve are disjoint and nested within each other. The logarithmic spiral or the pictured Archimedean spiral provide examples of curves whose curvature is monotonic for the entire curve.
Majang has a basic VSO word order,Getachew 2014, p. 193 though allowing some flexibility for focus, etc. The language makes extensive use of relative clauses, including for circumstances where English would use adjectives.Unseth(1989) A recent studyJoswig (2015a) states that Majang is characterized by a strong morphological ergative-absolutive system, and a conjoint-disjoint distinction which is based on the presence or absence of an absolutive noun phrase directly following the verb.
Let X be a topological space and A, B be two subspaces whose interiors cover X. (The interiors of A and B need not be disjoint.) The Mayer–Vietoris sequence in singular homology for the triad (X, A, B) is a long exact sequence relating the singular homology groups (with coefficient group the integers Z) of the spaces X, A, B, and the intersection A∩B. There is an unreduced and a reduced version.
A chef's knife In cooking, a chef's knife, also known as a cook's knife, is a cutting tool used in food preparation. The chef's knife was originally designed primarily to slice and disjoint large cuts of beef. Today it is the primary general-utility knife for most western cooks. A chef's knife generally has a blade eight inches (20 centimeters) in length and in width, although individual models range from in length.
Angluin defines a pattern to be "a string of constant symbols from Σ and variable symbols from a disjoint set". The language of such a pattern is the set of all its nonempty ground instances i.e. all strings resulting from consistent replacement of its variable symbols by nonempty strings of constant symbols.The language of a pattern with at least two occurrences of the same variable is not regular due to the pumping lemma.
Tweedy's crab-eating rat (Ichthyomys tweedii) is a species of rodent in the family Cricetidae. It is found in two disjoint regions in western Ecuador and central Panama. The species is found near fast-flowing streams in primary and secondary forest, and is known from elevations of 900 to 1700 m. It is presumed that like other members of its genus, it nocturnal and semiaquatic, and feeds on freshwater invertebrates, such as crabs.
Conversely, if X is a Hausdorff second-countable manifold, it must be σ-compact.Stack Exchange, Hausdorff locally compact and second countable is sigma-compact A manifold need not be connected, but every manifold M is a disjoint union of connected manifolds. These are just the connected components of M, which are open sets since manifolds are locally- connected. Being locally path connected, a manifold is path-connected if and only if it is connected.
Vu was an Erdős Lecturer at Hebrew University of Jerusalem in 2007. In 2008 he was awarded the Pólya Prize of the Society for Industrial and Applied Mathematics for his work on concentration of measure.Pólya Prize, SIAM. In 2012, Vu was awarded the Fulkerson Prize (jointly with Anders Johansson and Jeff Kahn) for determining the threshold of edge density above which a random graph can be covered by disjoint copies of a given smaller graph.
County Road 44 runs from State Highway 7 (MN 7) in Minnetrista to County 110 in Mound, passing between Halsted Bay and Priests Bay in Lake Minnetonka. County Road 46 is two disjoint sections of East 46th Street in Minneapolis. The first section runs from Lyndale Avenue South (County 22) to Cedar Avenue (County 152). This route was previously old State Highway 190 before Highway 190 was shifted to another route, then decommissioned.
There are 11 points on the circle sharing a color with a (including a itself), each of which is involved with 2 pairs. This means there are 21 pairs other than (a, b) which include the same color as a, and the same holds true for b. The worst that can happen is that these two sets are disjoint, so we can take d = 42 in the lemma. This gives : e p (d+1) \approx 0.966<1.
The relationships generated by this representation aggregate into six exhaustive and disjoint categories that match the four relational models, while the remaining two correspond to the asocial and null interactions defined in RMT. The model can be generalised to the presence of N social actions. This mapping allows one to infer that the four relational models form an exhaustive set of all possible dyadic relationships based on social coordination, thus explaining why there could exist just four relational models.
The writing style can be further divided as "looped", "italic" or "connected". The cursive method is used with many alphabets due to infrequent pen lifting and beliefs that it increases writing speed. In some alphabets, many or all letters in a word are connected, sometimes making a word one single complex stroke. A 2013 study found that speed of writing cursive is the same, regardless if the children first learned print handwriting or first learned disjoint handwriting.
In matroid theory, the dual of a matroid M is another matroid M^\ast that has the same elements as M, and in which a set is independent if and only if M has a basis set disjoint from it.... Matroid duals go back to the original paper by Hassler Whitney defining matroids.. Reprinted in , pp. 55–79. See in particular section 11, "Dual matroids", pp. 521–524. They generalize to matroids the notions of plane graph duality.
Consider a collection of disjoint and bounded subregions of the underlying space. By definition, the number of points of a Poisson point process in each bounded subregion will be completely independent of all the others. This property is known under several names such as complete randomness, complete independence, or independent scattering and is common to all Poisson point processes. In other words, there is a lack of interaction between different regions and the points in general,W. Feller.
In the mathematical field of graph theory, the bull graph is a planar undirected graph with 5 vertices and 5 edges, in the form of a triangle with two disjoint pendant edges. It has chromatic number 3, chromatic index 3, radius 2, diameter 3 and girth 3. It is also a self-complementary graph, a block graph, a split graph, an interval graph, a claw-free graph, a 1-vertex- connected graph and a 1-edge-connected graph.
Chernoff bounds have very useful applications in set balancing and packet routing in sparse networks. The set balancing problem arises while designing statistical experiments. Typically while designing a statistical experiment, given the features of each participant in the experiment, we need to know how to divide the participants into 2 disjoint groups such that each feature is roughly as balanced as possible between the two groups. Refer to this book section for more info on the problem.
It can be expressed as an application of a Cauchy principal value improper integral. For distributions in several variables, singular supports allow one to define wave front sets and understand Huygens' principle in terms of mathematical analysis. Singular supports may also be used to understand phenomena special to distribution theory, such as attempts to 'multiply' distributions (squaring the Dirac delta function fails – essentially because the singular supports of the distributions to be multiplied should be disjoint).
Klivans is the author of the book The Mathematics of Chip-Firing (CRC Press, 2018). Her research contributions include a disproof of a 50-year-old conjecture of Richard Stanley that every abstract simplicial complex whose face ring is a Cohen–Macaulay ring can be partitioned into disjoint intervals, each including a facet of the complex. Such a partition generalizes a shelling and (if it always existed) would have been helpful in understanding the -vectors of these complexes.
A topological space is zero-dimensional according to the Lebesgue covering dimension if every finite open cover has a refinement that is also an open cover by disjoint sets. A topological space is totally disconnected if it has no nontrivial connected subsets; for points in the plane, being totally disconnected is equivalent to being zero-dimensional. The Denjoy–Riesz theorem states that every compact totally disconnected subset of the plane is a subset of a Jordan arc..
The monochromatic triangle problem takes as input an n-node undirected graph G(V,E) with node set V and edge set E. The output is a Boolean value, true if the edge set E of G can be partitioned into two disjoint sets E1 and E2, such that both of the two subgraphs G1(V,E1) and G2(V,E2) are triangle-free graphs, and false otherwise. This decision problem is NP-complete.. A1.1: GT6, pg.191.
For the 75 prefectures after the 1871/72 wave of prefectural mergers, see the List of Japanese prefectures. For a complete list of not only -fu/-ken, but all -fu/-han/-ken at two points in time, see the List of Japanese prefectures by population#1868 to 1871, it also indicates the (often disjoint) territorial extent of the prefectures and domains in this period by listing the provinces the prefectures/domains extended to and the number of exclaves.
First a supervised learning algorithm is trained based on the labeled data only. This classifier is then applied to the unlabeled data to generate more labeled examples as input for the supervised learning algorithm. Generally only the labels the classifier is most confident in are added at each step. Co-training is an extension of self-training in which multiple classifiers are trained on different (ideally disjoint) sets of features and generate labeled examples for one another.
The result is the Coxeter graph. This construction exhibits the Coxeter graph as an induced subgraph of the Kneser graph . The Coxeter graph may also be constructed from the smaller distance- regular Heawood graph by constructing a vertex for each 6-cycle in the Heawood graph and an edge for each disjoint pair of 6-cycles.. The Coxeter graph may be derived from the Hoffman-Singleton graph. Take any vertex v in the Hoffman- Singleton graph.
In contrast to the group lasso problem, where features are grouped into disjoint blocks, it may be the case that grouped features are overlapping or have a nested structure. Such generalizations of group lasso have been considered in a variety of contexts. For overlapping groups one common approach is known as latent group lasso which introduces latent variables to account for overlap. Nested group structures are studied in hierarchical structure prediction and with directed acyclic graphs.
Opuntia trichophora is a species of cactus in the genus Opuntia, more commonly known as prickly pears or nopal. O. trichophora is distributed throughout parts of New Mexico, Colorado, and Utah, and may have disjoint populations in Wyoming, southern Montana, and southern Idaho. Opuntia trichophora is a diploid (2n=22) but has sometimes been treated as a variety of Opuntia polyacantha a tetraploid (2n=44). O. trichophora tends to have longer spines than O. polycantha or O. macrorhiza.
The sacrifice: they stored a given "item" of data (e.g., the information pertaining to a product in a product database) over several relations, and it takes time to assemble disjoint parts for a query. Many of Amazon's services demanded mostly primary-key reads on their data, and with speed a top priority, putting these pieces together was extremely taxing. Content with compromising storage efficiency, Amazon's response was Dynamo: a highly available key-value store built for internal use.
Jeffery Westbrook and Robert Tarjan (1992) developed an efficient data structure for this problem based on disjoint-set data structures. Specifically, it processes n vertex additions and m edge additions in O(m α(m, n)) total time, where α is the inverse Ackermann function. This time bound is proved to be optimal. Uzi Vishkin and Robert Tarjan (1985) designed a parallel algorithm on CRCW PRAM that runs in O(log n) time with n + m processors.
State Route 43 (SR 43) is a primary state highway in the U.S. state of Virginia. The state highway consists of two disjoint segments that have a total length of . The southern portion of the state highway runs from U.S. Route 29 Business (US 29 Bus.) in Altavista north to the Blue Ridge Parkway at Peaks of Otter. The northern segment has a length of between the Blue Ridge Parkway near Buchanan and US 220 in Eagle Rock.
The combination of path compression, splitting, or halving, with union by size or by rank, reduces the running time for operations of any type, up to of which are MakeSet operations, to \Theta(m\alpha(n)). This makes the amortized running time of each operation \Theta(\alpha(n)). This is asymptotically optimal, meaning that every disjoint set data structure must use \Omega(\alpha(n)) amortized time per operation. Here, the function \alpha(n) is the inverse Ackermann function.
A demo for Union-Find when using Kruskal's algorithm to find minimum spanning tree. Disjoint-set data structures model the partitioning of a set, for example to keep track of the connected components of an undirected graph. This model can then be used to determine whether two vertices belong to the same component, or whether adding an edge between them would result in a cycle. The Union–Find algorithm is used in high-performance implementations of unification.
A forest is an undirected graph in which any two vertices are connected by at most one path. Equivalently, a forest is an undirected acyclic graph. Equivalently, a forest is an undirected graph, all of whose connected components are trees; in other words, the graph consists of a disjoint union of trees. As special cases, the order-zero graph (a forest consisting of zero trees), a single tree, and an edgeless graph, are examples of forests.
The affine plane of order three is a configuration. When embedded in some ambient space it is called the Hesse configuration. It is not realizable in the Euclidean plane but is realizable in the complex projective plane as the nine inflection points of an elliptic curve with the 12 lines incident with triples of these. The 12 lines can be partitioned into four classes of three lines apiece where, in each class the lines are mutually disjoint.
Most local government areas are a single contiguous area (possibly including islands). However, Aboriginal Shires are often defined as a number of disjoint areas each containing an Indigenous community. In the case of the Aboriginal Shire of Mapoon, there are three areas all within the locality of Mapoon (which is otherwise within the Shire of Cook), two to the north and south of the Wenlock River's mouth at the Gulf of Carpentaria and a third further up river.
In mathematics, a packing in a hypergraph is a partition of the set of the hypergraph's edges into a number of disjoint subsets such that no pair of edges in each subset share any vertex. There are two famous algorithms to achieve asymptotically optimal packing in k-uniform hypergraphs. One of them is a random greedy algorithm which was proposed by Joel Spencer. He used a branching process to formally prove the optimal achievable bound under some side conditions.
Marwaris and Bengalis, despite coexisting in Kolkata for many generations, have led largely disjoint lives. In Bengali literature and art, the Marwari appears typically as a stereotype, a money-making reactionary. Saraogi's writing is in Hindi, albeit neither overly Sanskritised, nor informed by the popular Hindi film industry. Though she often uses Bengali expressions in her novels, especially in the speech of Bengali characters, even in her oeuvre the breach between the Hindi- and Bengali-speaking communities remains unbridged.
This is a disjoint union over all possible binary partitions of A. It is straightforward to show that multiplication is associative and commutative (up to isomorphism), and distributive over addition. As for the generating series, (F · G)(x) = F(x)G(x). The diagram below shows one possible (F · G)-structure on a set with five elements. The F-structure (red) picks up three elements of the base set, and the G-structure (light blue) takes the rest.
Packet Protection Scheme (1+1) This protection scheme is similar in a sense to Ring-based path protection and Dedicated Backup Path Protection (DBPP) schemes described before. Here, same traffic is transmitted over two, link and/or node disjoint, LSPs; primary and backup. The transmission is done by the head-end LSR. The tail-end LSR then receives and compares both traffics; when a failure occurs, the tail-end detects it and switches the traffic to the secondary LSP.
This is because the algorithm generates p-cycles that have only one straddling span. The key feature of the SLA is the ability to find the p-cyles quickly. The Algorithm works by finding the shortest path between the nodes of a span, and than find another shortest path between the same set of the nodes that is disjoint from the first route. The p-cycle is than created by combining the previously found two routes into one.
Suppose that S is a compact Riemann surface of genus g > 1\. The Fenchel–Nielsen coordinates depend on a choice of 6g − 6 curves on S, as follows. The Riemann surface S can be divided up into 2g − 2 pairs of pants by cutting along 3g − 3 disjoint simple closed curves. For each of these 3g − 3 curves γ, choose an arc crossing it that ends in other boundary components of the pairs of pants with boundary containing γ.
The fundamental group of the figure eight is the free group generated by a and b A rose is a wedge sum of circles. That is, the rose is the quotient space C/S, where C is a disjoint union of circles and S a set consisting of one point from each circle. As a cell complex, a rose has a single vertex, and one edge for each circle. This makes it a simple example of a topological graph.
The infinite symmetric product SP(X) of a topological space X with given basepoint e is the quotient of the disjoint union of all powers X, X2, X3, ... obtained by identifying points (x1,...,xn) with (x1,...,xn,e) and identifying any point with any other point given by permuting its coordinates. In other words its underlying set is the free commutative monoid generated by X (with unit e), and is the abelianization of the James reduced product.
In theoretical computer science a simulation preorder is a relation between state transition systems associating systems which behave in the same way in the sense that one system simulates the other. Intuitively, a system simulates another system if it can match all of its moves. The basic definition relates states within one transition system, but this is easily adapted to relate two separate transition systems by building a system consisting of the disjoint union of the corresponding components.
The database is partitioned into disjoint subsets each assigned to a single-threaded execution engine assigned to one core on one node. Each engine has exclusive access to all of the data in its partition. Because it is single-threaded, only one transaction at a time can access the data stored on that partition. No physical locks or latches are included in the system, and once it is started, no transaction stalls waiting for another transaction to complete.
Given a directed graph G = (V, E) and two vertices s and t, we are to find the maximum number of paths from s to t. This problem has several variants: 1\. The paths must be edge-disjoint. This problem can be transformed to a maximum flow problem by constructing a network N = (V, E) from G, with s and t being the source and the sink of N respectively, and assigning each edge a capacity of 1.
In alias analysis, we divide the program's memory into alias classes. Alias classes are disjoint sets of locations that cannot alias to one another. For the discussion here, it is assumed that the optimizations done here occur on a low-level intermediate representation of the program. This is to say that the program has been compiled into binary operations, jumps, moves between registers, moves from registers to memory, moves from memory to registers, branches, and function calls/returns.
This is the case because new memory allocations must be disjoint from all other memory allocations. #Each record field of each record type has its own alias class, in general, because the typing discipline usually only allows for records of the same type to alias. Since all records of a type will be stored in an identical format in memory, a field can only alias to itself. #Similarly, each array of a given type has its own alias class.
County Road 26 (non-CSAH) is a road in Minnetrista that begins at County 110 and extends westward to the Carver County line, where it continues as an unnumbered road for approximately 1/4 mile before it ends at Carver County Road 20. County Road 27 is two disjoint sections of Stinson Boulevard in Minneapolis. The northern section runs from 37th Avenue NE to St. Anthony Boulevard. The southern section runs from New Brighton Boulevard to Hennepin Avenue.
Formulated as a problem in graph theory, the pairs of people sitting next to each other at a single meal can be represented as a disjoint union of cycle graphs C_x+C_y+C_z+\cdots of the specified lengths, with one cycle for each of the dining tables. This union of cycles is a 2-regular graph, and every 2-regular graph has this form. If G is this 2-regular graph and has n vertices, the question is whether the complete graph K_n can be represented as an edge-disjoint union of copies of G. In order for a solution to exist, the total number of conference participants (or equivalently, the total capacity of the tables, or the total number of vertices of the given cycle graphs) must be an odd number. For, at each meal, each participant sits next to two neighbors, so the total number of neighbors of each participant must be even, and this is only possible when the total number of participants is odd.
Kirkman's schoolgirl problem, of grouping fifteen schoolgirls into rows of three in seven different ways so that each pair of girls appears once in each triple, is a special case of the Oberwolfach problem, OP(3^5). The problem of Hamiltonian decomposition of a complete graph K_n is another special case, OP(n). Alspach's conjecture, on the decomposition of a complete graph into cycles of given sizes, is related to the Oberwolfach problem, but neither is a special case of the other. If G is a 2-regular graph, with n vertices, formed from a disjoint union of cycles of certain lengths, then a solution to the Oberwolfach problem for G would also provide a decomposition of the complete graph into (n-1)/2 copies of each of the cycles of G. However, not every decomposition of K_n into this many cycles of each size can be grouped into disjoint cycles that form copies of G, and on the other hand not every instance of Alspach's conjecture involves sets of cycles that have (n-1)/2 copies of each cycle.
A hierarchical clustering of a collection of objects may be formalized as a maximal family of sets of the objects in which no two sets cross. That is, for every two sets S and T in the family, either S and T are disjoint or one is a subset of the other, and no more sets can be added to the family while preserving this property. If T is an unrooted binary tree, it defines a hierarchical clustering of its leaves: for each edge (u,v) in T there is a cluster consisting of the leaves that are closer to u than to v, and these sets together with the empty set and the set of all leaves form a maximal non-crossing family. Conversely, from any maximal non-crossing family of sets over a set of n elements, one can form a unique unrooted binary tree that has a node for each triple (A,B,C) of disjoint sets in the family that together cover all of the elements.
One can define a set of disjoint planes in this way from the circles of the packing, and a second set of disjoint planes defined by the circles that circumscribe each triangular gap between three of the circles in the packing. These two sets of planes meet at right angles, and form the generators of a reflection group whose fundamental domain can be viewed as a hyperbolic manifold. By Mostow rigidity, the hyperbolic structure of this domain is uniquely determined, up to isometry of the hyperbolic space; these isometries, when viewed in terms of their actions on the Euclidean plane on the boundary of the half-plane model, translate to Möbius transformations. There is also a more elementary proof of the same uniqueness property, based on the maximum principle and on the observation that, in the triangle connecting the centers of three mutually tangent circles, the angle formed at the center of one of the circles is monotone decreasing in its radius and monotone increasing in the two other radii.
This constraint affects even the title, which would conventionally be spelt Revenantes. An English translation by Ian Monk was published in 1996 as The Exeter Text: Jewels, Secrets, Sex in the collection Three. It has been remarked by Jacques Roubaud that these two novels draw words from two disjoint sets of the French language, and that a third novel would be possible, made from the words not used so far (those containing both "e" and a vowel other than "e").
In mathematics, the Vitali covering lemma is a combinatorial and geometric result commonly used in measure theory of Euclidean spaces. This lemma is an intermediate step, of independent interest, in the proof of the Vitali covering theorem. The covering theorem is credited to the Italian mathematician Giuseppe Vitali.. The theorem states that it is possible to cover, up to a Lebesgue-negligible set, a given subset E of Rd by a disjoint family extracted from a Vitali covering of E.
In Riemannian geometry, the Cheeger isoperimetric constant of a compact Riemannian manifold M is a positive real number h(M) defined in terms of the minimal area of a hypersurface that divides M into two disjoint pieces. In 1970, Jeff Cheeger proved an inequality that related the first nontrivial eigenvalue of the Laplace–Beltrami operator on M to h(M). This proved to be a very influential idea in Riemannian geometry and global analysis and inspired an analogous theory for graphs.
One says a manifold N is obtained from M by attaching j-handles if the union of M with finitely many j-handles is diffeomorphic to N. The definition of a handle decomposition is then as in the introduction. Thus, a manifold has a handle decomposition with only 0-handles if it is diffeomorphic to a disjoint union of balls. A connected manifold containing handles of only two types (i.e.: 0-handles and j-handles for some fixed j) is called a handlebody.
A cobordism (W; M, N). In mathematics, cobordism is a fundamental equivalence relation on the class of compact manifolds of the same dimension, set up using the concept of the boundary (French bord, giving cobordism) of a manifold. Two manifolds of the same dimension are cobordant if their disjoint union is the boundary of a compact manifold one dimension higher. The boundary of an (n + 1)-dimensional manifold W is an n-dimensional manifold ∂W that is closed, i.e., with empty boundary.
Since the total number of neighbors of all vertices is just the number of edges in the graph, the algorithm takes time linear in the number of edges, its input size.. Partition refinement also forms a key step in lexicographic breadth-first search, a graph search algorithm with applications in the recognition of chordal graphs and several other important classes of graphs. Again, the disjoint set elements are vertices and the set represent sets of neighbors, so the algorithm takes linear time...
Composition, also called substitution, is more complicated again. The basic idea is to replace components of F with G-structures, forming (F∘G). As with multiplication, this is done by splitting the input set A; the disjoint subsets are given to G to make G-structures, and the set of subsets is given to F, to make the F-structure linking the G-structures. It is required for G to map the empty set to itself, in order for composition to work.
During the calculation of the access nodes, the search space (all visited nodes towards the top of the hierarchy) for each node can be stored without including transit nodes. When performing a query, those search spaces for start- and target node are checked for an intersection. If those spaces are disjoint, transit node routing can be used because the up- and down-paths must meet at a transit node. Otherwise there could be a shortest path without a transit node.
MD 77 links Thurmont with western Carroll County through the communities of Graceham, Rocky Ridge, and Detour in the Monocacy River valley. MD 77 was constructed from Thurmont east to Detour in the 1920s and early 1930s. A disjoint section of MD 77 was built between Cavetown and Foxville in the late 1930s. The portions of the modern highway between Foxville and Thurmont and from Detour to Keymar were county highways until they were designated part of MD 77 in 1956.
Operadic composition in the little 2-disks operad. Operadic composition in the operad of symmetries. A little disks operad or, little balls operad or, more specifically, the little n-disks operad is a topological operad defined in terms of configurations of disjoint n-dimensional disks inside a unit n-disk centered in the origin of Rn. The operadic composition for little 2-disks is illustrated in the figure.Giovanni Giachetta, Luigi Mangiarotti, Gennadi Sardanashvily (2005) Geometric and Algebraic Topological Methods in Quantum Mechanics, , pp.
In some cases, the properties of the graphs in a minor-closed family may be closely connected to the properties of their excluded minors. For example a minor-closed graph family F has bounded pathwidth if and only if its forbidden minors include a forest,. F has bounded tree-depth if and only if its forbidden minors include a disjoint union of path graphs, F has bounded treewidth if and only if its forbidden minors include a planar graph,, Theorem 9, p.
In mathematics, the are three disjoint connected open sets of the plane or open unit square with the counterintuitive property that they all have the same boundary. In other words, for any point selected on the boundary of one of the lakes, the other two lakes' boundaries also contain that point. More than two sets with the same boundary are said to have the Wada property; examples include Wada basins in dynamical systems. This property is rare in real-world systems.
Another class of graphs in which the minimum clique cover can be found in polynomial time are the triangle-free graphs. In these graphs, every clique cover consists of a matching (a set of disjoint pairs of adjacent vertices) together with singleton sets for the remaining unmatched vertices. The number of cliques equals the number of vertices minus the number of matched pairs. Therefore, in triangle-free graphs, the minimum clique cover can be found by using an algorithm for maximum matching.
His work was a beginning to the algebra of sets, again not a concept available to Boole as a familiar model. His pioneering efforts encountered specific difficulties, and the treatment of addition was an obvious difficulty in the early days. Boole replaced the operation of multiplication by the word "and" and addition by the word "or". But in Boole's original system, + was a partial operation: in the language of set theory it would correspond only to disjoint union of subsets.
The objects need not be boxes, and the separators need not be axis-parallel: :: Let C be a collection of possible orientations of hyperplanes (i.e. C = {horizontal,vertical}). Given N d-objects, such that every two disjoint object are separated by a hyperplane with an orientation from C, whose interiors are k-thick, there exists a hyperplane with an orientation from C such that at least: (N + 1 − k)/O(C) of the d-objects interiors lie entirely to each side of the hyperplane.
The separate sections are denoted by numbers in the statute, which correspond in signage to the last digit in a hyphenated route number. For example, LA 560-3 is section 3 of LA 560. Similar as it may sound, this method is different from state route legislative definitions in states such as California, where state routes are often defined as existing in disjoint sections; but in these cases, there is often a linear continuity of a route through cosigning or implied connections made via other routes.
The Catawba Reservation (), located in two disjoint sections in York County, South Carolina east of Rock Hill; a total of , it reported a 2010 census population of 841 inhabitants. It also has a congressionally established service area in North Carolina, covering Mecklenburg, Cabarrus, Gaston, Union, Cleveland and Rutherford counties. The Catawba also owns a site in Kings Mountain, North Carolina, which will be used for a casino and mixed-use entertainment complex. Today the Catawba earns most of its revenue from Federal/State funds.
This results from the axiom of foundation – or the axiom of regularity – which enacts such a prohibition (cf. p. 190 in Being and Event). (This axiom states that every non-empty set A contains an element y that is disjoint from A.) Badiou's philosophy draws two major implications from this prohibition. Firstly, it secures the inexistence of the 'one': there cannot be a grand overarching set, and thus it is fallacious to conceive of a grand cosmos, a whole Nature, or a Being of God.
In graph theory, a branch of mathematics, a linear forest is a kind of forest formed from the disjoint union of path graphs. It is an undirected graph with no cycles in which every vertex has degree at most two. Linear forests are the same thing as claw-free forests. They are the graphs whose Colin de Verdière graph invariant is at most 1.. The linear arboricity of a graph is the minimum number of linear forests into which the graph can be partitioned.
Reginn after he had been killed by Sigurd on the 11th century Ramsund carving in Södermanland, Sweden. Reginsmál (Old Norse: 'The Lay of Reginn') is an Eddic poem interspersed with prose found in the Codex Regius manuscript. It is closely associated with Fáfnismál, the poem that immediately follows it in the Codex, and it is likely that the two of them were intended to be read together. The poem, if regarded as a single unit, is disjoint and fragmentary, consisting of stanzas both in ljóðaháttr and fornyrðislag.
A transaction executing under snapshot isolation appears to operate on a personal snapshot of the database, taken at the start of the transaction. When the transaction concludes, it will successfully commit only if the values updated by the transaction have not been changed externally since the snapshot was taken. Such a write–write conflict will cause the transaction to abort. In a write skew anomaly, two transactions (T1 and T2) concurrently read an overlapping data set (e.g. values V1 and V2), concurrently make disjoint updates (e.g.
The state highway was extended north from Rutledge to MD 146 near Taylor in 1938. A disjoint segment of MD 152 was constructed from US 40 south to Magnolia in 1940. At its southern end, instead of continuing southeast across the Pennsylvania Railroad (now Amtrak), the state highway curved to the southwest along what is now Fort Hoyle Road to end where Fort Hoyle Road turns north. Only two years later, the southern segment of MD 152 was widened and resurfaced as a military access project.
Suppose W is a set system, that is, a collection of subsets of a set U. The collection W is a sunflower (or \Delta-system) if there is a subset S of U such that for each distinct A and B in W, we have A \cap B = S. In other words, W is a sunflower if the pairwise intersection of each set in W is constant. Note that this intersection, S, may be empty; a collection of disjoint subsets is also a sunflower.
The single linkage algorithm is composed of the following steps: # Begin with the disjoint clustering having level L(0) = 0 and sequence number m=0. # Find the most similar pair of clusters in the current clustering, say pair (r), (s), according to d[(r),(s)] = \min d[(i),(j)]where the minimum is over all pairs of clusters in the current clustering. # Increment the sequence number: m = m + 1. Merge clusters (r) and (s) into a single cluster to form the next clustering m.
It is commonly stated as: : In a given instance of the stable- roommates problem (SRP), each of 2n participants ranks the others in strict order of preference. A matching is a set of n disjoint pairs of participants. A matching M in an instance of SRP is stable if there are no two participants x and y, each of whom prefers the other to their partner in M. Such a pair is said to block M, or to be a blocking pair with respect to M.
The complete linkage clustering algorithm consists of the following steps: # Begin with the disjoint clustering having level L(0) = 0 and sequence number m=0. # Find the most similar pair of clusters in the current clustering, say pair (r), (s), according to d[(r),(s)] = \min d[(i),(j)]where the minimum is over all pairs of clusters in the current clustering. # Increment the sequence number: m = m + 1. Merge clusters (r) and (s) into a single cluster to form the next clustering m.
The Petersen graph O3 is a well known non- Hamiltonian graph, but all odd graphs On for n ≥ 4 are known to have a Hamiltonian cycle. As the odd graphs are vertex-transitive, they are thus one of the special cases for which a positive answer to Lovász' conjecture is known. Biggs conjectured more generally that the edges of On can be partitioned into \lfloor n/2\rfloor edge-disjoint Hamiltonian cycles. When n is odd, the leftover edges must then form a perfect matching.
Horoscope is a ballet created in 1937 by Frederick Ashton with scenery by Sophie Fedorovitch and music by Constant Lambert.Frederick Ashton and His Ballets It is based on astrological themes, and is reminiscent of Gustav Holst's The Planets in its musical exploration of the mystical. The story of the ballet concerns a young man and woman who were born in the disjoint Sun signs of Leo and Virgo. However, both have their Moon in Gemini, and they are able to overcome their fate and become lovers.
The degeneracy of a graph with arboricity a is at least equal to a, and at most equal to 2a-1. The coloring number of a graph, also known as its Szekeres-Wilf number is always equal to its degeneracy plus 1 . The strength of a graph is a fractional value whose integer part gives the maximum number of disjoint spanning trees that can be drawn in a graph. It is the packing problem that is dual to the covering problem raised by the arboricity.
An additional section of the state highway was built on Madonna Road from the junction with MD 23 north to Nelson Mill Road in 1939. MD 146 was truncated at MD 23 when the Madonna Road segment was transferred to county maintenance in 1955. The section of the state highway south of Loch Raven Reservoir was originally designated MD 144 but became a disjoint segment of MD 146 by 1940. MD 144 was later reused for bypassed sections of U.S. Route 40 between Cumberland and Baltimore.
Every non-empty set x contains a member y such that x and y are disjoint sets. : \forall x [\exists a ( a \in x) \Rightarrow \exists y ( y \in x \land \lnot \exists z (z \in y \land z \in x))]. or in modern notation: \forall x\,(x eq \varnothing \Rightarrow \exists y \in x\,(y \cap x = \varnothing)). This (along with the Axiom of Pairing) implies, for example, that no set is an element of itself and that every set has an ordinal rank.
This has impact on continuity of the resulting curve or its higher derivatives; for instance, it allows the creation of corners in an otherwise smooth NURBS curve. A number of coinciding knots is sometimes referred to as a knot with a certain multiplicity. Knots with multiplicity two or three are known as double or triple knots. The multiplicity of a knot is limited to the degree of the curve; since a higher multiplicity would split the curve into disjoint parts and it would leave control points unused.
For example, the Hopf link is formed by two circles that each pass through the disk spanned by the other. It forms the simplest example of a pair of linked curves, but it is possible for curves to be linked in other more complicated ways. If two curves are not linked, then it is possible to find a topological disk in space, having the first curve as its boundary and disjoint from the second curve. Conversely if such a disk exists then the curves are necessarily unlinked.
The term cycle may also refer to an element of the cycle space of a graph. There are many cycle spaces, one for each coefficient field or ring. The most common is the binary cycle space (usually called simply the cycle space), which consists of the edge sets that have even degree at every vertex; it forms a vector space over the two-element field. By Veblen's theorem, every element of the cycle space may be formed as an edge-disjoint union of simple cycles.
Given a simple closed geodesic on an oriented hyperbolic surface and a real number t, one can cut the manifold along the geodesic, slide the edges a distance t to the left, and glue them back. This gives a new hyperbolic surface, and the (possibly discontinuous) map between them is an example of a left earthquake. More generally one can do the same construction with a finite number of disjoint simple geodesics, each with a real number attached to it. The result is called a simple earthquake.
The \lceil r\cdot n\rceil is the smallest possible when n is an integer. An r-partite hypergraph is an r-uniform hypergraph in which the vertices are partitioned into r disjoint sets and each hyperedge contains exactly one vertex of each set (so a 2-partite hypergraph is a just bipartite graph). Let n be any positive integer. Any family of rn-r+1 fractional-matchings (=colors) of size at least n in an r-partite hypergraph has a rainbow-fractional-matching of size n.
Raoul Bott used Morse–Bott theory in his original proof of the Bott periodicity theorem. Round functions are examples of Morse–Bott functions, where the critical sets are (disjoint unions of) circles. Morse homology can also be formulated for Morse–Bott functions; the differential in Morse–Bott homology is computed by a spectral sequence. Frederic Bourgeois sketched an approach in the course of his work on a Morse–Bott version of symplectic field theory, but this work was never published due to substantial analytic difficulties.
Specifically, the court questioned whether a unique file uploaded to multiple trackers would result in non-overlapping swarms, or whether any scenario existed where even though the same file was distributed, disjoint swarms arose that would never interact. Due to a lack of evidence, the court ultimately concluded that simply downloading the same copyrighted file was not proof the users identified acted in concert, and therefore could not be combined into a single case. Order denying plaintiff's motion for leave to file an amended complaint.
MD 471 was constructed in two disjoint segments in 1932: the current length of the state highway along Indian Bridge Road and St. Andrews Church Road between MD 5 and Fairgrounds Road near Leonardtown. The latter segment of MD 471 was removed from the state highway system in 1956 but returned when MD 4 was extended west from California to Leonardtown in 1982. The current portion of MD 471 was also transferred to county control in 1956 but returned to the state highway system by 1963.
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions such as disjoint unions, direct sums, coproducts, pushouts and direct limits. Limits and colimits, like the strongly related notions of universal properties and adjoint functors, exist at a high level of abstraction. In order to understand them, it is helpful to first study the specific examples these concepts are meant to generalize.
In two dimensions, the convex hull is sometimes partitioned into two parts, the upper hull and the lower hull, stretching between the leftmost and rightmost points of the hull. More generally, for convex hulls in any dimension, one can partition the boundary of the hull into upward-facing points (points for which an upward ray is disjoint from the hull), downward-facing points, and extreme points. For three-dimensional hulls, the upward-facing and downward-facing parts of the boundary form topological disks., p. 6.
For a Boolean algebra A and a commutative monoid M, a map μ : A → M is a measure, if μ(a)=0 if and only if a=0, and μ(a ∨ b)=μ(a)+μ(b) whenever a and b are disjoint (that is, a ∧ b=0), for any a, b in A. We say in addition that μ is a Vaught measure (after Robert Lawson Vaught), or V-measure, if for all c in A and all x,y in M such that μ(c)=x+y, there are disjoint a, b in A such that c=a ∨ b, μ(a)=x, and μ(b)=y. An element e in a commutative monoid M is measurable (with respect to M), if there are a Boolean algebra A and a V-measure μ : A → M such that μ(1)=e---we say that μ measures e. We say that M is measurable, if any element of M is measurable (with respect to M). Of course, every measurable monoid is a conical refinement monoid. Hans Dobbertin proved in 1983 that any conical refinement monoid with at most ℵ1 elements is measurable.
In theoretical computer science and network routing, Suurballe's algorithm is an algorithm for finding two disjoint paths in a nonnegatively-weighted directed graph, so that both paths connect the same pair of vertices and have minimum total length.. The algorithm was conceived by John W. Suurballe and published in 1974.. The main idea of Suurballe's algorithm is to use Dijkstra's algorithm to find one path, to modify the weights of the graph edges, and then to run Dijkstra's algorithm a second time. The output of the algorithm is formed by combining these two paths, discarding edges that are traversed in opposite directions by the paths, and using the remaining edges to form the two paths to return as the output. The modification to the weights is similar to the weight modification in Johnson's algorithm, and preserves the non-negativity of the weights while allowing the second instance of Dijkstra's algorithm to find the correct second path. The problem of finding two disjoint paths of minimum weight can be seen as a special case of a minimum cost flow problem, where in this case there are two units of "flow" and nodes have unit "capacity".
In combinatorial optimization, the set TSP, also known as the generalized TSP, group TSP, One-of-a-Set TSP, Multiple Choice TSP or Covering Salesman Problem, is a generalization of the Traveling salesman problem (TSP), whereby it is required to find a shortest tour in a graph which visits all specified subsets of the vertices of a graph. The subsets of vertices must be disjoint. The ordinary TSP is a special case of the set TSP when all subsets to be visited are singletons. Therefore, the set TSP is also NP-hard.
In mathematics, in the areas of order theory and combinatorics, Dilworth's theorem characterizes the width of any finite partially ordered set in terms of a partition of the order into a minimum number of chains. It is named for the mathematician . An antichain in a partially ordered set is a set of elements no two of which are comparable to each other, and a chain is a set of elements every two of which are comparable. A chain decomposition is a partition of the elements of the order into disjoint chains.
In particular, every uncountable Polish space has the perfect set property, and can be written as the disjoint union of a perfect set and a countable open set. The axiom of choice implies the existence of sets of reals that do not have the perfect set property, such as Bernstein sets. However, in Solovay's model, which satisfies all axioms of ZF but not the axiom of choice, every set of reals has the perfect set property, so the use of the axiom of choice is necessary. Every analytic set has the perfect set property.
This is the basic idea of Budan's theorem and Budan–Fourier theorem. By repeating the division of an interval into two intervals, one gets eventually a list of disjoint intervals containing together all real roots of the polynomial, and containing each exactly one real root. Descartes rule of signs and homographic transformations of the variable are, nowadays, the basis of the fastest algorithms for computer computation of real roots of polynomials (see Real-root isolation). Descartes himself used the transformation for using his rule for getting information of the number of negative roots.
An IP tunnel is an Internet Protocol (IP) network communications channel between two networks. It is used to transport another network protocol by encapsulation of its packets. IP tunnels are often used for connecting two disjoint IP networks that don't have a native routing path to each other, via an underlying routable protocol across an intermediate transport network. In conjunction with the IPsec protocol they may be used to create a virtual private network between two or more private networks across a public network such as the Internet.
It is immediately apparent from the Leibniz quote above that there are implications for sampling. Deming observed that in any forecasting activity, the population is that of future events while the sampling frame is, inevitably, some subset of historical events. Deming held that the disjoint nature of population and sampling frame was inherently problematic once the existence of special-cause variation was admitted, rejecting the general use of probability and conventional statistics in such situations. He articulated the difficulty as the distinction between analytic and enumerative statistical studies.
MD 103 originally followed all of Montgomery Road from US 1 in Elkridge, which followed Old Washington Road, to US 29 (Old Columbia Pike) in Ellicott City. The first section of MD 103 was paved in concrete from US 1 to west of Landing Road by 1923. The next section was constructed as a concrete road from there west to the highway's intersection with Meadowridge Road at Miller's Corner in 1924 and 1925. A disjoint section of MD 103 was built from US 29 east to near New Cut Road in 1928.
A disjoint segment of MD 32 was constructed on a new alignment from MD 175 in Odenton to MD 178 in Crownsville in the early 1970s. The Patuxent Freeway was built from Fort Meade to Columbia in the mid-1980s and from Fort Meade to Millersville in the late 1980s and early 1990s. The freeway was completed from Columbia to Clarksville in the mid-1990s and through Fort George G. Meade in 2005. Future plans call for MD 32 to be upgraded to a freeway from Clarksville to West Friendship.
In this case the encoder can fill them with arbitrary useful image data, even from future frames, and thereby serve the same purpose as the b-frames of the MPEG formats. Similar macroblocks can be assigned to one of up to four (even spatially disjoint) segments and thereby share parameters like the reference frame used, quantizer step size, or filter settings. VP8 offers two different adjustable deblocking filters that are integrated into the codec loops (in- loop filtering). Many coding tools use probabilities that are calculated continuously from recent context, starting at each intra frames.
This bijection then expands to the bijection :X = A + B + A + B + ... + Z. Substituting the right hand side for X in Y = B + X gives the bijection :Y = B + A + B + A + ... + Z. Switching every adjacent pair B + A yields :Y = A + B + A + B + ... + Z. Composing the bijection for X with the inverse of the bijection for Y then yields :X = Y. This argument depended on the bijections A + B = B + A and A + (B + C) = (A + B) + C as well as the well-definedness of infinite disjoint union.
In topology and related branches of mathematics, a connected space is a topological space that cannot be represented as the union of two or more disjoint non-empty open subsets. Connectedness is one of the principal topological properties that are used to distinguish topological spaces. A subset of a topological space X is a connected set if it is a connected space when viewed as a subspace of X. Some related but stronger conditions are path connected, simply connected, and n-connected. Another related notion is locally connected, which neither implies nor follows from connectedness.
Each Euler curve divides the plane into two regions or "zones": the interior, which symbolically represents the elements of the set, and the exterior, which represents all elements that are not members of the set. The sizes or shapes of the curves are not important; the significance of the diagram is in how they overlap. The spatial relationships between the regions bounded by each curve (overlap, containment or neither) corresponds to set- theoretic relationships (intersection, subset and disjointness). Curves whose interior zones do not intersect represent disjoint sets.
Before the 1940s, the fashion industry in America was largely disjoint. While there were many events ongoing at any given time, there was no way of learning about these events in one place. The Fashion Calendar was create as an aggregation of these events to not only advertise them and their participants, but also to facilitate scheduling. Once the calendar began to attract a larger following, subscribing to the Calendar indicated that one was an active participant in the fashion community, and became a key entry into American fashion.
The single-player campaign consists of 20 levels, which each level being unlocked after the previous level has been completed. Unlike traditional campaigns in strategy games, levels are completely disjoint from one another, with no "carry over" from one level to the next. Instead, for each level, the player is provided with a fixed set of units (at a certain experience level) and items. The levels are tied together by a brief (but humorous) story which revolves around Maal, the protagonist, and his quest to discover the true nature of the threat to his homeland.
The points x and y, separated by their respective neighbourhoods U and V. Points x and y in a topological space X can be separated by neighbourhoods if there exists a neighbourhood U of x and a neighbourhood V of y such that U and V are disjoint (U\cap V=\empty). X is a Hausdorff space if all distinct points in X are pairwise neighbourhood- separable. This condition is the third separation axiom (after T_0,T_1), which is why Hausdorff spaces are also called T_2 spaces. The name separated space is also used.
A mirrored pair of hinge functions with a knot at x=3.1 Hinge functions are a key part of MARS models. A hinge function takes the form : \max(0,x-c) or : \max(0,c-x) where c is a constant, called the knot. The figure on the right shows a mirrored pair of hinge functions with a knot at 3.1. A hinge function is zero for part of its range, so can be used to partition the data into disjoint regions, each of which can be treated independently.
A poset P is connected if P has rank ≤ 1, or, given any two proper faces F and G, there is a sequence of proper faces :H1, H2, ... ,Hk such that F = H1, G = Hk, and each Hi, i < k, is incident with its successor. The above condition ensures that a pair of disjoint triangles abc and xyz is not a (single) polytope. A poset P is strongly connected if every section of P (including P itself) is connected. With this additional requirement, two pyramids that share just a vertex are also excluded.
Bolo tie circa 1988. Private collection of Hopi silver overlay of Marek Wojciech Ługowski (Lugowski) Sekaquaptewa used a unique combination of traditional silver or gold overlay with contemporary design of his own. Combined, his jewelry comprises stylized or preserved traditional Hopi pottery motifs, as well as lapidary texture and color inserted through the use of semi-precious stones and abalone shell. Using stone and shell is unusual for Hopi silversmiths, and is more typical of the Zuni and other Pueblo people, as well as the ethnographically disjoint Diné (Navajo) silversmiths—usually turquoise.
One example is the "Rank–rank hypergeometric overlap" approach, which is designed to compare ranking of the genes that are at the "top" of two ordered lists of differentially expressed genes. A similar approach is taken by the "Rank Biased Overlap (RBO)", which also implements an adjustable probability, p, to customize the weight assigned at a desired depth of ranking. These approaches have the advantages of addressing disjoint sets, sets of different sizes, and top-weightedness (taking into account the absolute ranking position, which may be ignored in standard non-weighted rank correlation approaches).
In countries with more individualistic views such as America, happiness is viewed as infinite, attainable, and internally experienced. In collectivistic cultures such as Japan, emotions such as happiness are very relational, include a myriad of social and external factors, and reside in shared experiences with other people. Uchida, Townsend, Markus, & Bergseiker (2009) suggest that Japanese contexts reflect a conjoint model meaning that emotions derive from multiple sources and involve assessing the relationship between others and the self. However, in American contexts, a disjoint model is demonstrated through emotions being experienced individually and through self-reflection.
The rule of sum is an intuitive principle stating that if there are a possible outcomes for an event (or ways to do something) and b possible outcomes for another event (or ways to do another thing), and the two events cannot both occur (or the two things can't both be done), then there are a + b total possible outcomes for the events (or total possible ways to do one of the things). More formally, the sum of the sizes of two disjoint sets is equal to the size of their union.
Since chromatic number is an upper bound on the order of the maximum clique, the latter invariant is also at most degeneracy plus one. By using a greedy coloring algorithm on an ordering with optimal coloring number, one can graph color a k-degenerate graph using at most k + 1 colors.; . A k-vertex-connected graph is a graph that cannot be partitioned into more than one component by the removal of fewer than k vertices, or equivalently a graph in which each pair of vertices can be connected by k vertex-disjoint paths.
Having a formal ontology at your disposal, especially when it consists of a Formal upper layer enriched with concrete domain-independent 'middle layer' concepts, can really aid the creation of a domain specific ontology. It allows the modeller to focus on the content of the domain specific ontology without having to worry on the exact higher structure or abstract philosophical framework that gives his ontology a rigid backbone. Disjoint axioms at the higher level will prevent many of the commonly made ontological mistakes made when creating the detailed layer of the ontology.
The fractional version of the incidence coloring was first introduced by Yang in 2007. An r-tuple incidence k-coloring of a graph G is the assignment of r colors to each incidence of graph G from a set of k colors such that the adjacent incidences are given disjoint sets of colors.Yang, D (2012), "Fractional incidence coloring and star arboricity of graphs", Ars Combinatoria - Waterloo then Winnipeg 105, pp. 213–224 By definition, it is obvious that 1-tuple incidence k-coloring is an incidence k-coloring too.
Furthermore, every closed subset of a compact space is compact, and every compact subspace of a Hausdorff space is closed. Closed sets also give a useful characterization of compactness: a topological space X is compact if and only if every collection of nonempty closed subsets of X with empty intersection admits a finite subcollection with empty intersection. A topological space X is disconnected if there exist disjoint, nonempty, open subsets A and B of X whose union is X. Furthermore, X is totally disconnected if it has an open basis consisting of closed sets.
View south along SR 43 at SR 1315 in Buchanan SR 43 has two disjoint sections. SR 43 begins at an intersection with US 29 Business (Main Street) in the town of Altavista. The terminus is just north of the Roanoke River, which forms the boundary between Campbell and Pittsylvania counties. The junction is also adjacent to the crossing of Norfolk Southern Railway's Altavista and Danville district rail lines. SR 43 heads northwest as Bedford Avenue to the town limit at the highway's diamond interchange with US 29 (Wards Road).
The positions containing this symbol must all be in different rows and columns, and furthermore the other symbol in these positions must all be distinct. Hence, when viewed as a pair of Latin squares, the positions containing one symbol in the first square correspond to a transversal in the second square (and vice versa). A given Latin square of order n possesses an orthogonal mate if and only if it has n disjoint transversals. The Cayley table (without borders) of any group of odd order forms a Latin square which possesses an orthogonal mate.
Each node in a disjoint-set forest consists of a pointer and some auxiliary information, either a size or a rank (but not both). The pointers are used to make parent pointer trees, where each node that is not the root of a tree points to its parent. To distinguish root nodes from others, their parent pointers have invalid values, such as a circular reference to the node or a sentinel value. Each tree represents a set stored in the forest, with the members of the set being the nodes in the tree.
The MakeSet operation adds a new element. This element is placed into a new set containing only the new element, and the new set is added to the data structure. If the data structure is instead viewed as a partition of a set, then the MakeSet operation enlarges the set by adding the new element, and it extends the existing partition by putting the new element into a new subset containing only the new element. In a disjoint-set forest, MakeSet initializes the node's parent pointer and the node's size or rank.
The Banach–Tarski paradox shows that there is no way to define volume in three dimensions unless one of the following four concessions is made: # The volume of a set might change when it is rotated. # The volume of the union of two disjoint sets might be different from the sum of their volumes. # Some sets might be tagged "non- measurable", and one would need to check whether a set is "measurable" before talking about its volume. # The axioms of ZFC (Zermelo–Fraenkel set theory with the axiom of choice) might have to be altered.
In singularity theory, there is a different meaning, of a decomposition of a topological space X into disjoint subsets each of which is a topological manifold (so that in particular a stratification defines a partition of the topological space). This is not a useful notion when unrestricted; but when the various strata are defined by some recognisable set of conditions (for example being locally closed), and fit together manageably, this idea is often applied in geometry. Hassler Whitney and René Thom first defined formal conditions for stratification. See Whitney stratification and topologically stratified space.
In an equilateral triangle the area of the Malfatti circles (left) is approximately 1% smaller than the three area-maximizing circles (right). posed the problem of cutting three cylindrical columns out of a triangular prism of marble, maximizing the total volume of the columns. He assumed that the solution to this problem was given by three tangent circles within the triangular cross-section of the wedge. That is, more abstractly, he conjectured that the three Malfatti circles have the maximum total area of any three disjoint circles within a given triangle.
It can be easily shown that conjugacy is an equivalence relation and therefore partitions G into equivalence classes. (This means that every element of the group belongs to precisely one conjugacy class, and the classes Cl(a) and Cl(b) are equal if and only if a and b are conjugate, and disjoint otherwise.) The equivalence class that contains the element a in G is : and is called the conjugacy class' of . The ' of is the number of distinct (nonequivalent) conjugacy classes. All elements belonging to the same conjugacy class have the same order.
A map with twelve pentagonal faces In topology and graph theory, a map is a subdivision of a surface such as the Euclidean plane into interior-disjoint regions, formed by embedding a graph onto the surface and forming connected components (faces) of the complement of the graph. That is, it is a tessellation of the surface. A map graph is a graph derived from a map by creating a vertex for each face and an edge for each pair of faces that meet at a vertex or edge of the embedded graph.
The graph should be drawn in the plane with each vertex as a point, each edge as a curve connecting its two endpoints, and no vertex placed on an edge that it is not incident to. A crossing is counted whenever two edges that are disjoint in the graph have a nonempty intersection in the plane. The question is then, what is the minimum number of crossings in such a drawing? Turán's formulation of this problem is often recognized as one of the first studies of the crossing numbers of graphs.
Another class of problems asks whether copies of a given polyomino can tile a rectangle, and if so, what rectangles they can tile.Golomb, Polyominoes, chapter 8 These problems have been extensively studied for particular polyominoes, and tables of results for individual polyominoes are available. Klarner and Göbel showed that for any polyomino there is a finite set of prime rectangles it tiles, such that all other rectangles it tiles can be tiled by those prime rectangles. Kamenetsky and Cooke showed how various disjoint (called "holey") polyominoes can tile rectangles.
An Eulerian graph is one in which all vertices have even degree; Eulerian graphs may be disconnected. For planar graphs, the properties of being bipartite and Eulerian are dual: a planar graph is bipartite if and only if its dual graph is Eulerian. As Welsh showed, this duality extends to binary matroids: a binary matroid is bipartite if and only if its dual matroid is an Eulerian matroid, a matroid that can be partitioned into disjoint circuits. For matroids that are not binary, the duality between Eulerian and bipartite matroids may break down.
A pair of pants as a plane domain (in blue, with the boundary in red) As said in the lede, a pair of pants is any surface which is homeomorphic to a sphere with three holes, which formally are three open disks with pairwise disjoint closures removed from the sphere. Thus a pair of pants is a compact surface of genus zero with three boundary components. The Euler characteristic of a pair of pants is equal to −1. the only other surface with this property is the punctured torus (a torus minus an open disk).
That is, a proper edge coloring is the same thing as a partition of the graph into disjoint matchings. If the size of a maximum matching in a given graph is small, then many matchings will be needed in order to cover all of the edges of the graph. Expressed more formally, this reasoning implies that if a graph has edges in total, and if at most edges may belong to a maximum matching, then every edge coloring of the graph must use at least different colors., p. 134.
IEEE 802.1CB Frame Replication and Elimination for Reliability (FRER) sends duplicate copies of each frame over multiple disjoint paths, to provide proactive seamless redundancy for control applications that cannot tolerate packet losses. The packet replication can use traffic class and path information to minimize network congestion. Each replicated frame has a sequence identification number, used to re-order and merge frames and to discard duplicates. FRER requires centralized configuration management and needs to be used with 802.1Qcc and 802.1Qca. Industrial fault-tolerance HSR and PRP specified in IEC 62439-3 are supported.
A bipartite graph is a simple graph in which the vertex set can be partitioned into two sets, W and X, so that no two vertices in W share a common edge and no two vertices in X share a common edge. Alternatively, it is a graph with a chromatic number of 2. In a complete bipartite graph, the vertex set is the union of two disjoint sets, W and X, so that every vertex in W is adjacent to every vertex in X but there are no edges within W or X.
All these relations can be explained in terms of their characterisation by forbidden induced subgraphs. A cograph is a graph with no induced path on four vertices, P4, and a threshold graph is a graph with no induced P4, C4 nor 2K2. C4 is a cycle of four vertices and 2K2 is its complement, that is, two disjoint edges. This also explains why threshold graphs are closed under taking complements; the P4 is self-complementary, hence if a graph is P4-, C4\- and 2K2-free, its complement is as well.
In the disjoint-set data structure, m represents the number of operations while n represents the number of elements; in the minimum spanning tree algorithm, m represents the number of edges while n represents the number of vertices. Several slightly different definitions of exist; for example, is sometimes replaced by n, and the floor function is sometimes replaced by a ceiling. Other studies might define an inverse function of one where m is set to a constant, such that the inverse applies to a particular row. The inverse of the Ackermann function is primitive recursive.
It also has a countable dense subset, namely the set of rational numbers. It is a theorem that any linear continuum with a countable dense subset and no maximum or minimum element is order-isomorphic to the real line. The real line also satisfies the countable chain condition: every collection of mutually disjoint, nonempty open intervals in is countable. In order theory, the famous Suslin problem asks whether every linear continuum satisfying the countable chain condition that has no maximum or minimum element is necessarily order- isomorphic to .
Bucklodge Road was extended as a concrete road to the underpass of the railroad at Boyds east of the southern end of MD 121 in 1928. A disjoint segment of macadam road was laid from MD 118 at Old Germantown west to Schaeffer Road in 1930 and extended west to near Little Seneca Creek by 1933. A third section of macadam road was also built from the modern MD 117-MD 124 intersection west to the hamlet of Clopper east of Great Seneca Creek by 1933. All three segments were later marked as MD 117.
In mathematics, two positive (or signed or complex) measures μ and ν defined on a measurable space (Ω, Σ) are called singular if there exist two disjoint sets A and B in Σ whose union is Ω such that μ is zero on all measurable subsets of B while ν is zero on all measurable subsets of A. This is denoted by \mu \perp u. A refined form of Lebesgue's decomposition theorem decomposes a singular measure into a singular continuous measure and a discrete measure. See below for examples.
The Swiss-cheese operad. The Swiss-cheese operad is a two-colored topological operad defined in terms of configurations of disjoint n-dimensional disks inside a unit n-semidisk and n-dimensional semidisks, centered at the base of the semidisk and sitting inside the unit semidisk. The operadic composition comes from gluing configurations of "little" disks inside the unit disk into the "little" disks in another unit semidisk and configurations of "little" disks and semidisks inside the unit semidisk into the other unit semidisk. The Swiss-cheese operad was defined by Alexander A. Voronov.
It is rich in resonant alliterations, internal and disjoint rhymes without however becoming overly musical and without impeding its central function – its narrative propensity". - John Fizer, Professor-Emeritus at Rutgers University "With a foundation in the international community of letters, Winter Letters reminds us of the universal power of art". -Judith Baumel about the collection Winter Letters "Whoever believes in the power of language and of imagery to first stun then capture rather than explain reality will also love these poems. Makhno gift is his meticulous riffs fed by memory and imagination.
It follows from Waldhausen's Theorem that every reducible splitting of an irreducible manifold is stabilized. A Heegaard splitting is weakly reducible if there are disjoint essential simple closed curves \alpha and \beta on H where \alpha bounds a disk in V and \beta bounds a disk in W. A splitting is strongly irreducible if it is not weakly reducible. A Heegaard splitting is minimal or minimal genus if there is no other splitting of the ambient three-manifold of lower genus. The minimal value g of the splitting surface is the Heegaard genus of M.
Morton and Hilbert curves of level 6 (45=1024 cells in the recursive square partition) plotting each address as different color in the RGB standard, and using Geohash labels. The neighborhoods have similar colors, but each curve offers different pattern of grouping similars in smaller scales. If a curve is not injective, then one can find two intersecting subcurves of the curve, each obtained by considering the images of two disjoint segments from the curve's domain (the unit line segment). The two subcurves intersect if the intersection of the two images is non-empty.
For the precise definition, suppose that is a set of nodes. Using the reflexivity of partial orders, we can identify any tree on a subset of with its partial order - a subset of . The set of all relations that form a well-founded tree on a subset of is defined in stages , so that }. For each ordinal number , let belong to the -th stage if and only if is equal to : where is a subset of such that elements of are pairwise disjoint, and is a node that does not belong to .
A projective line in this plane consists of all projective points (which are lines) contained in a plane passing through . As the intersection of two planes passing through is a line passing through , the intersection of two distinct projective lines consists of a single projective point. The plane defines a projective line which is called the line at infinity of . By identifying each point of with the corresponding projective point, one can thus say that the projective plane is the disjoint union of and the (projective) line at infinity.
In order to identify the location of HGT events, genome spectral approaches decompose a gene tree into substructures (such as bipartitions or quartets) and identify those that are consistent or inconsistent with the species tree. Bipartitions Removing one edge from a reference tree produces two unconnected sub-trees, each a disjoint set of nodes--a bipartition. If a bipartition is present in both the gene and the species trees, it is compatible; otherwise, it is conflicting. These conflicts can indicate an HGT event or may be the result of uncertainty in gene tree inference.
The goal of cluster analysis is to group elements into disjoint subsets, or clusters, based on similarity between elements, so that elements in the same cluster are highly similar to each other (homogeneity), while elements from different clusters have low similarity to each other (separation). Similarity graph is one of the models to represent the similarity between elements, and in turn facilitate generating of clusters. To construct a similarity graph from similarity data, represent elements as vertices, and elicit edges between vertices when the similarity value between them is above some threshold.
An oriented graph is a finite directed graph obtained from a simple undirected graph by assigning an orientation to each edge. Equivalently, it is a directed graph that has no self-loops, no parallel edges, and no two-edge cycles. The first neighborhood of a vertex v (also called its open neighborhood) consists of all vertices at distance one from v, and the second neighborhood of v consists of all vertices at distance two from v. These two neighborhoods form disjoint sets, neither of which contains v itself.
A set of unary relations Pi for i in some set I is called independent if for every two disjoint finite subsets A and B of I there is some element x such that Pi(x) is true for i in A and false for i in B. Independence can be expressed by a set of first-order statements. The theory of a countable number of independent unary relations is complete, but has no atomic models. It is also an example of a theory that is superstable but not totally transcendental.
The pseudocode below determines the lowest common ancestor of each pair in P, given the root r of a tree in which the children of node n are in the set n.children. For this offline algorithm, the set P must be specified in advance. It uses the MakeSet, Find, and Union functions of a disjoint-set forest. MakeSet(u) removes u to a singleton set, Find(u) returns the standard representative of the set containing u, and Union(u,v) merges the set containing u with the set containing v.
SuperPascal is secure in that it should enable its compiler and runtime system to detect as many cases as possible in which the language concepts break down and produce meaningless results. SuperPascal imposes restrictions on the use of variables that enable a single-pass compiler to check that parallel processes are disjoint, even if the processes use procedures with global variables, eliminating time-dependent errors. Several features in Pascal were ambiguous or insecure and were omitted from SuperPascal, such as labels and `goto` statements, pointers and forward declarations.
Later authors changed the interpretation, commonly reading it as exclusive or, or in set theory terms symmetric difference; this step means that addition is always defined. In fact, there is the other possibility, that + should be read as disjunction. This other possibility extends from the disjoint union case, where exclusive or and non-exclusive or both give the same answer. Handling this ambiguity was an early problem of the theory, reflecting the modern use of both Boolean rings and Boolean algebras (which are simply different aspects of one type of structure).
The parallel composition Pc = Pc(X,Y) of two TTGs X and Y is a TTG created from the disjoint union of graphs X and Y by merging the sources of X and Y to create the source of Pc and merging the sinks of X and Y to create the sink of Pc. The series composition Sc = Sc(X,Y) of two TTGs X and Y is a TTG created from the disjoint union of graphs X and Y by merging the sink of X with the source of Y. The source of X becomes the source of Sc and the sink of Y becomes the sink of Sc. A two-terminal series- parallel graph (TTSPG) is a graph that may be constructed by a sequence of series and parallel compositions starting from a set of copies of a single- edge graph K2 with assigned terminals. Definition 1. Finally, a graph is called series-parallel (SP-graph), if it is a TTSPG when some two of its vertices are regarded as source and sink. In a similar way one may define series-parallel digraphs, constructed from copies of single-arc graphs, with arcs directed from the source to the sink.
Kulpa obtained his M.Sc. in Electronics at the Electronics Faculty of the Warsaw Technical University, and his PhD in Computer Science from the Institute of Computer Science of the Polish Academy of Sciences, Warsaw, Poland. After his studies Kulpa started working for the Institute of Automatic Control of the Polish Academy of Sciences (PAS) in the late 1960s. There he met the computer scientist Ryszard S. Michalski. With Michalski he wrote his first English articles, entitled "A System of Programs for the Synthesis of Switching Circuits Using the Method of Disjoint Stars" presented at the 1971 IFIP Congress in Ljubljana, Yugoslavia.
Discovery Net workflows are represented and stored using DPML (Discovery Process Markup Language), an XML-based representation language for workflow graphs supporting both a data flow model of computation (for analytical workflows) and a control flow model (for orchestrating multiple disjoint workflows). As with most modern workflow systems, the system supported a drag-and-drop visual interface enabling users to easily construct their applications by connecting nodes together. Within DPML, each node in a workflow graph represents an executable component (e.g. a computational tool or a wrapper that can extract data from a particular data source).
His literary legacy is punctuated by his modernist tendencies evinced throughout his oeuvre. His poems reveal the desolate internal landscape of modern humanity and, as in “Crow's eye view poem” (Ogamdo si je1ho), utilize an anti-realist technique to condense the themes of anxiety and fear. His stories disjoint the form of traditional fiction to show the conditions of the lives of modern people. “Wings” (Nalgae), for example, utilizes a stream-of-consciousness technique to express these conditions in terms of the alienation of modern people, who are fragmented commodities unable to relate to quotidian (daily) realities.
Maryland Route 316 (MD 316) is a state highway in the U.S. state of Maryland. Known as Appleton Road, the highway runs from MD 279 in Elkton north to MD 277 near Elk Mills in northeastern Cecil County. MD 316 was constructed in the early 1910s from Elkton to Elk Mills and in the early 1920s north of Elk Mills. In the early 1960s, the disjoint northern segment of the highway was transferred to the county and the highway's present southern terminus was established when MD 279 moved to a new alignment north of Elkton in the early 1960s.
A cluster graph with clusters (complete subgraphs) of sizes 1, 2, 3, 4, 4, 5, and 6 In graph theory, a branch of mathematics, a cluster graph is a graph formed from the disjoint union of complete graphs. Equivalently, a graph is a cluster graph if and only if it has no three-vertex induced path; for this reason, the cluster graphs are also called P3-free graphs. They are the complement graphs of the complete multipartite graphsCluster graphs, Information System on Graph Classes and their Inclusions, accessed 2016-06-26. and the 2-leaf powers..
Swanson linking example diagram Swanson linking is a term proposed in 2003Stegmann J, Grohmann G. Hypothesis generation guided by co-word clustering. Scientometrics. 2003;56:111–135. As quoted by Bekhuis that refers to connecting two pieces of knowledge previously thought to be unrelated. For example, it may be known that illness A is caused by chemical B, and that drug C is known to reduce the amount of chemical B in the body. However, because the respective articles were published separately from one another (called "disjoint data"), the relationship between illness A and drug C may be unknown.
Quorums are used to guarantee the safety properties of Gbcast by ensuring that there is a single globally agreed-upon sequence of group views and multicast messages and by preventing progress in more than one partition if a group becomes fragmented into two or more partitions (disjoint subsets of members that can communicate with other members of their subsets, but not with members of other subsets). Quorums are defined for a specific view. Given view i with n members {A,B,C….}, a quorum of the view is any majority subset of the members of that view.
In the above example, each vertex of H has exactly 2 preimages in C. Hence H is a 2-fold cover or a double cover of C. For any graph G, it is possible to construct the bipartite double cover of G, which is a bipartite graph and a double cover of G. The bipartite double cover of G is the tensor product of graphs G × K2: :File:Covering-graph-2.svg If G is already bipartite, its bipartite double cover consists of two disjoint copies of G. A graph may have many different double covers other than the bipartite double cover.
A section of TM is a vector field on M, and the dual bundle to TM is the cotangent bundle, which is the disjoint union of the cotangent spaces of M . By definition, a manifold M is parallelizable if and only if the tangent bundle is trivial. By definition, a manifold M is framed if and only if the tangent bundle TM is stably trivial, meaning that for some trivial bundle E the Whitney sum TM\oplus E is trivial. For example, the n-dimensional sphere Sn is framed for all n, but parallelizable only for (by results of Bott-Milnor and Kervaire).
The university of Pristina was founded in the Socialist Autonomous Province of Kosovo, Socialist Republic of Serbia, Yugoslavia, in Pristina, the first academic year being 1969-1970, and functioning until 1999. However, because of political upheaval, war, consecutive expulsion of faculty of one ethnicity or the other, extensive differences between the ethnicities, it separated into two disjoint institutions using the same name, albeit simply to reflect ethnic identity. Albanian-language activity continues to this day in Pristina, whilst the Serbian one, Univezitet u Prištini, has been located in Northern Mitrovica, where it still maintains its place in the Serbian Education System.
A permutation can be decomposed into one or more disjoint cycles, that is, the orbits, which are found by repeatedly tracing the application of the permutation on some elements. For example, the permutation \sigma defined by \sigma(7) = 7 has a 1-cycle, (\,7\,) while the permutation \pi defined by \pi(2) = 3 and \pi(3) = 2 has a 2-cycle (\,2\,3\,) (for details on the syntax, see below). In general, a cycle of length k, that is, consisting of k elements, is called a k-cycle. An element in a 1-cycle (\,x\,) is called a fixed point of the permutation.
The zeroth law establishes thermal equilibrium as an equivalence relationship. An equivalence relationship on a set (such as the set of all systems each in its own state of internal thermodynamic equilibrium) divides that set into a collection of distinct subsets ("disjoint subsets") where any member of the set is a member of one and only one such subset. In the case of the zeroth law, these subsets consist of systems which are in mutual equilibrium. This partitioning allows any member of the subset to be uniquely "tagged" with a label identifying the subset to which it belongs.
Since V is an unknotted solid torus, S^3 \setminus V is a tubular neighbourhood of an unknot J. The 2-component link K' \cup J together with the embedding f is called the pattern associated to the satellite operation. A convention: people usually demand that the embedding f \colon V \to S^3 is untwisted in the sense that f must send the standard longitude of V to the standard longitude of f(V). Said another way, given any two disjoint curves c_1,c_2 \subset V, f preserves their linking numbers i.e.: lk(f(c_1),f(c_2))=lk(c_1,c_2).
A space in which all components are one-point sets is called totally disconnected. Related to this property, a space X is called totally separated if, for any two distinct elements x and y of X, there exist disjoint open sets U containing x and V containing y such that X is the union of U and V. Clearly, any totally separated space is totally disconnected, but the converse does not hold. For example take two copies of the rational numbers Q, and identify them at every point except zero. The resulting space, with the quotient topology, is totally disconnected.
Additionally, however, for events A and B, let the conditional event A → B be represented as the following infinite union of disjoint sets: :[(A ∩ B) × Ω × Ω × Ω × …] ∪ :[A′ × (A ∩ B) × Ω × Ω × Ω × …] ∪ :[A′ × A ′ × (A ∩ B) × Ω × Ω × Ω × …] ∪ …. The motivation for this representation of conditional events will be explained shortly. Note that the construction can be iterated; A and B can themselves be conditional events. Intuitively, unconditional event A ought to be representable as conditional event Ω → A. And indeed: because Ω ∩ A = A and Ω′ = ∅, the infinite union representing Ω → A reduces to A × Ω × Ω × Ω × ….
The natural sum of α and β is often denoted by α⊕β or α#β, and the natural product by α⊗β or α⨳β. The natural operations come up in the theory of well partial orders; given two well partial orders S and T, of types (maximum linearizations) o(S) and o(T), the type of the disjoint union is o(S)⊕o(T), while the type of the direct product is o(S)⊗o(T).D. H. J. De Jongh and R. Parikh, Well-partial orderings and hierarchies, Indag. Math. 39 (1977), 195–206.
Available here One may take this relation as a definition of the natural operations by choosing S and T to be ordinals α and β; so α⊕β is the maximum order type of a total order extending the disjoint union (as a partial order) of α and β; while α⊗β is the maximum order type of a total order extending the direct product (as a partial order) of α and β.Philip W. Carruth, Arithmetic of ordinals with applications to the theory of ordered Abelian groups, Bull. Amer. Math. Soc. 48 (1942), 262–271. See Theorem 1.
The covering lemma can be used as intermediate step in the proof of the following basic form of the Vitali covering theorem. Actually, a little more is needed, namely the precised form of the covering lemma obtained in the "proof of the infinite version". :Theorem. For every subset E of Rd and every Vitali cover of E by a collection F of closed balls, there exists a disjoint subcollection G which covers E up to a Lebesgue-negligible set. Without loss of generality, one can assume that all balls in F are nondegenerate and have radius ≤ 1\.
If all the radii are not random, but common positive constant, then the resulting model is known as the Gilbert disk (Boolean) model.Balister, Paul and Sarkar, Amites and Bollobás, Béla, Percolation, connectivity, coverage and colouring of random geometric graphs, Handbook of Large-Scale Random Networks, 117–142, 2008 A Boolean model as a coverage model in a wireless network. Simulation of four Poisson–Boolean (constant-radius or Gilbert disk) models as the density increases with largest clusters in red. Instead of placing disks on the plane, one may assign a disjoint (or non-overlapping) subregion to each node.
Then the plane is partitioned into a collection of disjoint subregions. For example, each subregion may consist of the collection of all the locations of this plane that are closer to some point of the underlying point pattern than any other point of the point pattern. This mathematical structure is known as a Voronoi tessellation and may represent, for example, the association cells in a cellular network where users associate with the closest base station. Instead of placing a disk or a Voronoi cell on a point, one could place a cell defined from the information theoretic channels described above.
There are several competing theories trying to make sense of this apparent paradox. Doctor James White postulates that people with higher IQs are more critical of information and thus less likely to accept facts at face value. While marketing campaigns against drugs may deter individuals with lower IQs from using drugs with disjoint arguments or over-exaggeration of negative consequences, people with a higher IQ will seek to verify the validity of such claims in their immediate environment. White also eludes to an often-overlooked problem of people with higher IQ, the lack of adequate challenges and intellectual stimulation.
A second, disjoint portion of MD 45 was assigned from north of Parkton to the Pennsylvania state line when the Baltimore-Harrisburg Expressway was completed between Parkton and the York area in 1959. When the final section of the US 111 freeway was completed between Hereford and Parkton in 1960, MD 45 was assigned to the stretch of York Road between those communities. US 111 and I-83 were co-signed on the freeway between Towson and Harrisburg until US 111 was decommissioned in 1963; the MD 45 designation was extended south into Baltimore to US 1 at that time.
A system of imprimitivity is homogeneous of multiplicity n, where 1 ≤ n ≤ ω if and only if the corresponding projection-valued measure π on X is homogeneous of multiplicity n. In fact, X breaks up into a countable disjoint family {Xn} 1 ≤ n ≤ ω of Borel sets such that π is homogeneous of multiplicity n on Xn. It is also easy to show Xn is G invariant. Lemma. Any system of imprimitivity is an orthogonal direct sum of homogeneous ones. It can be shown that if the action of G on X is transitive, then any system of imprimitivity on X is homogeneous.
Austroasiatic languages The homeland of the Austroasiatic languages (e.g. Vietnamese, Cambodian) which are found from Southeast Asia to India is hypothesized to be located in "the hills of southern Yunnan in China," between 4000 BCE and 2000 BCE, with influences from Aryan and Dravidian languages at the Western edge of its expanse in India, and influence from Chinese at the Eastern edge of the regions where it is found. The disjoint distribution of Austroasiatic languages suggests that they were once spoken in most of the areas where the Kra–Dai languages (e.g. Thai, Lao) are now dominant.
The point location problem is a fundamental topic of computational geometry. It finds applications in areas that deal with processing geometrical data: computer graphics, geographic information systems (GIS), motion planning, and computer aided design (CAD). In its most general form, the problem is, given a partition of the space into disjoint regions, to determine the region where a query point lies. As an example application, each time one clicks a mouse to follow a link in a web browser, this problem must be solved in order to determine which area of the computer screen is under the mouse pointer.
In mathematics, economics and computer science, particularly in the fields of combinatorics, game theory and algorithms, the stable-roommate problem (SRP) is the problem of finding a stable matching for an even-sized set. A matching is a separation of the set into disjoint pairs ("roommates"). The matching is stable if there are no two elements which are not roommates and which both prefer each other to their roommate under the matching. This is distinct from the stable-marriage problem in that the stable-roommates problem allows matches between any two elements, not just between classes of "men" and "women".
A cut set is formed by allowing all but one of the tree branches to be short circuit. The cut set consists of the tree branch which was not short-circuited and any of the links which are not short-circuited by the other tree branches. A cut set of a graph produces two disjoint subgraphs, that is, it cuts the graph into two parts, and is the minimum set of branches needed to do so. The set of network equations are formed by equating the node pair voltages to the algebraic sum of the cut set branch voltages.
Thus, every properly embedded compact surface without 2-sphere components is related to an incompressible surface through a sequence of compressions. Sometimes we drop the condition that S be compressible. If D were to bound a disk inside S (which is always the case if S is incompressible, for example), then compressing S along D would result in a disjoint union of a sphere and a surface homeomorphic to S. The resulting surface with the sphere deleted might or might not be isotopic to S, and it will be if S is incompressible and M is irreducible.
Let A, B, and C designate three distinct (but not necessarily disjoint) sets of attributes of a relation. Suppose all three of the following conditions hold: # A → B # It is not the case that B → A # B → C Then the functional dependency A → C (which follows from 1 and 3 by the axiom of transitivity) is a transitive dependency. In database normalization, one of the important features of third normal form is that it excludes certain types of transitive dependencies. E.F. Codd, the inventor of the relational model, introduced the concepts of transitive dependence and third normal form in 1971.
So the family of separable states is the closed convex hull of pure product states. We will make use of the following variant of Hahn–Banach theorem: Theorem Let S_1 and S_2 be disjoint convex closed sets in a real Banach space and one of them is compact, then there exists a bounded functional f separating the two sets. This is a generalization of the fact that, in real Euclidean space, given a convex set and a point outside, there always exists an affine subspace separating the two. The affine subspace manifests itself as the functional f.
A ray, in an infinite graph, is a semi-infinite path: a connected infinite subgraph in which one vertex has degree one and the rest have degree two. defined two rays r0 and r1 to be equivalent if there exists a ray r2 that includes infinitely many vertices from each of them. This is an equivalence relation, and its equivalence classes (sets of mutually equivalent rays) are called the ends of the graph. defined a thick end of a graph to be an end that contains infinitely many rays that are pairwise disjoint from each other.
He went on to tackle the problem of large sets of disjoint Steiner triple systems. Zhu Lie (), a professor of mathematics at Soochow University working also in combinatorial mathematics, realized the importance of his work and suggested that he submit it to the international journal Journal of Combinatorial Theory, Series A. He wrote to its editorial board that he had essentially solved the problem, and the editors replied to him that if what he said was true, it would be a major achievement. (Many leaders in the field had worked on the problem starting from in 1917.
To any group is associated its Burnside ring which uses a ring to describe the various ways the group can act on a finite set. The Burnside ring's additive group is the free abelian group whose basis are the transitive actions of the group and whose addition is the disjoint union of the action. Expressing an action in terms of the basis is decomposing an action into its transitive constituents. The multiplication is easily expressed in terms of the representation ring: the multiplication in the Burnside ring is formed by writing the tensor product of two permutation modules as a permutation module.
The distinction between star-domination and usual domination is more substantial when their fractional variants are considered. A domatic partition is a partition of the vertices into disjoint dominating sets. The domatic number is the maximum size of a domatic partition. An eternal dominating set is a dynamic version of domination in which a vertex v in dominating set D is chosen and replaced with a neighbor u (u is not in D) such that the modified D is also a dominating set and this process can be repeated over any infinite sequence of choices of vertices v.
An incidence structure consists of a set whose elements are called points, a disjoint set whose elements are called lines and an incidence relation between them, that is, a subset of whose elements are called flags.Technically this is a rank two incidence structure, where rank refers to the number of types of objects under consideration (here, points and lines). Higher ranked structures are also studied, but several authors limit themselves to the rank two case, and we shall do so here. If is a flag, we say that is incident with or that is incident with (the relation is symmetric), and write .
In general, a closed manifold need not be a boundary: cobordism theory is the study of the difference between all closed manifolds and those that are boundaries. The theory was originally developed by René Thom for smooth manifolds (i.e., differentiable), but there are now also versions for piecewise linear and topological manifolds. A cobordism between manifolds M and N is a compact manifold W whose boundary is the disjoint union of M and N, \partial W=M \sqcup N. Cobordisms are studied both for the equivalence relation that they generate, and as objects in their own right.
A topological quantum field theory is a monoidal functor from a category of cobordisms to a category of vector spaces. That is, it is a functor whose value on a disjoint union of manifolds is equivalent to the tensor product of its values on each of the constituent manifolds. In low dimensions, the bordism question is relatively trivial, but the category of cobordism is not. For instance, the disk bounding the circle corresponds to a null-ary operation, while the cylinder corresponds to a 1-ary operation and the pair of pants to a binary operation.
This puts them into set B. We then take the points belonging to B and operate on them with σ2. They will now still be in B, but the set of these points will be disjoint from the previous set. We proceed in this manner, using σ3τ on the A points from C2 (after centring it) and σ4 on its B points, and so on. In this way, we have mapped all points from the big figure (except some fixed points) in a one-to-one manner to B type points not too far from the centre, and within the big figure.
In the design of algorithms, partition refinement is a technique for representing a partition of a set as a data structure that allows the partition to be refined by splitting its sets into a larger number of smaller sets. In that sense it is dual to the union-find data structure, which also maintains a partition into disjoint sets but in which the operations merge pairs of sets. Partition refinement forms a key component of several efficient algorithms on graphs and finite automata, including DFA minimization, the Coffman–Graham algorithm for parallel scheduling, and lexicographic breadth- first search of graphs....
A Schottky group is called classical if all the disjoint Jordan curves corresponding to some set of generators can be chosen to be circles. gave an indirect and non-constructive proof of the existence of non-classical Schottky groups, and gave an explicit example of one. It has been shown by that all finitely generated classical Schottky groups have limit sets of Hausdorff dimension bounded above strictly by a universal constant less than 2. Conversely, has proved that there exists a universal lower bound on the Hausdorff dimension of limit sets of all non- classical Schottky groups.
The line graph of the Petersen graph, another graph of this type, has a property analogous to the cages in that it is the smallest possible graph in which the largest clique has three vertices, each vertex is in exactly two edge-disjoint cliques, and the shortest cycle with edges from distinct cliques has length five. A more complicated expansion process applies to planar graphs. Let G be a planar graph embedded in the plane in such a way that every face is a quadrilateral, such as the graph of a cube. Necessarily, if G has n vertices, it has n-2 faces.
The glowworm depends on a variable local-decision domain, which is bounded above by a circular sensor range, to identify its neighbors and compute its movements. Each glowworm selects a neighbor that has a Luciferin value more than its own, using a probabilistic mechanism, and moves towards it. These movements that are based only on local information enable the swarm of glowworms to split into disjoint subgroups, exhibit simultaneous taxis-behavior towards, and rendezvous at multiple optimums (not necessarily equal) of a given multi-modal function. The algorithm was tested on a custom designed system of robots called Kinbots.
City greatly improved in the second half and equalised through Jones following good work by Walters and Pennant. Stoke pressed for the winning goal and hit the crossbar twice as West Ham held out for a 1–1 draw. After the win in the League Cup against Fulham Stoke made the long trip north to face Newcastle United at a strange 4.10pm kick off on a Sunday. Stoke gave a surprise start to Salif Diao while Faye and Delap dropped to the bench, these changes seemed to disjoint Stoke as they put in an abject first half performance.
As an example of an application of ACω, here is a proof (from ZF + ACω) that every infinite set is Dedekind-infinite: :Let X be infinite. For each natural number n, let An be the set of all 2n-element subsets of X. Since X is infinite, each An is nonempty. The first application of ACω yields a sequence (Bn : n = 0,1,2,3,...) where each Bn is a subset of X with 2n elements. :The sets Bn are not necessarily disjoint, but we can define :: C0 = B0 ::Cn = the difference between Bn and the union of all Cj, j < n.
An agreement forest for two unrooted -trees and is a partition } of the taxon set satisfying the following conditions: # and are isomorphic for every and # the subtrees in and are vertex-disjoint subtrees of and , respectively. The set partition } is identified with the forest of restricted subtrees }, with either or (the choice of it begin irrelevant because of condition 1). Therefore, an agreement forest can either be seen as a partition of the taxon set or as a forest (in the classical graph-theoretic sense) of restricted subtrees. The size of an agreement forest is simply its number of components.
Firstly, the ranges of the encryption function under any two distinct keys are disjoint (with overwhelming probability). The second property says that it can be checked efficiently whether a given ciphertext has been encrypted under a given key. With these two properties the receiver, after obtaining the labels for all circuit-input wires, can evaluate each gate by first finding out which of the four ciphertexts has been encrypted with his label keys, and then decrypting to obtain the label of the output wire. This is done obliviously as all the receiver learns during the evaluation are encodings of the bits.
The Colin de Verdière graph invariant is an integer defined for any graph using algebraic graph theory. The graphs with Colin de Verdière graph invariant at most μ, for any fixed constant μ, form a minor-closed family, and the first few of these are well-known: the graphs with μ ≤ 1 are the linear forests (disjoint unions of paths), the graphs with μ ≤ 2 are the outerplanar graphs, and the graphs with μ ≤ 3 are the planar graphs. As conjectured and proved, the graphs with μ ≤ 4 are exactly the linklessly embeddable graphs. A linkless apex graph that is not YΔY reducible.
A handlebody can be defined as an orientable 3-manifold-with-boundary containing pairwise disjoint, properly embedded 2-discs such that the manifold resulting from cutting along the discs is a 3-ball. It's instructive to imagine how to reverse this process to get a handlebody. (Sometimes the orientability hypothesis is dropped from this last definition, and one gets a more general kind of handlebody with a non- orientable handle.) The genus of a handlebody is the genus of its boundary surface. Up to homeomorphism, there is exactly one handlebody of any non- negative integer genus.
The following example shows how Suurballe's algorithm finds the shortest pair of disjoint paths from A to F. 900px Figure A illustrates a weighted graph G. Figure B calculates the shortest path P1 from A to F (A–B–D–F). Figure C illustrates the shortest path tree T rooted at A, and the computed distances from A to every vertex (). Figure D shows the residual graph Gt with the updated cost of each edge and the edges of path 'P1 reversed. Figure E calculates path P2 in the residual graph Gt (A–C–D–B–E–F).
The weight of any path from to in the modified system of weights equals the weight in the original graph, minus . Therefore, the shortest two disjoint paths under the modified weights are the same paths as the shortest two paths in the original graph, although they have different weights. Suurballe's algorithm may be seen as a special case of the successive shortest paths method for finding a minimum cost flow with total flow amount two from to . The modification to the weights does not affect the sequence of paths found by this method, only their weights.
A shallow minor of a graph G is a minor in which the edges of G that were contracted to form the minor form a collection of disjoint subgraphs with low diameter. Shallow minors interpolate between the theories of graph minors and subgraphs, in that shallow minors with high depth coincide with the usual type of graph minor, while the shallow minors with depth zero are exactly the subgraphs.. They also allow the theory of graph minors to be extended to classes of graphs such as the 1-planar graphs that are not closed under taking minors., pp. 319–321.
Meta-positron emission tomography (PET) analysis has lent support toward a division of the hippocampus between caudal and rostral regions. Scans have demonstrated a uniform variation in blood flow distribution within the hippocampus (and the medial temporal lobe broadly) during the separate processes of episodic encoding and retrieval. In the hippocampal encoding/retrieval (HIPER) model, episodic encoding is found to take place within the rostral region of the hippocampus whereas retrieval takes place in the caudal region. However, the divide between these regions need not be disjoint, as functional magnetic resonance imaging (fMRI) data has demonstrated encoding processes occurring within the caudal region.
The intersection of Cs and Ct is the set of lines on F that cuts the disjoint lines Ls and Lt. Consider the linear span of Ls and Lt : it is an hyperplane into P4 that cuts F into a smooth cubic surface. By well known results on a cubic surface, the number of lines that cuts two disjoints lines is 5, thus we get (Cs) 2 =Cs Ct=5. As K is numerically equivalent to 3Cs, we obtain K 2 =45. c) The natural composite map: S -> G(2,5) -> P9 is the canonical map of S. It is an embedding.
Ending laminations were introduced by . Suppose that a hyperbolic 3-manifold has a geometrically tame end of the form S×[0,1) for some compact surface S without boundary, so that S can be thought of as the "points at infinity" of the end. The ending lamination of this end is (roughly) a lamination on the surface S, in other words a closed subset of S that is written as the disjoint union of geodesics of S. It is characterized by the following property. Suppose that there is a sequence of closed geodesics on S whose lifts tends to infinity in the end.
In cooperative game theory, a hedonic game Haris Aziz and Rahul Savani, "Hedonic Games". Chapter 15 in: (also known as a hedonic coalition formation game) is a game that models the formation of coalitions (groups) of players when players have preferences over which group they belong to. A hedonic game is specified by giving a finite set of players, and, for each player, a preference ranking over all coalitions (subsets) of players that the player belongs to. The outcome of a hedonic game consists of a partition of the players into disjoint coalitions, that is, each player is assigned a unique group.
A maximum independent set in a penny graph is a subset of the pennies, no two of which touch each other. Finding maximum independent sets is NP-hard for arbitrary graphs, and remains NP-hard on penny graphs. It is an instance of the maximum disjoint set problem, in which one must find large subsets of non- overlapping regions of the plane. However, as with planar graphs more generally, Baker's technique provides a polynomial-time approximation scheme for this problem.. In 1983, Paul Erdős asked for the largest number such that every -vertex penny graph has an independent set of at least vertices.
During learning, fusion ART formulates recognition categories of input patterns across multiple channels. The knowledge that fusion ART discovers during learning, is compatible with symbolic rule-based representation. Specifically, the recognition categories learned by the F_2 category nodes are compatible with a class of IF-THEN rules that maps a set of input attributes (antecedents) in one pattern channel to a disjoint set of output attributes (consequents) in another channel. Due to this compatibility, at any point of the incremental learning process, instructions in the form of IF-THEN rules can be readily translated into the recognition categories of a fusion ART system.
Skorobogatov has developed a technique capable of detecting malicious insertions into chips. New York University Tandon School of Engineering researchers have developed a way to corroborate a chip's operation using verifiable computing whereby "manufactured for sale" chips contain an embedded verification module that proves the chip's calculations are correct and an associated external module validates the embedded verification module. Another technique developed by researchers at University College London (UCL) relies on distributing trust between multiple identical chips from disjoint supply chains. Assuming that at least one of those chips remains honest the security of the device is preserved.
Conversely, from any bramble of order k, one may construct a haven of the same order, by defining β(X) (for each choice of X) to be the X-flap that includes all of the subgraphs in the bramble that are disjoint from X. The requirement that the subgraphs in the bramble all touch each other can be used to show that this X-flap exists, and that all of the flaps β(X) chosen in this way touch each other. Thus, a graph has a bramble of order k if and only if it has a haven of order k.
Basic Formal Ontology (BFO) is a top-level ontology developed by Barry Smith and his associates for the purposes of promoting interoperability among domain ontologies built in its terms through a process of downward population. A guide to building BFO-conformant domain ontologies was published by MIT Press in 2015. The standard is currently under development. The structure of BFO is based on a division of entities into two disjoint categories of continuant and occurrent, the former comprehending for example objects and spatial regions, the latter comprehending processes conceived as extended through (or as spanning) time.
If the task to be scheduled are numbered from 1 to n, let t_i denote the repeat time for task i. Then the density of a pinwheel scheduling problem is \textstyle\sum 1/t_i. For a solution to exist, it is necessary that the density is at most 1. This condition on density is also sufficient for a schedule to exist in the special case that all repeat times are multiples of each other (for instance, if all are powers of two), because in this case one can solve the problem using a disjoint covering system.
County Road 13 is Brockton Lane in Dayton between the County Road 81 / County Road 101 junction and County Road 12. This road was previously State Highway 101 (MN 101) until Highway 101 was rerouted to its current route about 1 mile west in 1968. County Road 14 is Zane Avenue North from Brooklyn Boulevard in Brooklyn Park to West River Road in Champlin. County Road 14 originally extended southeast along Brooklyn Boulevard, through Brooklyn Center, to the Minneapolis city line, before old State Highway 152 (MN 152) was created in 1934. County Road 15 is divided into two disjoint sections.
The first segment was built in 1925 and 1926; the road was completed to Comus in 1927 and 1928. A disjoint segment of MD 109 was built as a concrete road along Westerly Road between Edwards Ferry Road and Willard Road on the southern edge of Poolesville in 1929 and 1930. The county highway between MD 95 at Comus and U.S. Route 240 (now MD 355) at Hyattstown was reconstructed as a macadam road by the Maryland State Roads Commission between 1936 and 1938. MD 109's bridge across Little Bennett Creek and its interchange with Washington National Pike (now I-270) were constructed between 1951 and 1953.
Consider a graph G = (V, E), where V denotes the set of n vertices and E the set of edges. For a (k,v) balanced partition problem, the objective is to partition G into k components of at most size v · (n/k), while minimizing the capacity of the edges between separate components. Also, given G and an integer k > 1, partition V into k parts (subsets) V1, V2, ..., Vk such that the parts are disjoint and have equal size, and the number of edges with endpoints in different parts is minimized. Such partition problems have been discussed in literature as bicriteria-approximation or resource augmentation approaches.
TS`, followed by full table options such as `center` to center the table on a line or `box` to draw a box around it (boxes in tbl are drawn with overlapping hyphens and underscores; there were no line drawing commands at the time of creation. Disjoint edges can be observed upon close inspection). Succeeding lines set up the formatting of each cell in the table with one character flags, such as `c` to center data in its cell, hyphens to draw horizontal rules, vertical bars to draw vertical rules, and carets to span cells vertically. The last formatting ends a period indicate cell data follows.
The state highway had previously met MD 117 west of that highway's railroad underpass at a T intersection through which MD 117 formed the east-west primary road through the intersection. MD 121's northern terminus was rolled back to the northern end of the I-270 interchange in 2007 when county-maintained Stringtown Road was completed from the I-270 interchange to MD 355. The disconnected part of Clarksburg Road west of Stringtown Road and south of MD 355 became MD 121A. MD 121 has had three disjoint segments in Germantown and Dawsonville that have never connected with the main Boyds-Clarksburg route.
Suppose all agents have piecewise-constant valuations. This means that, for each agent, the cake is partitioned into finitely many subsets, and the agent's value density in each subset is constant. For this case, Aziz and Ye present a randomized algorithm that is more economically-efficient: Constrained Serial Dictatorship is truthful in expectation, robust proportional, and satisfies a property called unanimity: if each agent's most preferred 1/n length of the cake is disjoint from other agents, then each agent gets their most preferred 1/n length of the cake. This is a weak form of efficiency that is not satisfied by the mechanisms based on exact division.
"Can a ball be decomposed into a finite number of point sets and reassembled into two balls identical to the original?" The Banach–Tarski paradox is a theorem in set-theoretic geometry, which states the following: Given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets, which can then be put back together in a different way to yield two identical copies of the original ball. Indeed, the reassembly process involves only moving the pieces around and rotating them without changing their shape. However, the pieces themselves are not "solids" in the usual sense, but infinite scatterings of points.
The proof of the Cantor–Bernstein–Schroeder theorem might be seen as antecedent of the Eilenberg–Mazur swindle. In fact, the ideas are quite similar. If there are injections of sets from X to Y and from Y to X, this means that formally we have X=Y+A and Y=X+B for some sets A and B, where + means disjoint union and = means there is a bijection between two sets. Expanding the former with the latter, :X = X + A + B. In this bijection, let Z consist of those elements of the left hand side that correspond to an element of X on the right hand side.
That is, there is no bi- continuous map from R3 → R3 that carries C onto A. To show this, suppose there was such a map h : R3 → R3, and consider a loop k that is interlocked with the necklace. k cannot be continuously shrunk to a point without touching A because two loops cannot be continuously unlinked. Now consider any loop j disjoint from C. j can be shrunk to a point without touching C because we can simply move it through the gap intervals. However, the loop g = h−1(k) is a loop that cannot be shrunk to a point without touching C, which contradicts the previous statement.
In this case, the elements of set Ω are called elements of type . Consequently, in a geometry of rank , each maximal flag has exactly elements. An incidence geometry of rank 2 is commonly called an incidence structure with elements of type 1 called points and elements of type 2 called blocks (or lines in some situations). More formally, :An incidence structure is a triple D = (V, B, ) where V and B are any two disjoint sets and is a binary relation between V and B, that is, ⊆ V × B. The elements of V will be called points, those of B blocks and those of flags. .
One can continue adding 10-cell rings adjacent to the previous ones, but it's more instructive to construct a second torus, disjoint from the one above, from the remaining 60 cells, that interlocks with the first. The 120-cell, like the 3-sphere, is the union of these two (Clifford) tori. If the center ring of the first torus is a meridian great circle as defined above, the center ring of the second torus is the equatorial great circle that is centered on the meridian circle. Also note that the spiraling shell of 50 cells around a center ring can be either left handed or right handed.
Once we have this fact, Tychonoff's theorem can be applied; we then use the finite intersection property (FIP) definition of compactness. The proof itself (due to J. L. Kelley) follows: Let {Ai} be an indexed family of nonempty sets, for i ranging in I (where I is an arbitrary indexing set). We wish to show that the cartesian product of these sets is nonempty. Now, for each i, take Xi to be Ai with the index i itself tacked on (renaming the indices using the disjoint union if necessary, we may assume that i is not a member of Ai, so simply take Xi = Ai ∪ {i}).
More generally, every planar graph of minimum degree at least three either has an edge of total degree at most 12, or at least 60 edges that (like the edges in the triakis icosahedron) connect vertices of degrees 3 and 10. If all triangular faces of a polyhedron are vertex-disjoint, there exists an edge with smaller total degree, at most eight. Generalizations of the theorem are also known for graph embeddings onto surfaces with higher genus. The theorem cannot be generalized to all planar graphs, as the complete bipartite graphs K_{1,n-1} and K_{2,n-2} have edges with unbounded total degree.
North of the bridge, Durant Road splits north to continue along the rail line while SR 154's divided highway curves to the northwest along Craig Avenue, which passes through a commercial area and intersects another rail spur. The state highway reduces to a two-lane street at Chestnut Street, after which the highway passes between the baseball and football fields of Covington High School. SR 154 continues through a residential area to its southern segment's northern terminus at Locust Street. The northern segment of SR 154 begins at Lexington Avenue on the edge of downtown Covington; the route's disjoint sections are connected by Locust Street and Lexington Avenue.
The first section of modern MD 84 to be constructed was between modern MD 75 and MD 800B; this section and MD 800B form a segment of MD 75's original alignment that was constructed as a concrete road between 1921 and 1923. The first sections of MD 84 proper, which was then named Uniontown Road, were built in 1924 and 1925; a short section of macadam road was constructed from old MD 75 to Roop Branch and concrete from there to Uniontown. Baust Church Road was improved to a modern highway around 1936 and designated MD 630 by 1939. MD 630 became a disjoint section of MD 84 in 1951.
Size of Baird's beaked whale compared to an average human Size of Arnoux's beaked whale compared to an average human Size of B. minimus beaked whale compared to a human The two established species have very similar features and would be indistinguishable at sea if they did not exist in disjoint locations. Both whales reach similar sizes, have bulbous melons, and long prominent beaks. Their lower jaw is longer than the upper, and the front teeth are visible even when the mouth is fully closed. The Baird's and Arnoux beaked whales in the family Ziphiidae are the only whales in this family that both sexes have erupted teeth.
The Schläfli double six In geometry, the Schläfli double six is a configuration of 30 points and 12 lines, introduced by . The lines of the configuration can be partitioned into two subsets of six lines: each line is disjoint from (skew with) the lines in its own subset of six lines, and intersects all but one of the lines in the other subset of six lines. Each of the 12 lines of the configuration contains five intersection points, and each of these 30 intersection points belongs to exactly two lines, one from each subset, so in the notation of configurations the Schläfli double six is written 125302.
Brookville Road dates to the 19th century as a connection between the Georgetown and Rockville Turnpike (now Wisconsin Avenue) in Tennallytown and the Union Turnpike, also known as the 7th Street Pike and now Georgia Avenue, north of Silver Spring. The Union Turnpike connected Washington with Brookeville in northern Montgomery County. Disjoint segments of the road's course in the District of Columbia remain as Belt Road between Chevy Chase and Tenleytown. North of MD 186's terminus, the route is followed by modern MD 410 to Rock Creek, then is lost before resuming as Brookville Road in Silver Spring, which ends at Seminary Road west of MD 97.
For instance, the figure shows the vertices of the graph placed on a cycle, with the internal diagonals of the cycle forming a matching. By subdividing the cycle edges into two matchings, we can partition the Heawood graph into three perfect matchings (that is, 3-color its edges) in eight different ways. Every two perfect matchings, and every two Hamiltonian cycles, can be transformed into each other by a symmetry of the graph.. There are 28 six-vertex cycles in the Heawood graph. Each 6-cycle is disjoint from exactly three other 6-cycles; among these three 6-cycles, each one is the symmetric difference of the other two.
Cubic graphs arise naturally in topology in several ways. For example, if one considers a graph to be a 1-dimensional CW complex, cubic graphs are generic in that most 1-cell attaching maps are disjoint from the 0-skeleton of the graph. Cubic graphs are also formed as the graphs of simple polyhedra in three dimensions, polyhedra such as the regular dodecahedron with the property that three faces meet at every vertex. Representation of a planar embedding as a graph-encoded map An arbitrary graph embedding on a two-dimensional surface may be represented as a cubic graph structure known as a graph-encoded map.
If X is the circle S^1, the mapping cone C_f can be considered as the quotient space of the disjoint union of Y with the disk D^2 formed by identifying each point x on the boundary of D^2 to the point f(x) in Y. Consider, for example, the case where Y is the disk D^2, and f\colon S^1 \to Y = D^2 is the standard inclusion of the circle S^1 as the boundary of D^2. Then the mapping cone C_f is homeomorphic to two disks joined on their boundary, which is topologically the sphere S^2.
This is called a pants decomposition for the surface, and the curves are called the cuffs of the decomposition. This decomposition is not unique, but by quantifying the argument one sees that all pants decompositions of a given surface have the same number of curves, which is exactly the complexity. For connected surfaces a pants decomposition has exactly 2g - 2 + k pants. A collection of simple closed curves on a surface is a pants decomposition if and only if they are disjoint, no two of them are homotopic and none is homotopic to a boundary component, and the collection is maximal for these properties.
By the Robertson–Seymour theorem, because they form a minor-closed family of graphs, the apex graphs have a forbidden graph characterization. There are only finitely many graphs that are neither apex graphs nor have another non-apex graph as a minor. These graphs are forbidden minors for the property of being an apex graph. Any other graph G is an apex graph if and only if none of the forbidden minors is a minor of G. These forbidden minors include the seven graphs of the Petersen family, three disconnected graphs formed from the disjoint unions of two of K5 and K3,3, and many other graphs.
There are many other equivalent statements of the axiom of choice. These are equivalent in the sense that, in the presence of other basic axioms of set theory, they imply the axiom of choice and are implied by it. One variation avoids the use of choice functions by, in effect, replacing each choice function with its range. :Given any set X of pairwise disjoint non-empty sets, there exists at least one set C that contains exactly one element in common with each of the sets in X.. According to , this was the formulation of the axiom of choice which was originally given by .
Different orderings of the vertices of a graph may cause the greedy coloring to use different numbers of colors, ranging from the optimal number of colors to, in some cases, a number of colors that is proportional to the number of vertices in the graph. For instance, a crown graph (a graph formed from two disjoint sets of vertices } and } by connecting to whenever ) can be a particularly bad case for greedy coloring. With the vertex ordering , a greedy coloring will use colors, one color for each pair . However, the optimal number of colors for this graph is two, one color for the vertices and another for the vertices .
The extended lift matroid L0(Ω) has for its ground set the set E0, which is the union of E with an extra point e0. The lift matroid L(Ω) is the extended lift matroid restricted to E. The extra point acts exactly like an unbalanced loop or a half-edge, so we describe only the lift matroid. An edge set is independent if it contains either no circles or just one circle, which is unbalanced. A circuit is a balanced circle, a pair of unbalanced circles that are either disjoint or have just a common vertex, or a theta graph whose circles are all unbalanced.
If a graph H has an embedding into the projective plane, then it necessarily has a planar cover, given by the preimage of H in the orientable double cover of the projective plane, which is a sphere. proved, conversely, that if a connected graph H has a two-ply planar cover then H must have an embedding into the projective plane.; , Theorem 2, p. 2 The assumption that H is connected is necessary here, because a disjoint union of projective-planar graphs may not itself be projective-planarFor instance, the two Kuratowski graphs are projective-planar but any union of two of them is not .
OneSource serves developers, integrators, managers, and community of interest (COI) participants as a focus point for searching, navigating, annotating, semantic matching, and mapping data terms extracted from military standards, COI vocabularies, programs of record, and other schemas and data sources. OneSource is based upon a United States Air Force researched and developed triplestore knowledge base architecture, which allows XML Schema, Web Ontology Language, relational database, spreadsheet, and even custom data models to be handled and presented in the same manner. Initial capability was released in 2006. Version 2 was released in 2008 with the previously disjoint matching and mapping capabilities fully integrated for use in a web browser.
By the characterization of interval graphs as AT-free chordal graphs, interval graphs are strongly chordal graphs and hence perfect graphs. Their complements belong to the class of comparability graphs, and the comparability relations are precisely the interval orders. Based on the fact that a graph is an interval graph if and only if it is chordal and its complement is a comparability graph, we have: A graph and its complement are interval graphs if and only if it is both a split graph and a permutation graph. The interval graphs that have an interval representation in which every two intervals are either disjoint or nested are the trivially perfect graphs.
Maryland Route 144 (MD 144) is a collection of state highways in the U.S. state of Maryland. These highways are sections of old alignment of U.S. Route 40 (US 40) between Cumberland and Baltimore. Along with US 40 Scenic, US 40 Alternate, and a few sections of county-maintained highway, MD 144 is assigned to what was once the main highway between the two cities, connecting those endpoints with Hancock, Hagerstown, Frederick, New Market, Mount Airy, Ellicott City, and Catonsville. MD 144 has seven disjoint sections of mainline highway that pass through mountainous areas of Allegany and Washington counties and the rolling Piedmont of Frederick, Carroll, Howard, and Baltimore counties.
The loop braid group is a mathematical group structure that is used in some models of theoretical physics to model the exchange of particles with loop- like topologies within three dimensions of space and time. The basic operations which generate a loop braid group for n loops are exchanges of two adjacent loops, and passing one adjacent loop through another. The topology forces these generators to satisfy some relations, which determine the group. To be precise, the loop braid group on n loops is defined as the motion group of n disjoint circles embedded in a compact three-dimensional "box" diffeomorphic to the three-dimensional disk.
Any two skew lines of these 27 belong to a unique Schläfli double six configuration, a set of 12 lines whose intersection graph is a crown graph in which the two lines have disjoint neighborhoods. Correspondingly, in the Schläfli graph, each edge uv belongs uniquely to a subgraph in the form of a Cartesian product of complete graphs K6 \square K2 in such a way that u and v belong to different K6 subgraphs of the product. The Schläfli graph has a total of 36 subgraphs of this form, one of which consists of the zero-one vectors in the eight-dimensional representation described above.
Typical business systems IDEF9, or integrated definition for business constraint discovery, is designed to assist in the discovery and analysis of constraints in a business system. A primary motivation driving the development of IDEF9 was an acknowledgment that the collection of constraints that forge an enterprise system is generally poorly defined. The knowledge of what constraints exist and how those constraints interact is incomplete, disjoint, distributed, and often completely unknown. Just as living organisms do not need to be aware of the genetic or autonomous constraints that govern certain behaviors, organizations can (and most do) perform well without explicit knowledge of the glue that structures the system.
In topological graph theory, a mathematical discipline, a linkless embedding of an undirected graph is an embedding of the graph into Euclidean space in such a way that no two cycles of the graph are linked. A flat embedding is an embedding with the property that every cycle is the boundary of a topological disk whose interior is disjoint from the graph. A linklessly embeddable graph is a graph that has a linkless or flat embedding; these graphs form a three- dimensional analogue of the planar graphs.. Complementarily, an intrinsically linked graph is a graph that does not have a linkless embedding. Flat embeddings are automatically linkless, but not vice versa.
The reason for this is that if an open set containing a point intersects the closure of a set, it necessarily intersects the set itself, hence a neighborhood can intersect at most the same number of closures (it may intersect fewer, since two distinct, indeed disjoint, sets can have the same closure). The converse, however, can fail if the closures of the sets are not distinct. For example, in the finite complement topology on R the collection of all open sets is not locally finite, but the collection of all closures of these sets is locally finite (since the only closures are R and the empty set).
The University of Pristina was founded in the Socialist Autonomous Province of Kosovo, Socialist Republic of Serbia, Yugoslavia, in the city of Pristina, for the academic year 1969–1970 and functioned until 1999. However, owing to political upheaval, war, successive mutual expulsions of faculty of one ethnicity or the other, and resultant pervasive ethnic-based polarisation, there came to be two disjoint institutions using the same name, albeit idiosyncratically to reflect ethnic identity. Albanian-language activity continues at the original location (University of Pristina), whilst the Serbian-language University of Priština has relocated to North Mitrovica, where it maintains its place within the Serbian education system.
Although the 9×9 grid with 3×3 regions is by far the most common, many other variations exist. Sample puzzles can be 4×4 grids with 2×2 regions; 5×5 grids with pentomino regions have been published under the name Logi-5; the World Puzzle Championship has featured a 6×6 grid with 2×3 regions and a 7×7 grid with six heptomino regions and a disjoint region. Larger grids are also possible, or different irregular shapes (under various names such as Suguru, Tectonic, Jigsaw Sudoku etc.). The Times offers a 12×12-grid "Dodeka Sudoku" with 12 regions of 4×3 squares.
A Steiner chain for two disjoint circles is a finite cyclic sequence of additional circles, each of which is tangent to the two given circles and to its two neighbors in the chain. Steiner's porism states that if two circles have a Steiner chain, they have infinitely many such chains. The chain is allowed to wrap more than once around the two circles, and can be characterized by a rational number p whose numerator is the number of circles in the chain and whose denominator is the number of times it wraps around. All chains for the same two circles have the same value of p.
County Road 20 is Blake Road between State Highway 7 (MN 7) and Excelsior Boulevard in Hopkins. A previous segment of County Road 20 extended southward along Blake Road from Excelsior Boulevard to Interlachen Boulevard, then following Interlachen Boulevard to Vernon Avenue near the interchange of Vernon Avenue with State Highway 100 (MN 100). Another previous segment was a disjoint segment in Minneapolis beginning at France Avenue and followed 44th Street, Sheridan Avenue, Lake Calhoun Parkway, 35th Street, and Hennepin Avenue, ending at the Lowry Hill Tunnel. County Road 21 is West 50th Street from France Avenue South to Lyndale Avenue South in Minneapolis.
This is more easily understood geometrically as any closed balls of radius k centered on distinct codewords being disjoint. These balls are also called Hamming spheres in this context. For example, consider the same 3 bit code consisting of two codewords "000" and "111". The Hamming space consists of 8 words 000, 001, 010, 011, 100, 101, 110 and 111. The codeword "000" and the single bit error words "001","010","100" are all less than or equal to the Hamming distance of 1 to "000". Likewise, codeword "111" and its single bit error words "110","101" and "011" are all within 1 Hamming distance of the original "111".
The result of many community discussions for over 30 years, the Lewis and Clark Bridge (known as the East End Bridge from its conception until completion of construction) is part of a new 6.5 mile (10.5 km) highway that connects the formerly disjoint sections of I-265 in Indiana and Kentucky. With the new section complete, I-265 forms a 3/4 beltway around the Louisville metropolitan area. Design A-15 was chosen over six alternatives for the I-265 connection, which includes the Lewis and Clark Bridge Bridge. A tunnel for the new highway was constructed under the historic Drumanard Estate in Kentucky because the property is listed on the National Register of Historic Places.
Referring to the quadratic form applied to the displacement vector of a point on the quasi-sphere from the centre (i.e. ) as the radial scalar square, in any pseudo-Euclidean space the quasi-spheres may be separated into three disjoint sets: those with positive radial scalar square, those with negative radial scalar square, those with zero radial scalar square. In a space with a positive-definite quadratic form (i.e. a Euclidean space), a quasi-sphere with negative radial scalar square is the empty set, one with zero radial scalar square consists of a single point, one with positive radial scalar square is a standard -sphere, and one with zero curvature is a hyperplane that is partitioned with the -spheres.
A family of closed sets called tiles forms a tessellation or tiling of a Euclidean space if their union is the whole space and every two distinct sets in the family have disjoint interiors. A tiling is said to be monohedral if all of the tiles are congruent to each other. Keller's conjecture concerns monohedral tilings in which all of the tiles are hypercubes of the same dimension as the space. As formulates the problem, a cube tiling is a tiling by congruent hypercubes in which the tiles are additionally required to all be translations of each other, without any rotation, or equivalently to have all of their sides parallel to the coordinate axes of the space.
The same rule holds with ∃ in place of ∀. For example, in ∀x ∀y (P(x) → Q(x,f(x),z)), x and y occur only bound,y occurs bound by rule 4, although it doesn't appear in any atomic subformula z occurs only free, and w is neither because it does not occur in the formula. Free and bound variables of a formula need not be disjoint sets: in the formula P(x) → ∀x Q(x), the first occurrence of x, as argument of P, is free while the second one, as argument of Q, is bound. A formula in first- order logic with no free variable occurrences is called a first-order sentence.
These multicasts are delivered to all the non-failed members listed in the view during which delivery is scheduled, a property referred to as virtual synchrony. Network partitions can split a group into two or more disjoint subgroups, creating the risk of split brain behavior, in which some group members take a decision (perhaps, to launch the rocket) without knowing that some other partition of the group has taken a different, conflicting decision. Gbcast offers protection against this threat: the protocol ensures that progress occurs only in a single primary partition of the group. Thus, should a network partition arise, at most one subgroup of members will continue operations, while the other is certain to stall and shut down.
The Mandelbrot set creates a fractal lake In geometry, and less formally, in most fractal-generating software, the fractal lake of an 'orbits' (or escape- time) fractal, is the part of the complex plane for which the orbit (a sequence of complex numbers) that is generated by iterating a given function does not "escape" from the unit circle. The lake may be connected or disjoint, and it may also have zero area. Orbits that are initialized inside the lake are either eventually captured by zero, captured by another point inside the unit circle, or may oscillate through a set of finite values indefinitely without ever converging to a fixed point. These points are described as being Inside the lake.
"The Dangers of Replication and a Solution"Proceedings of the 1999 ACM SIGMOD International Conference on Management of Data: SIGMOD '99, Philadelphia, PA, US; June 1–3, 1999, Volume 28; p. 3. He argued that unless the data splits in some natural way so that the database can be treated as n disjoint sub-databases, concurrency control conflicts will result in seriously degraded performance and the group of replicas will probably slow as a function of n. Gray suggested that the most common approaches are likely to result in degradation that scales as O(n³). His solution, which is to partition the data, is only viable in situations where data actually has a natural partitioning key.
Circularly permuted ssrA has been reported in three major lineages: i) all alphaproteobacteria and the primitive mitochondria of jakobid protists, ii) two disjoint groups of cyanobacteria (Gloeobacter and a clade containing Prochlorococcus and many Synechococcus), and iii) some members of the betaproteobacteria (Cupriavidus and some Rhodocyclales). All produce the same overall two-piece (acceptor and coding pieces) form, equivalent to the standard form nicked downstream of the reading frame. None retain more than two pseudoknots compared to the four (or more) of standard tmRNA. Alphaproteobacteria have two signature sequences: replacement of the typical T-loop sequence TΨCRANY with GGCRGUA, and the sequence AACAGAA in the large loop of the 3´-terminal pseudoknot.
A simple case in which a separator is guaranteed to exist is the following: :: Given a set of n disjoint axis- parallel squares in the plane, there is a rectangle R such that, at most 2n/3 of the squares are inside R, at most 2n/3 of the squares are outside R, and at most O(sqrt(n)) of the squares are not inside and not outside R (i.e. intersect the boundary of R). Thus, R is a geometric separator that separates the n squares into two subset ("inside R" and "outside R"), with a relatively small "loss" (the squares intersected by R are considered "lost" because they do not belong to any of the two subsets).
The above theorem can be generalized from disjoint rectangles to k-thick rectangles. Additionally, by induction on d, it is possible to generalize the above theorem to d dimensions and get the following theorem: :: Given N axis-parallel d-boxes whose interiors are k-thick, there exists an axis-parallel hyperplane such that at least: ::: \lfloor(N+1-k)/(2d)\rfloor :: of the d-box interiors lie to each side of the hyperplane. For the special case when k = N − 1 (i.e. each point is contained in at most N − 1 boxes), the following theorem holds: :: Given N axis-parallel d-boxes whose interiors are (N − 1)-thick, there exists an axis-parallel hyperplane that separates two of them.
That's why I decided to go with this unconventional method of composition." To record the soundtrack, Ushio and Yamada went to the real-life school the building in the film is based on, where Ushio recorded himself tapping and using the present objects in different ways, and later included those sounds in the soundtrack; he later remembered that Yamada "couldn't stop laughing" while he recorded. To compose, Ushio used a method Yamada qualified as "decalcomania", spilling ink on sheet music and folding it before putting the end result into music. He described it as "a way of representing how the two girls, that sort of disjoint between them and that gradual separation.
Additionally, C must have length exactly √8√n, as otherwise it could be improved by replacing one of its edges by the other two sides of a triangle. If the vertices in C are numbered (in clockwise order) from 1 to √8√n, and vertex i is matched up with vertex , then these matched pairs can be connected by vertex-disjoint paths within the disk, by a form of Menger's theorem for planar graphs. However, the total length of these paths would necessarily exceed n, a contradiction. With some additional work they show by a similar method that there exists a simple cycle separator of size at most (3/√2)√n, approximately 2.12√n.
A bramble of order four in a 3×3 grid graph, consisting of six mutually touching connected subgraphs In graph theory, a bramble for an undirected graph G is a family of connected subgraphs of G that all touch each other: for every pair of disjoint subgraphs, there must exist an edge in G that has one endpoint in each subgraph. The order of a bramble is the smallest size of a hitting set, a set of vertices of G that has a nonempty intersection with each of the subgraphs. Brambles may be used to characterize the treewidth of G.. In this reference, brambles are called "screens" and their order is called "thickness".
A four-coloring of a map of the states of the United States (ignoring lakes). In mathematics, the four color theorem, or the four color map theorem, states that given any separation of a plane into contiguous regions, producing a figure called a map, no more than four colors are required to color the regions of the map—so that no two adjacent regions have the same color. Two regions are called adjacent if they share a common boundary that is not a corner, where corners are the points shared by three or more regions.From this paper: Definitions: A planar map is a set of pairwise disjoint subsets of the plane, called regions.
A face of an embedded graph is an open 2-cell in the surface that is disjoint from the graph, but whose boundary is the union of some of the edges of the embedded graph. Let F be a face of an embedded graph G and let v0, v1, ..., vn−1,vn = v0 be the vertices lying on the boundary of F (in that circular order). A circular interval for F is a set of vertices of the form {va, va+1, ..., va+s} where a and s are integers and where subscripts are reduced modulo n. Let Λ be a finite list of circular intervals for F. We construct a new graph as follows.
Partition of the graph of a rhombic dodecahedron into two linear forests, showing that its linear arboricity is two In graph theory, a branch of mathematics, the linear arboricity of an undirected graph is the smallest number of linear forests its edges can be partitioned into. Here, a linear forest is an acyclic graph with maximum degree two; that is, it is a disjoint union of path graphs. Linear arboricity is a variant of arboricity, the minimum number of forests into which the edges can be partitioned. The linear arboricity of any graph of maximum degree \Delta is known to be at least \lceil\Delta/2\rceil and is conjectured to be at most \lceil(\Delta+1)/2\rceil.
Like Set, FinSet and FinOrd are topoi. As in Set, in FinSet the categorical product of two objects A and B is given by the cartesian product , the categorical sum is given by the disjoint union , and the exponential object BA is given by the set of all functions with domain A and codomain B. In FinOrd, the categorical product of two objects n and m is given by the ordinal product , the categorical sum is given by the ordinal sum , and the exponential object is given by the ordinal exponentiation nm. The subobject classifier in FinSet and FinOrd is the same as in Set. FinOrd is an example of a PRO.
If a transaction cannot be committed due to conflicting changes, it is typically aborted and re-executed from the beginning until it succeeds. The benefit of this optimistic approach is increased concurrency: no thread needs to wait for access to a resource, and different threads can safely and simultaneously modify disjoint parts of a data structure that would normally be protected under the same lock. However, in practice, STM systems also suffer a performance hit compared to fine-grained lock-based systems on small numbers of processors (1 to 4 depending on the application). This is due primarily to the overhead associated with maintaining the log and the time spent committing transactions.
Chuzhoy won the best paper award at the 2012 Symposium on Foundations of Computer Science for her paper with Shi Li on approximating the problem of connecting many given pairs of vertices in a graph by edge-disjoint paths. She is also known for her work showing a polynomial relation between the size of a grid graph minor of a graph and its treewidth. This connection between these two graph properties is a key component of the Robertson–Seymour theorem, is closely related to Halin's grid theorem for infinite graphs, and underlies the theory of bidimensionality for graph approximation algorithms. She was an Invited Speaker at the 2014 International Congress of Mathematicians, in Seoul.
A common format for biblical citations is Book chapter:verses, using a colon to delimit chapter from verse, as in: : "In the beginning, God created the heaven and the earth" (Gen. 1:1). Or, stated more formally,Five books have a single chapter: Obadiah, Philemon, 2 & 3 John, Jude. In many printed editions, the chapter number is omitted for these books, and references just use the verse numbers. :Book chapter for a chapter (John 3); :Book chapter1-chapter2 for a range of chapters (John 1-3); :book chapter:verse for a single verse (John 3:16); :book chapter:verse1-verse2 for a range of verses (John 3:16-17); :book chapter:verse1,verse2 for multiple disjoint verses (John 6:14, 44).
The red points form part of an ε-net for the Euclidean plane, where ε is the radius of the large yellow disks. The blue disks of half the radius are disjoint, and the yellow disks together cover the whole plane, satisfying the two definitional requirements on an ε-net. In the mathematical theory of metric spaces, ε-nets, ε-packings, ε-coverings, uniformly discrete sets, relatively dense sets, and Delone sets (named after Boris Delone) are several closely related definitions of well-spaced sets of points, and the packing radius and covering radius of these sets measure how well-spaced they are. These sets have applications in coding theory, approximation algorithms, and the theory of quasicrystals.
One can construct a simple earthquake map by cutting a surface along a finite number of disjoint simple closed geodesics, sliding the edges of each of these cut past each other by some amount, and closing the surface back up. One can imagine the surface being cut by strike- slip faults. An earthquake is a sort of limit of simple earthquakes, where one has an infinite number of geodesics, and instead of attaching a positive real number to each geodesic one puts a measure on them. In 2014, with Alex Eskin and with input from Amir Mohammadi, Mirzakhani proved that complex geodesics and their closures in moduli space are surprisingly regular, rather than irregular or fractal.
Marable studied the Autobiography manuscript "raw materials" archived by Haley's biographer, Anne Romaine, and described a critical element of the collaboration, Haley's writing tactic to capture the voice of his subject accurately, a disjoint system of data mining that included notes on scrap paper, in-depth interviews, and long "free style" discussions. Marable writes, "Malcolm also had a habit of scribbling notes to himself as he spoke." Haley would secretly "pocket these sketchy notes" and reassemble them in a sub rosa attempt to integrate Malcolm X's "subconscious reflections" into the "workable narrative". This is an example of Haley asserting authorial agency during the writing of the Autobiography, indicating that their relationship was fraught with minor power struggles.
A clique-sum of two planar graphs and the Wagner graph, forming a K5-minor- free graph. In graph theory, a branch of mathematics, a clique-sum is a way of combining two graphs by gluing them together at a clique, analogous to the connected sum operation in topology. If two graphs G and H each contain cliques of equal size, the clique-sum of G and H is formed from their disjoint union by identifying pairs of vertices in these two cliques to form a single shared clique, and then possibly deleting some of the clique edges. A k-clique-sum is a clique-sum in which both cliques have at most k vertices.
Malfatti circles In geometry, the Malfatti circles are three circles inside a given triangle such that each circle is tangent to the other two and to two sides of the triangle. They are named after Gian Francesco Malfatti, who made early studies of the problem of constructing these circles in the mistaken belief that they would have the largest possible total area of any three disjoint circles within the triangle. Malfatti's problem has been used to refer both to the problem of constructing the Malfatti circles and to the problem of finding three area-maximizing circles within a triangle. A simple construction of the Malfatti circles was given by , and many mathematicians have since studied the problem.
Steiner's construction of the Malfatti circles using bitangents Although much of the early work on the Malfatti circles used analytic geometry, provided the following simple synthetic construction. A circle that is tangent to two sides of a triangle, as the Malfatti circles are, must be centered on one of the angle bisectors of the triangle (green in the figure). These bisectors partition the triangle into three smaller triangles, and Steiner's construction of the Malfatti circles begins by drawing a different triple of circles (shown dashed in the figure) inscribed within each of these three smaller triangles. In general these circles are disjoint, so each pair of two circles has four bitangents (lines touching both).
The number of possible results in a boolean 9IM matrix is 29=512, and in a DE-9IM matrix is 39=6561. The percentage of these results that satisfy a specific predicate is determined as following, On usual applications the geometries intersects a priori, and the other relations are checked. The composite predicates "Intersects OR Disjoint" and "Equals OR Different" have the sum 100% (always true predicates), but "Covers OR CoveredBy" have 41%, that is not the sum, because they are not logical complements neither independent relations; idem "Contains OR Within", that have 21%. The sum 25%+12.5%=37.5% is obtained when ignoring overlapping of lines in "Crosses OR Overlaps", because the valid input sets are disjoints.
Finding ψ(G) is an optimization problem. The decision problem for complete coloring can be phrased as: :INSTANCE: a graph G=(V,E) and positive integer k :QUESTION: does there exist a partition of V into k or more disjoint sets V_1,V_2,\ldots,V_k such that each V_i is an independent set for G and such that for each pair of distinct sets V_i,V_j,V_i \cup V_j is not an independent set. Determining the achromatic number is NP-hard; determining if it is greater than a given number is NP-complete, as shown by Yannakakis and Gavril in 1978 by transformation from the minimum maximal matching problem. A1.1: GT5, pg.191.
Petersen graph as Kneser graph KG_{5,2} The Petersen graph is the complement of the line graph of K_5. It is also the Kneser graph KG_{5,2}; this means that it has one vertex for each 2-element subset of a 5-element set, and two vertices are connected by an edge if and only if the corresponding 2-element subsets are disjoint from each other. As a Kneser graph of the form KG_{2n-1,n-1} it is an example of an odd graph. Geometrically, the Petersen graph is the graph formed by the vertices and edges of the hemi-dodecahedron, that is, a dodecahedron with opposite points, lines and faces identified together.
The highway meets the western end of MD 372 (Wilkens Avenue) and passes between Rolling Road Golf Course on the east and the Catonsville Campus of the Community College of Baltimore County to the west. MD 166 temporarily expands to four lanes between Valley Road and Bloomsbury Avenue, between which the highway passes Catonsville High School. The highway narrows and passes along the edge of the Central Catonsville and Summit Park Historic District before veering north at Hilton Avenue and reaching its northern terminus at MD 144 (Frederick Road) west of downtown Catonsville. View north along MD 166 (Rolling Road) in Catonsville There are two disjoint sections of South Rolling Road that are not part of MD 166.
The extended lift matroid L0(G) has for its ground set the set E0 the union of edge set E with an extra point, which we denote e0. The lift matroid L(G) is the extended lift matroid restricted to E. The extra point acts exactly like a negative loop, so we describe only the lift matroid. An edge set is independent if it contains either no circles or just one circle, which is negative. (This is the same rule that is applied separately to each component in the signed-graphic matroid.) A matroid circuit is either a positive circle or a pair of negative circles that are either disjoint or have just a common vertex.
The anarboricity of a graph is the maximum number of edge-disjoint nonacyclic subgraphs into which the edges of the graph can be partitioned. The star arboricity of a graph is the size of the minimum forest, each tree of which is a star (tree with at most one non-leaf node), into which the edges of the graph can be partitioned. If a tree is not a star itself, its star arboricity is two, as can be seen by partitioning the edges into two subsets at odd and even distances from the tree root respectively. Therefore, the star arboricity of any graph is at least equal to the arboricity, and at most equal to twice the arboricity.
An equivalent procedure is to start with a regular octahedron and twist one face within its plane, without breaking any edges. With a 60° twist a triangular prism is formed; with a 120° twist there are two tetrahedra sharing the central vertex; any amount of twist between these two cases gives a Schönhardt polyhedron. Alternatively, the Schönhardt polyhedron can be formed by removing three disjoint tetrahedra from this convex hull: each of the removed tetrahedra is the convex hull of four vertices from the two triangles, two from each triangle. This removal causes the longer of the three connecting edges to be replaced by three new edges with concave dihedral angles, forming a nonconvex polyhedron.
The first portion of Old Washington Road to be constructed as a state highway was from MD 32 in Fenby south to Salem Bottom Road in 1935. By 1939, this road, which became one of two disjoint sections of MD 570, was extended south to Nicodemus Road. All of Old Washington Road from MD 32 to MD 26, except the east-west portion parallel to MD 26, became a state highway in 1956 when MD 97 was extended north through Westminster from Howard County. Construction on a new alignment for MD 97 from MD 26 to Westminster, New Washington Road, began in 1957 from the Westminster Bypass south to Fenby; this new highway was completed in 1960.
In it, the vertices can be connected by a path, such that every two edges in the path are at right angles to each other. A two-dimensional orthoscheme is a right triangle. A three-dimensional orthoscheme can be constructed from a cube by finding a path of three edges of the cube that do not all lie on the same square face, and forming the convex hull of the four points on this path. Dissection of a cube into six orthoschemes A dissection of a shape S (which may be any closed set in Euclidean space) is a representation of S as a union of other shapes whose interiors are disjoint from each other.
Possible methods include: the super-ko rule, time control, or placing an upper bound on the number of moves. This is also affected by the scoring method used since territory scoring penalizes extended play after the boundaries of the territories have been settled. Second, how to decide which player won the game; and whether draws (jigo) should be allowed. Possible terms to include in the score are: komi, prisoners captured during the game, stones in dead groups on the board at the end of the game, points of territory controlled by a player but not occupied by their stones, their living stones, the number of passes, and the number of disjoint living groups on the board.
The prison director's assignment of prisoner numbers to drawers can mathematically be described as a permutation of the numbers 1 to 100. Such a permutation is a one-to-one mapping of the set of natural numbers from 1 to 100 to itself. A sequence of numbers which after repeated application of the permutation returns to the first number is called a cycle of the permutation. Every permutation can be decomposed into disjoint cycles, that is, cycles which have no common elements. The permutation of the first example above can be written in cycle notation as :(1 ~ 7 ~ 5 )( 2 ~ 4 ~ 8) (3 ~ 6) and thus consists of two cycles of length 3 and one cycle of length 2.
In probability theory, a Lévy process, named after the French mathematician Paul Lévy, is a stochastic process with independent, stationary increments: it represents the motion of a point whose successive displacements are random, in which displacements in pairwise disjoint time intervals are independent, and displacements in different time intervals of the same length have identical probability distributions. A Lévy process may thus be viewed as the continuous-time analog of a random walk. The most well known examples of Lévy processes are the Wiener process, often called the Brownian motion process, and the Poisson process. Aside from Brownian motion with drift, all other proper (that is, not deterministic) Lévy processes have discontinuous paths.
The state diagram for a Mealy machine associates an output value with each transition edge, in contrast to the state diagram for a Moore machine, which associates an output value with each state. When the input and output alphabet are both , one can also associate to a Mealy Automata an Helix directed graph .Akhavi et al (2012) This graph has as vertices the couples of state and letters, every nodes are of out-degree one, and the successor of is the next state of the automata and the letter that the automata output when it is instate and it reads letter . This graph is a union of disjoint cycles if the automaton is bireversible.
In mathematics, a Swiss cheese is a compact subset of the complex plane obtained by removing from a closed disc some countable union of open discs, usually with some restriction on the centres and radii of the removed discs. Traditionally the deleted discs should have pairwise disjoint closures which are subsets of the interior of the starting disc, the sum of the radii of the deleted discs should be finite, and the Swiss cheese should have empty interior. This is the type of Swiss cheese originally introduced by the Swiss mathematician Alice Roth. More generally, a Swiss cheese may be all or part of Euclidean space Rn - or of an even more complicated manifold - with "holes" in it.
In topology, a branch of mathematics, the James reduced product or James construction J(X) of a topological space X with given basepoint e is the quotient of the disjoint union of all powers X, X2, X3, ... obtained by identifying points (x1,...,xk−1,e,xk+1,...,xn) with (x1,...,xk−1, xk+1,...,xn). In other words, its underlying set is the free monoid generated by X (with unit e). It was introduced by . For a connected CW complex X, the James reduced product J(X) has the same homotopy type as ΩΣX, the loop space of the suspension of X. The commutative analogue of the James reduced product is called the infinite symmetric product.
Per Martin-Löf constructed several type theories that were published at various times, some of them much later than the preprints with their description became accessible to the specialists (among others Jean-Yves Girard and Giovanni Sambin). The list below attempts to list all the theories that have been described in a printed form and to sketch the key features that distinguished them from each other. All of these theories had dependent products, dependent sums, disjoint unions, finite types and natural numbers. All the theories had the same reduction rules that did not include η-reduction either for dependent products or for dependent sums, except for MLTT79 where the η-reduction for dependent products is added.
The Bolyai–Gerwien theorem is a related but much simpler result: it states that one can accomplish such a decomposition of a simple polygon with finitely many polygonal pieces if both translations and rotations are allowed for the reassembly. It follows from a result of that it is possible to choose the pieces in such a way that they can be moved continuously while remaining disjoint to yield the square. Moreover, this stronger statement can be proved as well to be accomplished by means of translations only. These results should be compared with the much more paradoxical decompositions in three dimensions provided by the Banach–Tarski paradox; those decompositions can even change the volume of a set.
For finding real roots of a polynomial, the common strategy is to divide the real line (or an interval of it where root are searched) into disjoint intervals until having at most one root in each interval. Such a procedure is called root isolation, and a resulting interval that contains exactly one root is an isolating interval for this root. Wilkinson's polynomial shows that a very small modification of one coefficient of a polynomial may change dramatically not only the value of the roots, but also their nature (real or complex). Also, even with a good approximation, when one evaluates a polynomial at an approximate root, one may get a result that is far to be close to zero.
The initial proof was based on that of the Hirzebruch–Riemann–Roch theorem (1954), and involved cobordism theory and pseudodifferential operators. The idea of this first proof is roughly as follows. Consider the ring generated by pairs (X, V) where V is a smooth vector bundle on the compact smooth oriented manifold X, with relations that the sum and product of the ring on these generators are given by disjoint union and product of manifolds (with the obvious operations on the vector bundles), and any boundary of a manifold with vector bundle is 0. This is similar to the cobordism ring of oriented manifolds, except that the manifolds also have a vector bundle.
In the 1980s, Šiljak and his collaborators developed a large number of new and highly original concepts and methods for the decentralized control of uncertain large-scale interconnected systems. He broadened new notions of overlapping sub-systems and decompositions to formulate the inclusion principle. The principle described the process of expansion and contraction of dynamic systems that serve the purpose of rewriting overlapping decompositions as disjoint, which, in turn, allows the standard methods for control design. Structurally fixed modes, multiple controllers for reliable stabilization, decentralized optimization, and hierarchical, epsilon, and overlapping decompositions laid the foundation for a powerful and efficient approach to a broad set of problems in control design of large complex systems.
A generalization of Sperner's lemma to polytopes guarantees that, if this polytope is triangulated and labeled in an appropriate manner, there are at least n − d sub-simplices with a full labeling; each of these simplices corresponds to an (approximate) envy-free allocation in which each partner receives a different combination of pieces. However, the combinations might overlap: one partner might get the "morning" and "evening" shifts while another partner might get "evening" and "evening". Although these are different selections, they are incompatible. section 4 of proves that an envy-free division to two partners with disjoint pieces might be impossible if m = k = 2, but is always possible if m = 2 and k = 3 (i.e.
Then, the central projection maps a point to the intersection of the line with the projection plane. Such an intersection exists if and only if the point does not belong to the plane (, in green on the figure) that passes through and is parallel to . It follows that the lines passing through split in two disjoint subsets: the lines that are not contained in , which are in one to one correspondence with the points of , and those contained in , which are in one to one correspondence with the directions of parallel lines in . This suggests to define the points (called here projective points for clarity) of the projective plane as the lines passing through .
A standard fibered torus corresponding to (5,2) is obtained by gluing the top of the cylinder to the bottom by a 2/5 rotation counterclockwise. A Seifert manifold is a closed 3-manifold together with a decomposition into a disjoint union of circles (called fibers) such that each fiber has a tubular neighborhood that forms a standard fibered torus. A standard fibered torus corresponding to a pair of coprime integers (a,b) with a>0 is the surface bundle of the automorphism of a disk given by rotation by an angle of 2πb/a (with the natural fibering by circles). If a = 1 the middle fiber is called ordinary, while if a>1 the middle fiber is called exceptional.
The problem of determining whether two vertices in a graph are connected can be solved efficiently using a search algorithm, such as breadth-first search. More generally, it is easy to determine computationally whether a graph is connected (for example, by using a disjoint-set data structure), or to count the number of connected components. A simple algorithm might be written in pseudo-code as follows: #Begin at any arbitrary node of the graph, #Proceed from that node using either depth-first or breadth-first search, counting all nodes reached. #Once the graph has been entirely traversed, if the number of nodes counted is equal to the number of nodes of , the graph is connected; otherwise it is disconnected.
Since most routes with suffixes are very short highways of low importance, these highways are rarely signed with route markers. However, there are several numbered highways with several disjoint segments whose segments are denoted internally with letter suffixes but the segments are signed with the same numerical route marker; examples of these highways include MD 7, MD 18, MD 144, MD 648, MD 675, and MD 765. There is no duplication allowed between U.S. and Maryland state highways unless the two highways are related. The only signed example of duplicate numbers is MD 222, which is the old alignment of US 222 before the latter was rolled back from Perryville to Conowingo.
While graph isomorphism may be studied in a classical mathematical way, as exemplified by the Whitney theorem, it is recognized that it is a problem to be tackled with an algorithmic approach. The computational problem of determining whether two finite graphs are isomorphic is called the graph isomorphism problem. Its practical applications include primarily cheminformatics, mathematical chemistry (identification of chemical compounds), and electronic design automation (verification of equivalence of various representations of the design of an electronic circuit). The graph isomorphism problem is one of few standard problems in computational complexity theory belonging to NP, but not known to belong to either of its well-known (and, if P ≠ NP, disjoint) subsets: P and NP-complete.
Previous segments of County 16 include a westward extension along McGinty Road to Superior Boulevard in Wayzata and an eastward extension along County Road 5 and County Road 61, then east along Cedar Lake Road to Glenwood Avenue in downtown Minneapolis. This east segment was re-routed at various times due to the upgrade of U.S. Highway 12 (now Interstate 394) in downtown Minneapolis to freeway standards and to the construction of Interstate 494. County Road 17 is France Avenue South between Excelsior Boulevard in Minneapolis and Old Shakopee Road in Bloomington. There was a second disjoint section of County 17 that ran along France Avenue between Minnetonka Boulevard and Cedar Lake Road in Minneapolis.
Johnson's algorithm is a way to find the shortest paths between all pairs of vertices in an edge-weighted, directed graph. It allows some of the edge weights to be negative numbers, but no negative-weight cycles may exist. It works by using the Bellman–Ford algorithm to compute a transformation of the input graph that removes all negative weights, allowing Dijkstra's algorithm to be used on the transformed graph.. Section 25.3, "Johnson's algorithm for sparse graphs", pp. 636–640.. It is named after Donald B. Johnson, who first published the technique in 1977.. A similar reweighting technique is also used in Suurballe's algorithm for finding two disjoint paths of minimum total length between the same two vertices in a graph with non-negative edge weights..
The second edition adds material on several recent results in this area, in many cases inspired by the first edition of the book. Trevor Wilson proved the existence of a continuous motion from the one-ball assembly to the two-ball assembly, keeping the sets of the partition disjoint at all times; this question had been posed by de Groot in the first edition of the book. Miklós Laczkovich solved Tarski's circle-squaring problem, asking for a dissection of a disk to a square of the same area, in 1990. And Edward Marczewski had asked in 1930 whether the Banach–Tarski paradox could be achieved using only Baire sets; a positive answer was found in 1994 by Randall Dougherty and Matthew Foreman.
Heterogeneous System Architecture (HSA) is a cross-vendor set of specifications that allow for the integration of central processing units and graphics processors on the same bus, with shared memory and tasks. The HSA is being developed by the HSA Foundation, which includes (among many others) AMD and ARM. The platform's stated aim is to reduce communication latency between CPUs, GPUs and other compute devices, and make these various devices more compatible from a programmer's perspective, relieving the programmer of the task of planning the moving of data between devices' disjoint memories (as must currently be done with OpenCL or CUDA). CUDA and OpenCL as well as most other fairly advanced programming languages can use HSA to increase their execution performance.
Aspect is a term used across several religions and in theology to describe a particular manifestation or conception of a deity or other divine being. Depending on the religion, these might be disjoint or overlapping parts, or methods of perceiving or conceptualizing the deity in a particular context. In the Bahá'í Faith, this might be conceived as a Manifestation of God. In Christianity, Trinitarianism (see Trinity) is the belief in God as three distinct Persons in one Divinity, all of One Being, not confounding the Substance nor dividing the Essence: as such it would be false and indeed heretical (Sabellianism), from the perspective of orthodox Christianity, to conceive of one God manifested in three separate aspects or modes.G. T. Stokes, “Sabellianism,” ed.
The names are justified by analogy to the more commonly studied trees and forests. (A tree is a connected graph with no cycles; a forest is a disjoint union of trees.) Gabow and Tarjan. attribute the study of pseudoforests to Dantzig's 1963 book on linear programming, in which pseudoforests arise in the solution of certain network flow problems.. Pseudoforests also form graph-theoretic models of functions and occur in several algorithmic problems. Pseudoforests are sparse graphs – their number of edges is linearly bounded in terms of their number of vertices (in fact, they have at most as many edges as they have vertices) – and their matroid structure allows several other families of sparse graphs to be decomposed as unions of forests and pseudoforests.
The real numbers with the usual topology have the rational numbers as a countable dense subset which shows that the cardinality of a dense subset of a topological space may be strictly smaller than the cardinality of the space itself. The irrational numbers are another dense subset which shows that a topological space may have several disjoint dense subsets (in particular, two dense subsets may be each other's complements), and they need not even be of the same cardinality. Perhaps even more surprisingly, both the rationals and the irrationals have empty interiors, showing that dense sets need not contain any non-empty open set. The intersection of two dense open subsets of a topological space is again dense and open.
The size of the separator S constructed in this way is at most √8√n, or approximately 2.83√n. The vertices of the separator and the two disjoint subgraphs can be found in linear time. This proof of the separator theorem applies as well to weighted planar graphs, in which each vertex has a non-negative cost. The graph may be partitioned into three sets A, S, and B such that A and B each have at most 2/3 of the total cost and S has O(√n) vertices, with no edges from A to B.. By analysing a similar separator construction more carefully, shows that the bound on the size of S can be reduced to √6√n, or approximately 2.45√n.
Constructing the automorphism group is at least as difficult (in terms of its computational complexity) as solving the graph isomorphism problem, determining whether two given graphs correspond vertex-for-vertex and edge-for-edge. For, G and H are isomorphic if and only if the disconnected graph formed by the disjoint union of graphs G and H has an automorphism that swaps the two components.. In fact, just counting the automorphisms is polynomial-time equivalent to graph isomorphism. This drawing of the Petersen graph displays a subgroup of its symmetries, isomorphic to the dihedral group D5, but the graph has additional symmetries that are not present in the drawing. For example, since the graph is symmetric, all edges are equivalent.
The concept of bramble has also been defined for directed graphs. In a directed graph D, a bramble is a collection of strongly connected subgraphs of D that all touch each other: for every pair of disjoint elements X, Y of the bramble, there must exist an arc in D from X to Y and one from Y to X. The order of a bramble is the smallest size of a hitting set, a set of vertices of D that has a nonempty intersection with each of the elements of the bramble. The bramble number of D is the maximum order of a bramble in D. The bramble number of a directed graph is within a constant factor of its directed treewidth.
Brian Alspach coauthored an article with T.D. Parsons titled A construction for vertex –transitive graph published in the Canadian Journal of Mathematics (April, 1982). Alspach's conjecture, posed by Alspach in 1981, concerns the characterization of disjoint cycle covers of complete graphs with prescribed cycle lengths. With Heather Gavlas Jordon, in 2001, Alspach proved a special case, on the decomposition of complete graphs into cycles that all have the same length. This is possible if and only if the complete graph has an odd number of vertices (so its degree is even), the given cycle length is at most the number of vertices (so that cycles of that length exist), and the given length divides the number of edges of the graph.
By the precised form of the covering lemma, there exists a disjoint subcollection G of F such that every ball B ∈ F intersects a ball C ∈ G for which B ⊂ 5 C. Let r > 0 be given, and let Z denote the set of points z ∈ E that are not contained in any ball from G and belong to the open ball B(r) of radius r, centered at 0. It is enough to show that Z is Lebesgue-negligible, for every given r. Let G denote the subcollection of those balls in G that meet B(r). Consider the partition of G into sets Gn, n ≥ 0, consisting of balls that have radius in (2−n−1, 2−n].
Paving matroids were initially studied by , in their equivalent formulation in terms of d-partitions; Hartmanis called them generalized partition lattices. In their 1970 book Combinatorial Geometries, Henry Crapo and Gian-Carlo Rota observed that these structures were matroidal; the name "paving matroid" was introduced by following a suggestion of Rota. The simpler structure of paving matroids, compared to arbitrary matroids, has allowed some facts about them to be proven that remain elusive in the more general case. An example is Rota's basis conjecture, the statement that a set of n disjoint bases in a rank-n matroid can be arranged into an n × n matrix so that the rows of the matrix are the given bases and the columns are also bases.
Building on the work of Felix Hausdorff, in 1924 Stefan Banach and Alfred Tarski proved that given a solid ball in 3‑dimensional space, there exists a decomposition of the ball into a finite number of disjoint subsets that can be reassembled together in a different way to yield two identical copies of the original ball. Banach and Tarski proved that, using isometric transformations, the result of taking apart and reassembling a two-dimensional figure would necessarily have the same area as the original. This would make creating two unit squares out of one impossible. But in a 1929 paper, von Neumann proved that paradoxical decompositions could use a group of transformations that include as a subgroup a free group with two generators.
If M is a binary matroid, then so is its dual, and so is every minor of M. Additionally, the direct sum of binary matroids is binary. define a bipartite matroid to be a matroid in which every circuit has even cardinality, and an Eulerian matroid to be a matroid in which the elements can be partitioned into disjoint circuits. Within the class of graphic matroids, these two properties describe the matroids of bipartite graphs and Eulerian graphs (not-necessarily-connected graphs in which all vertices have even degree), respectively. For planar graphs (and therefore also for the graphic matroids of planar graphs) these two properties are dual: a planar graph or its matroid is bipartite if and only if its dual is Eulerian.
The relation "contains" can be represented by a bipartite graph. The vertices of the graph are divided into two disjoint sets, one representing the subsets in and another representing the elements in X. If a subset contains an element, an edge connects the corresponding vertices in the graph. In the graph representation, an exact cover is a selection of vertices corresponding to subsets such that each vertex corresponding to an element is connected to exactly one selected vertex. For example, the relation "contains" in the detailed example above can be represented by a bipartite graph with 6+7 = 13 vertices: 300px Again, the subcollection = {B, D, F} is an exact cover, since each element is contained in exactly one selected subset, i.e.
Lipton and J. Naughton presented an adaptive random sampling algorithm for database queryingRichard J. Lipton, Jeffrey F. Naughton (1990) "Query Size Estimation By Adaptive Sampling", "PODS '90: Proceedings of the ninth ACM SIGACT-SIGMOD-SIGART symposium on Principles of database systems"Richard J. Lipton, Jeffrey F. Naughton, Donovan A. Schneider (1990) "SIGMOD '90: Proceedings of the 1990 ACM SIGMOD international conference on Management of data " which is applicable to any query for which answers to the query can be partitioned into disjoint subsets. Unlike most sampling estimation algorithms—which statically determine the number of samples needed—their algorithm decides the number of samples based on the sizes of the samples, and tends to keep the running time constant (as opposed to linear in the number of samples).
Pugh An Improved Closing Lemma and a General Density Theorem, American Journal of Mathematics, Band 89, 1967, S.1010–1021, "Closing Lemma" by Christian Bonatti in Scholarpedia The lemma states: Let f be a diffeomorphism of a compact manifold with a nonwandering point x.Wandering points were introduced by George Birkhoff to describe dissipative systems (with chaotic behavior). In the case of a dynamical system given by a map f, a point wanders if it has a neighborhood U which is disjoint to all of the iterations of the map on it: f^n(U) \cap U = \varnothing.\, Then there is (in the space of diffeomorphisms, equipped with the C^1 topology) in a neighborhood of f a diffeomorphism g for which x is a periodic point.
Unicode blocks are identified by unique names, which use only ASCII characters and are usually descriptive of the nature of the symbols, in English; such as "Tibetan" or "Supplemental Arrows-A". (When comparing block names, one is supposed to equate uppercase with lowercase letters, and ignore any whitespace, hyphens, and underbars; so the last name is equivalent to "supplemental_arrows__a" and "SUPPLEMENTALARROWSA".} Blocks are pairwise disjoint, that is, they do not overlap. The starting code point and the size (number of code points) of each block are always multiples of 16; therefore, in the hexadecimal notation, the starting (smallest) point is U+xxx0 and the ending (largest) point is U+yyyF, where xxx and yyy are three or more hexadecimal digits.
A perfectly normal space is a topological space X in which every two disjoint closed sets E and F can be precisely separated by a continuous function f from X to the real line R: the preimages of {0} and {1} under f are, respectively, E and F. (In this definition, the real line can be replaced with the unit interval [0,1].) It turns out that X is perfectly normal if and only if X is normal and every closed set is a Gδ set. Equivalently, X is perfectly normal if and only if every closed set is a zero set. Every perfectly normal space is automatically completely normal. A Hausdorff perfectly normal space X is a T6 space, or perfectly T4 space.
A partition of a set S is a family of non-empty disjoint subsets of S that have S as their union. A partition, together with a total order on the sets of the partition, gives a structure called by Richard P. Stanley an ordered partition. and by Theodore Motzkin a list of sets.. An ordered partition of a finite set may be written as a finite sequence of the sets in the partition: for instance, the three ordered partitions of the set {a, b} are :{a}, {b}, :{b}, {a}, and :{a, b}. In a strict weak ordering, the equivalence classes of incomparability give a set partition, in which the sets inherit a total ordering from their elements, giving rise to an ordered partition.
One can also compose two cobordisms when the end of the first is equal to the start of the second. A n-dimensional topological quantum field theory (TQFT) is a monoidal functor from the category of n-cobordisms to the category of complex vector space (where multiplication is given by the tensor product). In particular, cobordisms between 1-dimensional manifolds (which are unions of circles) are compact surfaces whose boundary has been separated into two disjoint unions of circles. Two-dimensional TQFTs correspond to Frobenius algebras, where the circle (the only connected closed 1-manifold) maps to the underlying vector space of the algebra, while the pair of pants gives a product or coproduct, depending on how the boundary components are grouped – which is commutative or cocommutative.
But it is a pitfall that one needs to be aware of. Secondly, the guard conditions are also disjoint in addition to being exhaustive. Finally, note that guard conditions can ”peek” at the incoming tokens without actually consuming them — if the guards happen to be false or the action is not fired for some other reason, and if the token is not consumed by another action, then it remains where it is, and will be available for the next firing. (Or it will remain there forever, as in the case of the zero token in front of SplitDead, which is never removed because the actor is dead.) The Select actor below is another example of the use of guarded actions.
For the purposes of defining the crossing number, a drawing of an undirected graph is a mapping from the vertices of the graph to disjoint points in the plane, and from the edges of the graph to curves connecting their two endpoints. No vertex should be mapped onto an edge that it is not an endpoint of, and whenever two edges have curves that intersect (other than at a shared endpoint) their intersections should form a finite set of proper crossings, where the two curves are transverse. A crossing is counted separately for each of these crossing points, for each pair of edges that cross. The crossing number of a graph is then the minimum, over all such drawings, of the number of crossings in a drawing.
The existence of well-balanced orientations, together with Menger's theorem, immediately implies Robbins' theorem: by Menger's theorem, a 2-edge-connected graph has at least two edge- disjoint paths between every pair of vertices, from which it follows that any well-balanced orientation must be strongly connected. More generally this result implies that every -edge-connected undirected graph can be oriented to form a -edge-connected directed graph. A totally cyclic orientation of a graph is an orientation in which each edge belongs to a directed cycle. For connected graphs, this is the same thing as a strong orientation, but totally cyclic orientations may also be defined for disconnected graphs, and are the orientations in which each connected component of becomes strongly connected.
True gaps are where multiple disjoint sections of road have the same Interstate highway number and can reasonably be considered part of "one highway" in theory, based on the directness of connections via other highways, or based on future plans to fill in the gap in the Interstate, or simply based on the shortness of the gap. The sections are either not physically connected at all, or they are connected but the connection is not signed as part of the highway. This list does not include different highways that share the same number, such as the two different I-76s, I-84s, I-86s, I-87s, and I-88s, which were always intended as distinct highways and were never intended as a contiguous route.
Theorem. Let X be a non-empty compact Hausdorff space that satisfies the property that no one-point set is open. Then X is uncountable. Proof. We will show that if U ⊆ X is non-empty and open, and if x is a point of X, then there is a neighbourhood V ⊂ U whose closure does not contain x (x may or may not be in U). Choose y in U different from x (if x is in U, then there must exist such a y for otherwise U would be an open one point set; if x is not in U, this is possible since U is non-empty). Then by the Hausdorff condition, choose disjoint neighbourhoods W and K of x and y respectively.
F is the root and the remaining nodes are the leaves. The primary light tree is shown in solid lines and (directed-link-disjoint) the back up light tree is shown in dotted lines carrying traffic from source node to destinations. The ring based approach is also proposed to protect multicast session.C. Boworntummarat, L. Wuttisittikulkij, and S. Segkhoonthod, "Lighttree based Protection Strategies for Multicast Traffic in Transport WDM Mesh Networks with Multi-fiber Systems", in Proc. IEEE ICC'04, June 2004, vol. 3, pp.1791–1795 The segment protection scheme is another way to protect multicast connections.N. Singhal, L. sahasrabuddhe, and B. Mukherjee, "Provisioning of Survivable Multicast Sessions Against Single Link Failures in Optical WDM Mesh Networks", IEEE/OSA Journal of Lightwave Technology, vol.
Partitions of sets can be arranged in a partial order, showing that each partition of a set of size n "uses" one of the partitions of a set of size n-1. The 52 partitions of a set with 5 elements In general, Bn is the number of partitions of a set of size n. A partition of a set S is defined as a set of nonempty, pairwise disjoint subsets of S whose union is S. For example, B3 = 5 because the 3-element set {a, b, c} can be partitioned in 5 distinct ways: :{ {a}, {b}, {c} } :{ {a}, {b, c} } :{ {b}, {a, c} } :{ {c}, {a, b} } :{ {a, b, c} }. B0 is 1 because there is exactly one partition of the empty set.
For instance, the Lebesgue measure of the interval in the real numbers is its length in the everyday sense of the word, specifically, 1. Technically, a measure is a function that assigns a non-negative real number or to (certain) subsets of a set (see Definition below). It must further be countably additive: the measure of a 'large' subset that can be decomposed into a finite (or countably infinite) number of 'smaller' disjoint subsets is equal to the sum of the measures of the "smaller" subsets. In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure.
There is a geographic region denoted by C ("cake"). A partition of C, denoted by X, is a list of disjoint subregions whose union is C: :C = X_1\sqcup\cdots\sqcup X_n There is a certain set of additional parameters (such as: obstacles, fixed points or probability density functions), denoted by P. There is a real-valued function denoted by G ("goal") on the set of all partitions. The map segmentation problem is to find: :\arg\min_X G(X_1,\dots,X_n \mid P) where the minimization is on the set of all partitions of C. Often, there are geometric shape constraints on the partitions, e.g., it may be required that each part be a convex set or a connected set or at least a measurable set.
In complex analysis, the theorem states that the finite zeros of the derivative r'(z) of a nonconstant rational function r(z) that are not multiple zeros are also the positions of equilibrium in the field of force due to particles of positive mass at the zeros of r(z) and particles of negative mass at the poles of r(z) , with masses numerically equal to the respective multiplicities, where each particle repels with a force equal to the mass times the inverse distance. Furthermore, if C1 and C2 are two disjoint circular regions which contain respectively all the zeros and all the poles of r(z) , then C1 and C2 also contain all the critical points of r(z) .
In males, the bulbospongiosus is located in the middle line of the perineum, in front of the anus. It consists of two symmetrical parts, united along the median line by a tendinous perineal raphe. It arises from the central tendinous point of the perineum and from the median perineal raphe in front. In females, there is no union, nor a tendinous perineal raphe; the parts are disjoint primarily and arise from the same central tendinous point of the perinium, which is the tendon that is formed at the point where the bulbospongiosus muscle, superficial transverse perineal muscle, and external anal sphincter muscle converge to form this major supportive structure of vagina and other organs, and from the clitoris in front.
Note that if α is a successor ordinal, then α is compact, in which case its one-point compactification α+1 is the disjoint union of α and a point. As topological spaces, all the ordinals are Hausdorff and even normal. They are also totally disconnected (connected components are points), scattered (every non-empty set has an isolated point; in this case, just take the smallest element), zero- dimensional (the topology has a clopen basis: here, write an open interval (β,γ) as the union of the clopen intervals (β,γ'+1)=[β+1,γ'] for γ'<γ). However, they are not extremally disconnected in general (there are open sets, for example the even numbers from ω, whose closure is not open).
6 An example of a non-regular open set is the set U = ∪ in R with its normal topology, since 1 is in the interior of the closure of U, but not in U. The regular open subsets of a space form a complete Boolean algebra. ; Relatively compact: A subset Y of a space X is relatively compact in X if the closure of Y in X is compact. ; Residual: If X is a space and A is a subset of X, then A is residual in X if the complement of A is meagre in X. Also called comeagre or comeager. ; Resolvable: A topological space is called resolvable if it is expressible as the union of two disjoint dense subsets.
Charles Street has a partial cloverleaf interchange with I-695 (Baltimore Beltway) before meeting a disjoint segment of Bellona Avenue at an intersection with traffic lights (for the better part of a decade traffic was managed by a roundabout) on the southern edge of Lutherville.Marney Kirk, New Traffic Lights at Charles Street Circle, December 14, 2010. Bellona Avenue heads west toward a ramp to westbound I-695 and east to receive an exit ramp from westbound I-695 and intersect MD 131 (Seminary Avenue) within the Lutherville Historic District. Although MD 139 ends at the intersection, Charles Street continues north one block as a residential street to its northern terminus, where the road heads west into an apartment complex as Nightingale Way.
The Johnson graph J(n,k) is closely related to the Johnson scheme, an association scheme in which each pair of -element sets is associated with a number, half the size of the symmetric difference of the two sets.. The Johnson graph has an edge for every pair of sets at distance one in the association scheme, and the distances in the association scheme are exactly the shortest path distances in the Johnson graph.The explicit identification of graphs with association schemes, in this way, can be seen in . The Johnson scheme is also related to another family of distance-transitive graphs, the odd graphs, whose vertices are k-element subsets of an (2k+1)-element set and whose edges correspond to disjoint pairs of subsets.
If the two circles α and β cross each other, another two circles γ and δ are each tangent to both α and β, and in addition γ and δ are tangent to each other, then the point of tangency between γ and δ necessarily lies on one of the two circles of antisimilitude. If α and β are disjoint and non-concentric, then the locus of points of tangency of γ and δ again forms two circles, but only one of these is the (unique) circle of antisimilitude. If α and β are tangent or concentric, then the locus of points of tangency degenerates to a single circle, which again is the circle of antisimilitude.Tangencies: Circular Angle Bisectors, The Geometry Junkyard, David Eppstein, 1999.
A tessellation of the plane or of any other space is a cover of the space by closed shapes, called tiles, that have disjoint interiors. Some of the tiles may be congruent to one or more others. If is the set of tiles in a tessellation, a set of shapes is called a set of prototiles if no two shapes in are congruent to each other, and every tile in is congruent to one of the shapes in .. It is possible to choose many different sets of prototiles for a tiling: translating or rotating any one of the prototiles produces another valid set of prototiles. However, every set of prototiles has the same cardinality, so the number of prototiles is well defined.
Master Thesis, University of Tübingen, 2018 K16 3-coloured as three Clebsch graphs. The edges of the complete graph K16 may be partitioned into three disjoint copies of the 5-regular Clebsch graph. Because the Clebsch graph is a triangle-free graph, this shows that there is a triangle-free three-coloring of the edges of K16; that is, that the Ramsey number R(3,3,3) describing the minimum number of vertices in a complete graph without a triangle-free three-coloring is at least 17. used this construction as part of their proof that R(3,3,3) = 17.. The 5-regular Clebsch graph may be colored with four colors, but not three: its largest independent set has five vertices, not enough to partition the graph into three independent color classes.
In mathematics, Veblen's theorem, introduced by , states that the set of edges of a finite graph can be written as a union of disjoint simple cycles if and only if every vertex has even degree. Thus, it is closely related to the theorem of that a finite graph has an Euler tour (a single non-simple cycle that covers the edges of the graph) if and only if it is connected and every vertex has even degree. Indeed, a representation of a graph as a union of simple cycles may be obtained from an Euler tour by repeatedly splitting the tour into smaller cycles whenever there is a repeated vertex. However, Veblen's theorem applies also to disconnected graphs, and can be generalized to infinite graphs in which every vertex has finite degree .
The Banach–Tarski paradox is that a ball can be decomposed into a finite number of point sets and reassembled into two balls identical to the original. In set theory, a paradoxical set is a set that has a paradoxical decomposition. A paradoxical decomposition of a set is two families of disjoint subsets, along with appropriate group actions that act on some universe (of which the set in question is a subset), such that each partition can be mapped back onto the entire set using only finitely many distinct functions (or compositions thereof) to accomplish the mapping. A set that admits such a paradoxical decomposition where the actions belong to a group G is called G-paradoxical or paradoxical with respect to G. Paradoxical sets exist as a consequence of the Axiom of Infinity.
Again, a double cover of the resulting graph may be extended in a straightforward way to a double cover of the original graph: every cycle of the split off graph corresponds either to a cycle of the original graph, or to a pair of cycles meeting at v. Thus, every minimal counterexample must be cubic. But if a cubic graph can have its edges 3-colored (say with the colors red, blue, and green), then the subgraph of red and blue edges, the subgraph of blue and green edges, and the subgraph of red and green edges each form a collection of disjoint cycles that together cover all edges of the graph twice. Therefore, every minimal counterexample must be a non-3-edge-colorable bridgeless cubic graph, that is, a snark.
In computer science, a tagged union, also called a variant, variant record, choice type, discriminated union, disjoint union, sum type or coproduct, is a data structure used to hold a value that could take on several different, but fixed, types. Only one of the types can be in use at any one time, and a tag field explicitly indicates which one is in use. It can be thought of as a type that has several "cases", each of which should be handled correctly when that type is manipulated. This is critical in defining recursive datatypes, in which some component of a value may have the same type as the value itself, for example in defining a type for representing trees, where it is necessary to distinguish multi-node subtrees and leafs.
The medial graph of the Herschel graph is a 4-regular planar graph with no Hamiltonian decomposition. The shaded regions correspond to the vertices of the underlying Herschel graph. The Herschel graph also provides an example of a polyhedral graph for which the medial graph cannot be decomposed into two edge-disjoint Hamiltonian cycles. The medial graph of the Herschel graph is a 4-regular graph with 18 vertices, one for each edge of the Herschel graph; two vertices are adjacent in the medial graph whenever the corresponding edges of the Herschel graph are consecutive on one of its faces.. It is 4-vertex-connected and essentially 6-edge-connected, meaning that for every partition of the vertices into two subsets of at least two vertices, there are at least six edges crossing the partition.
There was a sense that there was a disjoint between the central 'English feminist agenda' pushed by the militant headquarters and the needs of Welsh social, cultural and political views. The strains existed between the two organisation until the EFF was abandoned in 1914. Margaret Haig Mackworth, later Viscountess Rhondda, Welsh militant suffragette. In 1913 a Suffrage Pilgrimage was organised, to end with a rally in Hyde Park, London on 26 July. It was an attempt to remind the public of the larger constitutional and non-militant wing of the movement, and routes were planned from 17 British towns and cities, including Wales. Twenty-eight members from Welsh NUWSS branches left from Bangor on 2 July travelling through Wales where they were met with both support and hostility.
List builder in Huggle, a Wikipedia anti-vandalism tool A list builder,List builders, Microsoft Developer Network also known as a dual list, dual listbox,dual listbox demo, ZK documentationJSFiddle Dual ListBox disjoint listbox,disjointlistbox, Incr Tcl documentation list shuttle,rich:listShuttle component, JBoss shuttle,Shuttle and Reorder, Oracle Browser Look and Feel Guidelines swaplistswaplist package, tcllib package documentation and two sided multi selecttwo sided multi select, jquery plugin is a graphical control element in which a user can select a set of text values by moving values between two list boxes, one representing selected values and the other representing unselected ones. Moving values back and forth is usually accomplished through buttons reading "Add" and "Remove", rather than by dragging and dropping them. The widget can sometimes also include the ability to rearrange the selected values.
Orthogonality as a property of term rewriting systems describes where the reduction rules of the system are all left-linear, that is each variable occurs only once on the left hand side of each reduction rule, and there is no overlap between them. Orthogonal term rewriting systems have the consequent property that all reducible expressions (redexes) within a term are completely disjoint -- that is, the redexes share no common function symbol. For example, the term rewriting system with reduction rules : \rho_1\ :\ f(x, y) \rightarrow g(y) : \rho_2\ :\ h(y) \rightarrow f(g(y), y) is orthogonal -- it is easy to observe that each reduction rule is left-linear, and the left hand side of each reduction rule shares no function symbol in common, so there is no overlap. Orthogonal term rewriting systems are confluent.
The input to the algorithm is an undirected graph with vertex set , edge set , and (optionally) numerical weights on the edges in . The goal of the algorithm is to partition into two disjoint subsets and of equal (or nearly equal) size, in a way that minimizes the sum of the weights of the subset of edges that cross from to . If the graph is unweighted, then instead the goal is to minimize the number of crossing edges; this is equivalent to assigning weight one to each edge. The algorithm maintains and improves a partition, in each pass using a greedy algorithm to pair up vertices of with vertices of , so that moving the paired vertices from one side of the partition to the other will improve the partition.
Routing a permutation of the doubly-directed cube graph In mathematics, Szymanski's conjecture, named after , states that every permutation on the n-dimensional doubly directed hypercube graph can be routed with edge-disjoint paths. That is, if the permutation σ matches each vertex v to another vertex σ(v), then for each v there exists a path in the hypercube graph from v to σ(v) such that no two paths for two different vertices u and v use the same edge in the same direction. Through computer experiments it has been verified that the conjecture is true for n ≤ 4 . Although the conjecture remains open for n ≥ 5, in this case there exist permutations that require the use of paths that are not shortest paths in order to be routed .
The British had great success with two of these networks, but the third, used for German-Japanese naval coordination, remained unbroken because of a faulty assumption that it employed a simplified version of Enigma. After OP-20-G's Marshall Hall observed that certain metadata in Berlin-to-Tokyo transmissions used letter sets disjoint from those used in Tokyo-to-Berlin metadata, Gleason hypothesized that the corresponding unencrypted letters sets were A-M (in one direction) and N-Z (in the other), then devised novel statistical tests by which he confirmed this hypothesis. The result was routine decryption of this third network by 1944. (This work also involved deeper related to permutation groups and the graph isomorphism problem.) OP-20-G then turned to the Japanese navy's "Coral" cipher.
A Grammar is said to be SLR(1) if and only if, for each and every state s in the SLR(1) automaton, none of the following conditions are violated: # For any reducible rule A → a • Xb in state s (where X is some terminal), there must not exist some irreducible rule, B → a • in the same state s such that the follow set of B contains the terminal X. In more formal terms, the intersection of set containing the terminal X and the follow set of B must be empty. Violation of this rule is a Shift-Reduce Conflict. # For any two complete items A → a • and B → b • in s, Follow(A) and Follow(B) are disjoint (their intersection is the empty set). Violation of this rule is a Reduce-Reduce Conflict.
For, suppose that 2k − 1 colors are available in total, and that, on a single side of the bipartition, each vertex has available to it a different k-tuple of these colors than each other vertex. Then, each side of the bipartition must use at least k colors, because every set of k − 1 colors will be disjoint from the list of one vertex. Since at least k colors are used on one side and at least k are used on the other, there must be one color which is used on both sides, but this implies that two adjacent vertices have the same color. In particular, the utility graph K3,3 has list-chromatic number at least three, and the graph K10,10 has list-chromatic number at least four.
If we operate on any point in Euclidean 2-space by the various elements of H we get what is called the orbit of that point. All the points in the plane can thus be classed into orbits, of which there are an infinite number with the cardinality of the continuum. Using the axiom of choice, we can choose one point from each orbit and call the set of these points M. We exclude the origin, which is a fixed point in H. If we then operate on M by all the elements of H, we generate each point of the plane (except the origin) exactly once. If we operate on M by all the elements of A or of B, we get two disjoint sets whose union is all points but the origin.
The composition of permutations, when they are written in cyclic form, is obtained by juxtaposing the two permutations (with the second one written on the left) and then simplifying to a disjoint cycle form if desired. Thus, in cyclic notation the above product would be given by: :Q \cdot P = (1 5)(2 4) \cdot (1 2 4 3) = (1 4 3 5). Since function composition is associative, so is the product operation on permutations: (\sigma \cdot \pi) \cdot \rho = \sigma \cdot(\pi \cdot \rho). Therefore, products of two or more permutations are usually written without adding parentheses to express grouping; they are also usually written without a dot or other sign to indicate multiplication (the dots of the previous example were added for emphasis, so would simply be written as \sigma \pi \rho).
When an additional element is considered for incorporation into the sequence of stretches (list of disjoint heap structures) it either forms a new one-element stretch, or it combines the two rightmost stretches by becoming the parent of both their roots and forming a new stretch that replaces the two in the sequence. Which of the two happens depends only on the sizes of the stretches currently present (and ultimately only on the index of the element added); Dijkstra stipulated that stretches are combined if and only if their sizes are and L(k) for some k, i.e., consecutive Leonardo numbers; the new stretch will have size L(k+2). In either case, the new element must be sifted down to its correct place in the heap structure.
A lattice in the complex plane and its fundamental domain, with quotient a torus. Given an action of a group G on a topological space X by homeomorphisms, a fundamental domain for this action is a set D of representatives for the orbits. It is usually required to be a reasonably nice set topologically, in one of several precisely defined ways. One typical condition is that D is almost an open set, in the sense that D is the symmetric difference of an open set in G with a set of measure zero, for a certain (quasi)invariant measure on X. A fundamental domain always contains a free regular set U, an open set moved around by G into disjoint copies, and nearly as good as D in representing the orbits.
Now define an action of on the , and the linear subspace they span in , given by The last equality in , which follows from and the property of the gamma matrices, shows that the constitute a representation of since the commutation relations in are exactly those of . The action of can either be thought of as six-dimensional matrices multiplying the basis vectors , since the space in spanned by the is six-dimensional, or be thought of as the action by commutation on the . In the following, The and the are both (disjoint) subsets of the basis elements of Cℓ4(C), generated by the four-dimensional Dirac matrices in four spacetime dimensions. The Lie algebra of is thus embedded in Cℓ4(C) by as the real subspace of Cℓ4(C) spanned by the .
A graph that requires four colors in any coloring, and four connected subgraphs that, when contracted, form a complete graph, illustrating the case k = 4 of Hadwiger's conjecture In graph theory, the Hadwiger conjecture states that if G is loopless and has no K_t minor then its chromatic number satisfies \chi(G) < t. It is known to be true for 1 \leq t \leq 6. The conjecture is a generalization of the four-color theorem and is considered to be one of the most important and challenging open problems in the field. In more detail, if all proper colorings of an undirected graph G use k or more colors, then one can find k disjoint connected subgraphs of G such that each subgraph is connected by an edge to each other subgraph.
152 (abstract) As Bohm and his colleagues emphasized, in such an algebraic approach operators and operands are of the same type: "there is no need for the disjoint features of the present mathematical formalism [of quantum theory], namely the operators on the one hand and the state vectors on the other. Rather, one uses only a single type of object, the algebraic element"., and its introductory note More specifically, Frescura and Hiley showed how "the states of quantum theory become elements of the minimal ideals of the algebra and [..] the projection operators are just the idempotents which generate these ideals". In a 1981 preprint that remained unpublished for many years, Bohm, P.G. Davies and Hiley presented their algebraic approach in context with the work of Arthur Stanley Eddington.
For a graph G = (V, E), an independent set S is a maximal independent set if for v \in V, one of the following is true: # v \in S # N(v) \cap S eq \emptyset where N(v) denotes the neighbors of v The above can be restated as a vertex either belongs to the independent set or has at least one neighbor vertex that belongs to the independent set. As a result, every edge of the graph has at least one endpoint not in S. However, it is not true that every edge of the graph has at least one, or even one endpoint in S Notice that any neighbor to a vertex in the independent set S cannot be in S because these vertices are disjoint by the independent set definition.
The combinatorial characterization of a set X ⊂ ℝ3 as a solid involves representing X as an orientable cell complex so that the cells provide finite spatial addresses for points in an otherwise innumerable continuum. The class of semi-analytic bounded subsets of Euclidean space is closed under Boolean operations (standard and regularized) and exhibits the additional property that every semi-analytic set can be stratified into a collection of disjoint cells of dimensions 0,1,2,3. A triangulation of a semi-analytic set into a collection of points, line segments, triangular faces, and tetrahedral elements is an example of a stratification that is commonly used. The combinatorial model of solidity is then summarized by saying that in addition to being semi-analytic bounded subsets, solids are three-dimensional topological polyhedra, specifically three-dimensional orientable manifolds with boundary.
The circles defined by the Apollonian pursuit problem for the same two points A and B, but with varying ratios of the two speeds, are disjoint from each other and form a continuous family that cover the entire plane; this family of circles is known as a hyperbolic pencil. Another family of circles, the circles that pass through both A and B, are also called a pencil, or more specifically an elliptic pencil. These two pencils of Apollonian circles intersect each other at right angles and form the basis of the bipolar coordinate system. Within each pencil, any two circles have the same radical axis; the two radical axes of the two pencils are perpendicular, and the centers of the circles from one pencil lie on the radical axis of the other pencil.
Although it follows from ZFC that every measurable cardinal is inaccessible (and is ineffable, Ramsey, etc.), it is consistent with ZF that a measurable cardinal can be a successor cardinal. It follows from ZF + axiom of determinacy that ω1 is measurable, and that every subset of ω1 contains or is disjoint from a closed and unbounded subset. Ulam showed that the smallest cardinal κ that admits a non-trivial countably- additive two-valued measure must in fact admit a κ-additive measure. (If there were some collection of fewer than κ measure-0 subsets whose union was κ, then the induced measure on this collection would be a counterexample to the minimality of κ.) From there, one can prove (with the Axiom of Choice) that the least such cardinal must be inaccessible.
The theory of combinatorial species and its extension to analytic combinatorics provide a language for describing many important combinatorial classes, constructing new classes from combinations of previously defined ones, and automatically deriving their counting sequences. For example, two combinatorial classes may be combined by disjoint union, or by a Cartesian product construction in which the objects are ordered pairs of one object from each of two classes, and the size function is the sum of the sizes of each object in the pair. These operations respectively form the addition and multiplication operations of a semiring on the family of (isomorphism equivalence classes of) combinatorial classes, in which the zero object is the empty combinatorial class, and the unit is the class whose only object is the empty set..
A haven of order k in a graph G is a function β that maps each set X of fewer than k vertices to a connected component of G − X, in such a way that every two subsets β(X) and β(Y) touch each other. Thus, the set of images of β forms a bramble in G, with order k. Conversely, every bramble may be used to determine a haven: for each set X of size smaller than the order of the bramble, there is a unique connected component β(X) that contains all of the subgraphs in the bramble that are disjoint from X. As Seymour and Thomas showed, the order of a bramble (or equivalently, of a haven) characterizes treewidth: a graph has a bramble of order k if and only if it has treewidth at least .
The cycle space of a planar graph is the cut space of its dual graph, and vice versa. The minimum weight cycle basis for a planar graph is not necessarily the same as the basis formed by its bounded faces: it can include cycles that are not faces, and some faces may not be included as cycles in the minimum weight cycle basis. There exists a minimum weight cycle basis in which no two cycles cross each other: for every two cycles in the basis, either the cycles enclose disjoint subsets of the bounded faces, or one of the two cycles encloses the other one. Following the duality between cycle spaces and cut spaces, this basis for a planar graph corresponds to a Gomory–Hu tree of the dual graph, a minimum weight basis for its cut space..
The Reye configuration can be realized in three-dimensional projective space by taking the lines to be the 12 edges and four long diagonals of a cube, and the points as the eight vertices of the cube, its center, and the three points where groups of four parallel cube edges meet the plane at infinity. Two regular tetrahedra may be inscribed within a cube, forming a stella octangula; these two tetrahedra are perspective figures to each other in four different ways, and the other four points of the configuration are their centers of perspectivity. These two tetrahedra together with the tetrahedron of the remaining 4 points form a desmic system of three tetrahedra. Any two disjoint spheres in three dimensional space, with different radii, have two bitangent double cones, the apexes of which are called the centers of similitude.
They have the same pattern of point-line intersections as the Euclidean version of the configuration. The finite projective plane PG(2,7) has 57 points and 57 lines, and can be given coordinates based on the integers modulo 7. In this space, every conic C (the set of solutions to a two-variable quadratic equation modulo 7) has 28 secant lines through pairs of its points, 8 tangent lines through a single point, and 21 nonsecant lines that are disjoint from C. Dually, there are 28 points where pairs of tangent lines meet, 8 points on C, and 21 interior points that do not belong to any tangent line. The 21 nonsecant lines and 21 interior points form an instance of the Grünbaum–Rigby configuration, meaning that again these points and lines have the same pattern of intersections.
A genus \varphi assigns a number \Phi(X) to each manifold X such that #\Phi(X \sqcup Y) =\Phi(X) + \Phi(Y) (where \sqcup is the disjoint union); #\Phi(X \times Y) =\Phi(X)\Phi(Y); #\Phi(X)=0 if X is the boundary of a manifold with boundary. The manifolds and manifolds with boundary may be required to have additional structure; for example, they might be oriented, spin, stably complex, and so on (see list of cobordism theories for many more examples). The value \Phi(X) is in some ring, often the ring of rational numbers, though it can be other rings such as \Z/2\Z or the ring of modular forms. The conditions on \Phi can be rephrased as saying that \varphi is a ring homomorphism from the cobordism ring of manifolds (with additional structure) to another ring.
A subset of X is closed/open if and only if its preimage under fi is closed/open in Y_i for each i ∈ I. The final topology on X can be characterized by the following characteristic property: a function g from X to some space Z is continuous if and only if g \circ f_i is continuous for each i ∈ I. Characteristic property of the final topology By the universal property of the disjoint union topology we know that given any family of continuous maps fi : Yi -> X, there is a unique continuous map :f\colon \coprod_i Y_i \to X. If the family of maps fi covers X (i.e. each x in X lies in the image of some fi) then the map f will be a quotient map if and only if X has the final topology determined by the maps fi.
A rooted tree may be directed, called a directed rooted tree, either making all its edges point away from the root—in which case it is called an arborescence or out-tree—or making all its edges point towards the root—in which case it is called an anti-arborescence or in-tree. A rooted tree itself has been defined by some authors as a directed graph. A rooted forest is a disjoint union of rooted trees. A rooted forest may be directed, called a directed rooted forest, either making all its edges point away from the root in each rooted tree—in which case it is called a branching or out-forest—or making all its edges point towards the root in each rooted tree—in which case it is called an anti-branching or in-forest.
Since then, only five more of these values have been found.. In the same 1955 paper, Greenwood and Gleason also computed the multicolor Ramsey number R(3,3,3): the smallest number r such that, if a complete graph on r vertices has its edges colored with three colors, then it necessarily contains a monochromatic triangle. As they showed, R(3,3,3) = 17; this remains the only nontrivial multicolor Ramsey number whose exact value is known. As part of their proof, they used an algebraic construction to show that a 16-vertex complete graph can be decomposed into three disjoint copies of a triangle-free 5-regular graph with 16 vertices and 40 edges. . (sometimes called the Greenwood–Gleason graph).. Ronald Graham writes that the paper by Greenwood and Gleason "is now recognized as a classic in the development of Ramsey theory".
That is, it is formed from a minimal vertex series parallel graph by forgetting the orientation of each edge. The comparability graph of a series-parallel partial order is a cograph: the series and parallel composition operations of the partial order give rise to operations on the comparability graph that form the disjoint union of two subgraphs or that connect two subgraphs by all possible edges; these two operations are the basic operations from which cographs are defined. Conversely, every cograph is the comparability graph of a series-parallel partial order. If a partial order has a cograph as its comparability graph, then it must be a series-parallel partial order, because every other kind of partial order has an N suborder that would correspond to an induced four- vertex path in its comparability graph, and such paths are forbidden in cographs.
An apex graph is a graph that may be made planar by the removal of one vertex, and a k-apex graph is a graph that may be made planar by the removal of at most k vertices. A 1-planar graph is a graph that may be drawn in the plane with at most one simple crossing per edge, and a k-planar graph is a graph that may be drawn with at most k simple crossings per edge. A map graph is a graph formed from a set of finitely many simply- connected interior-disjoint regions in the plane by connecting two regions when they share at least one boundary point. When at most three regions meet at a point, the result is a planar graph, but when four or more regions meet at a point, the result can be nonplanar.
In particular, if S and T intersect only in the identity, then every element of ST has a unique expression as a product st with s in S and t in T. If S and T also commute, then ST is a group, and is called a Zappa–Szép product. Even further, if S or T is normal in ST, then ST coincides with the semidirect product of S and T. Finally, if both S and T are normal in ST, then ST coincides with the direct product of S and T. If S and T are subgroups whose intersection is the trivial subgroup (identity element) and additionally ST = G, then S is called a complement of T and vice versa. By a (locally unambiguous) abuse of terminology, two subgroups that intersect only on the (otherwise obligatory) identity are sometimes called disjoint.
This equation says that a tree consists of a single root and a set of (sub-)trees. The recursion does not need an explicit base case: it only generates trees in the context of being applied to some finite set. One way to think about this is that the Ar functor is being applied repeatedly to a "supply" of elements from the set -- each time, one element is taken by X, and the others distributed by E among the Ar subtrees, until there are no more elements to give to E. This shows that algebraic descriptions of species are quite different from type specifications in programming languages like Haskell. Likewise, the species P can be characterised as P = E(E+): "a partition is a pairwise disjoint set of nonempty sets (using up all the elements of the input set)".
If the column headings in a relational database table are divided into three disjoint groupings X, Y, and Z, then, in the context of a particular row, we can refer to the data beneath each group of headings as x, y, and z respectively. A multivalued dependency X \twoheadrightarrow Y signifies that if we choose any x actually occurring in the table (call this choice xc), and compile a list of all the xcyz combinations that occur in the table, we will find that xc is associated with the same y entries regardless of z. So essentially the presence of z provides no useful information to constrain the possible values of y. A trivial multivalued dependency X \twoheadrightarrow Y is one where either Y is a subset of X, or X and Y together form the whole set of attributes of the relation.
If G has genus g ≥ 1 then the ΣnC are closely related to the Jacobian variety J of C. More accurately for n taking values up to g they form a sequence of approximations to J from below: their images in J under addition on J (see theta-divisor) have dimension n and fill up J, with some identifications caused by special divisors. For g = n we have ΣgC actually birationally equivalent to J; the Jacobian is a blowing down of the symmetric product. That means that at the level of function fields it is possible to construct J by taking linearly disjoint copies of the function field of C, and within their compositum taking the fixed subfield of the symmetric group. This is the source of André Weil's technique of constructing J as an abstract variety from 'birational data'.
In order theory, a subset A of a partially ordered set P is a strong downwards antichain if it is an antichain in which no two distinct elements have a common lower bound in P, that is, :\forall x, y \in A \; [x eq y \rightarrow eg\exists z \in P \; [ z \leq x \land z \leq y . In the case where P is ordered by inclusion, and closed under subsets, but does not contain the empty set, this is simply a family of pairwise disjoint sets. A strong upwards antichain B is a subset of P in which no two distinct elements have a common upper bound in P. Authors will often omit the "upwards" and "downwards" term and merely refer to strong antichains. Unfortunately, there is no common convention as to which version is called a strong antichain.
An embedding of a graph into three-dimensional space consists of a mapping from the vertices of the graph to points in space, and from the edges of the graph to curves in space, such that each endpoint of each edge is mapped to an endpoint of the corresponding curve, and such that the curves for two different edges do not intersect except at a common endpoint of the edges. Any finite graph has a finite (though perhaps exponential) number of distinct simple cycles, and if the graph is embedded into three-dimensional space then each of these cycles forms a simple closed curve. One may compute the linking number of each disjoint pair of curves formed in this way; if all pairs of cycles have zero linking number, the embedding is said to be linkless.; ; .
In any topological space X, the empty set and the whole space X are both clopen. (regarding the real numbers and the empty set in R) (regarding topological spaces) Now consider the space X which consists of the union of the two open intervals (0,1) and (2,3) of R. The topology on X is inherited as the subspace topology from the ordinary topology on the real line R. In X, the set (0,1) is clopen, as is the set (2,3). This is a quite typical example: whenever a space is made up of a finite number of disjoint connected components in this way, the components will be clopen. Now let X be an infinite set under the discrete metricthat is, two points p, q in X have distance 1 if they're not the same point, and 0 otherwise.
In the case where a graph H can be obtained from a graph G by a sequence of lifting operations (on G) and then finding an isomorphic subgraph, we say that H is an immersion minor of G. There is yet another way of defining immersion minors, which is equivalent to the lifting operation. We say that H is an immersion minor of G if there exists an injective mapping from vertices in H to vertices in G where the images of adjacent elements of H are connected in G by edge- disjoint paths. The immersion minor relation is a well-quasi-ordering on the set of finite graphs and hence the result of Robertson and Seymour applies to immersion minors. This furthermore means that every immersion minor-closed family is characterized by a finite family of forbidden immersion minors.
Because path connected sets are connected, we have PC_x \subseteq C_x for all x in X. However the closure of a path connected set need not be path connected: for instance, the topologist's sine curve is the closure of the open subset U consisting of all points (x,y) with x > 0, and U, being homeomorphic to an interval on the real line, is certainly path connected. Moreover, the path components of the topologist's sine curve C are U, which is open but not closed, and C \setminus U, which is closed but not open. A space is locally path connected if and only if for all open subsets U, the path components of U are open. Therefore the path components of a locally path connected space give a partition of X into pairwise disjoint open sets.
Told through the eyes of Gertie Nevels, a woman torn from the woods and farmland to move with her children to join her husband living in World War II factory workers' housing in Detroit, it can be seen as a work of feminist fiction. Arnow herself disputed this characterization however, preferring to see it as an individual woman's struggle to survive in a harsh and changing world "A southern woman's view on the disjoint between feminism and individualism " in Oral Histories of the American South. Of her writing she said, "I am afflicted with too many words ... Like the characters in my books, I talk too much and tell things I shouldn't tell." Later works were published under the now-familiar byline Harriette Simpson Arnow, and most reissues of her earlier work use this form of her name.
A variety V is defined over k if every polynomial in kalg[x1, …, xn] that vanishes on V is the linear combination (over kalg) of polynomials in k[x1, …, xn] that vanish on V. A k-algebraic set is also an L-algebraic set for infinitely many subfields L of kalg. A field of definition of a variety V is a subfield L of kalg such that V is an L-variety defined over L. Equivalently, a k-variety V is a variety defined over k if and only if the function field k(V) of V is a regular extension of k, in the sense of Weil. That means every subset of k(V) that is linearly independent over k is also linearly independent over kalg. In other words those extensions of k are linearly disjoint.
In descriptive set theory, a subset of a Polish space has the perfect set property if it is either countable or has a nonempty perfect subset (Kechris 1995, p. 150). Note that having the perfect set property is not the same as being a perfect set. As nonempty perfect sets in a Polish space always have the cardinality of the continuum, and the reals form a Polish space, a set of reals with the perfect set property cannot be a counterexample to the continuum hypothesis, stated in the form that every uncountable set of reals has the cardinality of the continuum. The Cantor–Bendixson theorem states that closed sets of a Polish space X have the perfect set property in a particularly strong form: any closed subset of X may be written uniquely as the disjoint union of a perfect set and a countable set.
The graphs that can be built from a single vertex by pendant vertices and true twins, without any false twin operations, are special cases of the Ptolemaic graphs and include the block graphs. The graphs that can be built from a single vertex by false twin and true twin operations, without any pendant vertices, are the cographs, which are therefore distance-hereditary; the cographs are exactly the disjoint unions of diameter-2 distance-hereditary graphs. The neighborhood of any vertex in a distance-hereditary graph is a cograph. The transitive closure of the directed graph formed by choosing any set of orientations for the edges of any tree is distance-hereditary; the special case in which the tree is oriented consistently away from some vertex forms a subclass of distance-hereditary graphs known as the trivially perfect graphs, which are also called chordal cographs..
Euclid proves that is perfect by observing that the geometric series with ratio 2 starting at , with the same number of terms, is proportional to the original series; therefore, since the original series sums to , the second series sums to , and both series together add to , two times the supposed perfect number. However, these two series are disjoint from each other and (by the primality of ) exhaust all the divisors of , so has divisors that sum to , showing that it is perfect.. Over a millennium after Euclid, Alhazen conjectured that even perfect number is of the form where is prime, but he was not able to prove this result. It was not until the 18th century that Leonhard Euler proved that the formula will yield all the even perfect numbers.. Originally read to the Berlin Academy on February 23, 1747, and published posthumously. See in particular section 8, p. 88.
However, the space of realizations of locally-square spiral packings is infinite-dimensional, unlike the Doyle spirals which can be determined only by a constant number of parameters. It is also possible to describe spiraling systems of overlapping circles that cover the plane, rather than non-crossing circles that pack the plane, with each point of the plane covered by at most two circles except for points where three circles meet at 60^\circ angles, and with each circle surrounded by six others. These have many properties in common with the Doyle spirals. The Doyle spiral, in which the circle centers lie on logarithmic spirals and their radii increase geometrically in proportion to their distance from the central limit point, should be distinguished from a different spiral pattern of disjoint but non-tangent unit circles, also resembling certain forms of plant growth such as the seed heads of sunflowers.
The cycle double cover conjecture posits that in every bridgeless graph one can find a collection of cycles covering each edge twice, or equivalently that the graph can be embedded onto a surface in such a way that all faces of the embedding are simple cycles. Snarks form the difficult case for this conjecture: if it is true for snarks, it is true for all graphs.. In this connection, Branko Grünbaum conjectured that it was not possible to embed any snark onto a surface in such a way that all faces are simple cycles and such that every two faces either are disjoint or share only a single edge; however, a counterexample to Grünbaum's conjecture was found by Martin Kochol.... Work by Peter Tait established that the 4-color theorem is true if and only every snark is non-planar. Thus all snarks are non-planar.
In graph theory, the planar separator theorem is a form of isoperimetric inequality for planar graphs, that states that any planar graph can be split into smaller pieces by removing a small number of vertices. Specifically, the removal of O(√n) vertices from an n-vertex graph (where the O invokes big O notation) can partition the graph into disjoint subgraphs each of which has at most 2n/3 vertices. A weaker form of the separator theorem with O(√n log n) vertices in the separator instead of O(√n) was originally proven by , and the form with the tight asymptotic bound on the separator size was first proven by . Since their work, the separator theorem has been reproven in several different ways, the constant in the O(√n) term of the theorem has been improved, and it has been extended to certain classes of nonplanar graphs.
In many cases, statistical physics uses probability measures, but not all measures it uses are probability measures.A course in mathematics for students of physics, Volume 2 by Paul Bamberg, Shlomo Sternberg 1991 page 802The concept of probability in statistical physics by Yair M. Guttmann 1999 page 149 In mathematics, a probability measure is a real-valued function defined on a set of events in a probability space that satisfies measure properties such as countable additivity.An introduction to measure-theoretic probability by George G. Roussas 2004 page 47 The difference between a probability measure and the more general notion of measure (which includes concepts like area or volume) is that a probability measure must assign value 1 to the entire probability space. Intuitively, the additivity property says that the probability assigned to the union of two disjoint events by the measure should be the sum of the probabilities of the events, e.g.
The decomposition of a permutation into a product of transpositions is obtained for example by writing the permutation as a product of disjoint cycles, and then splitting iteratively each of the cycles of length 3 and longer into a product of a transposition and a cycle of length one less: :(a~b~c~d~\ldots~y~z) = (a~b)\cdot (b~c~d~\ldots~y~z). This means the initial request is to move a to b, b to c, y to z, and finally z to a. Instead one may roll the elements keeping a where it is by executing the right factor first (as usual in operator notation, and following the convention in the article on Permutations). This has moved z to the position of b, so after the first permutation, the elements a and z are not yet at their final positions.
Equivalently, the inverse linear operator (T − λ)−1, which is defined on the dense subset R, is not a bounded operator, and therefore cannot be extended to the whole of X. Then λ is said to be in the continuous spectrum, σc(T), of T. #T − λ is injective but does not have dense range. That is, there is some element x in X and a neighborhood N of x such that (T − λ)(y) is never in N. In this case, the map (T − λ)−1 x → x may be bounded or unbounded, but in any case does not admit a unique extension to a bounded linear map on all of X. Then λ is said to be in the residual spectrum of T, σr(T). So σ(T) is the disjoint union of these three sets, :\sigma(T) = \sigma_p (T) \cup \sigma_c (T) \cup \sigma_r (T).
A t-shallow minor of a graph G is defined to be a graph formed from G by contracting a collection of vertex-disjoint subgraphs of radius t, and deleting the remaining vertices of G. A family of graphs has bounded expansion if there exists a function f such that, in every t-shallow minor of a graph in the family, the ratio of edges to vertices is at most f(t).. Equivalent definitions of classes of bounded expansions are that all shallow minors have chromatic number bounded by a function of t, or that the given family has a bounded value of a topological parameter. Such a parameter is a graph invariant that is monotone under taking subgraphs, such that the parameter value can change only in a controlled way when a graph is subdivided, and such that a bounded parameter value implies that a graph has bounded degeneracy..
In mathematics, specifically group theory, the free product is an operation that takes two groups G and H and constructs a new The result contains both G and H as subgroups, is generated by the elements of these subgroups, and is the “universal” group having these properties, in the sense that any two homomorphisms from G and H into a group K factor uniquely through a homomorphism from to K. Unless one of the groups G and H is trivial, the free product is always infinite. The construction of a free product is similar in spirit to the construction of a free group (the universal group with a given set of generators). The free product is the coproduct in the category of groups. That is, the free product plays the same role in group theory that disjoint union plays in set theory, or that the direct sum plays in module theory.
A T3 space or regular Hausdorff space is a topological space that is both regular and a Hausdorff space. (A Hausdorff space or T2 space is a topological space in which any two distinct points are separated by neighbourhoods.) It turns out that a space is T3 if and only if it is both regular and T0. (A T0 or Kolmogorov space is a topological space in which any two distinct points are topologically distinguishable, i.e., for every pair of distinct points, at least one of them has an open neighborhood not containing the other.) Indeed, if a space is Hausdorff then it is T0, and each T0 regular space is Hausdorff: given two distinct points, at least one of them misses the closure of the other one, so (by regularity) there exist disjoint neighborhoods separating one point from (the closure of) the other.
In code optimization during the translation of computer programs into an executable form, and in mathematical reduction generally, a reduction strategy for a term rewriting system determines which reducible subterms (or reducible expressions, redexes) should be reduced (contracted) within a term; it may be the case that a term may contain multiple redexes which are disjoint from one another and that choosing to contract one redex before another may have no influence on the resulting reduced form of the term, or that the redexes in a term do overlap and that choosing to contract one of the overlapping redexes over the other may result in a different reduced form of the term. It is the choice of which redex at each step in the reduction to contract that determines the strategy chosen. This can be seen as a practical application of the theoretical notion of reduction strategy in lambda calculus.
In mathematics, and, more specifically in numerical analysis and computer algebra, real-root isolation of a polynomial consist of producing disjoint intervals of the real line, which contain each one (and only one) real root of the polynomial, and, together, contain all the real roots of the polynomial. Real-root isolation is useful because usual root-finding algorithms for computing the real roots of a polynomial may produce some real roots, but, cannot generally certify having found all real roots. In particular, if such an algorithm does not find any root, one does not know whether it is because there is no real root. Some algorithms compute all complex roots, but, as there are generally much fewer real roots than complex roots, most of their computation time is generally spent for computing non-real roots (in the average, a polynomial of degree has complex roots, and only real roots; see ).
The treatment of these themes include the romantic relationship and eventual marriage, once the girl becomes an adult via time-travel, of a 30-year-old engineer and an 11-year-old girl in The Door into Summer or the more overt intra-familial incest in To Sail Beyond the Sunset and Farnham's Freehold. Heinlein often posed situations where the nominal purpose of sexual taboos was irrelevant to a particular situation, due to future advances in technology. For example, in Time Enough for Love Heinlein describes a brother and sister (Joe and Llita) who were mirror twins, being complementary diploids with entirely disjoint genomes, and thus not at increased risk for unfavorable gene duplication due to consanguinity. In this instance, Llita and Joe were props used to explore the concept of incest, where the usual objection to incest—heightened risk of genetic defect in their children—was not a consideration.
The complete graphs on 1, 2, 3, and 4 vertices are all maximal planar and well- covered; their vertex connectivity is either unbounded or at most three, depending on details of the definition of vertex connectivity that are irrelevant for larger maximal planar graphs. There are no well-covered 5-connected maximal planar graphs, and there are only four 4-connected well- covered maximal planar graphs: the graphs of the regular octahedron, the pentagonal dipyramid, the snub disphenoid, and an irregular polyhedron (a nonconvex deltahedron) with 12 vertices, 30 edges, and 20 triangular faces. However, there are infinitely many 3-connected well-covered maximal planar graphs.. For instance, a well-covered 3-connected maximal planar graph may be obtained via the clique cover construction from any -vertex maximal planar graph in which there are disjoint triangle faces by adding new vertices, one within each of these faces.
In mathematics, the Weil reciprocity law is a result of André Weil holding in the function field K(C) of an algebraic curve C over an algebraically closed field K. Given functions f and g in K(C), i.e. rational functions on C, then :f((g)) = g((f)) where the notation has this meaning: (h) is the divisor of the function h, or in other words the formal sum of its zeroes and poles counted with multiplicity; and a function applied to a formal sum means the product (with multiplicities, poles counting as a negative multiplicity) of the values of the function at the points of the divisor. With this definition there must be the side-condition, that the divisors of f and g have disjoint support (which can be removed). In the case of the projective line, this can be proved by manipulations with the resultant of polynomials.
The cofinal relation over partially ordered sets ("posets") is reflexive: every poset is cofinal in itself. It is also transitive: if B is a cofinal subset of a poset A, and C is a cofinal subset of B (with the partial ordering of A applied to B), then C is also a cofinal subset of A. For a partially ordered set with maximal elements, every cofinal subset must contain all maximal elements, otherwise a maximal element that is not in the subset would fail to be less than or equal to any element of the subset, violating the definition of cofinal. For a partially ordered set with a greatest element, a subset is cofinal if and only if it contains that greatest element (this follows, since a greatest element is necessarily a maximal element). Partially ordered sets without greatest element or maximal elements admit disjoint cofinal subsets.
The western section begins at the west junction of U.S. Highway 12 in Wayzata and heads westward through Minnetonka Beach, Spring Park, Mound, and Minnetrista before crossing the Carver County line and continuing as Carver County Road 24. The eastern section is Gleason Lake Road from the east junction of U.S. 12 in Wayzata to Vicksburg Lane at the Plymouth / Minnetonka border. Previous sections of County Road 15 include a section connecting the two disjoint sections of County Road 15 along Shoreline Drive, Lake Street East, Superior Boulevard, and Wayzata Boulevard in Wayzata, and a section extending east of Vicksburg Lane along Gleason Lake Road and Sunset Trail in Plymouth to State Highway 55 (MN 55). County Road 16 begins at the intersection of Interstate 494 and County Road 5 and runs alongside Interstate 494 to McGinty Road, where it follows McGinty Road to County Road 101 in Wayzata.
The portion of MD 316 between its southern terminus and what is now Belle Hill Road was originally part of MD 279, which continued along Belle Hill Road to its current course. Both the Newark Road and the Barksdale Road, which followed what is now MD 316 north from the Newark Road toward the village of Barksdale north of the Baltimore & Ohio Railroad (B&O; Railroad, now CSX's Philadelphia Subdivision), were planned to be built by the state but were instead constructed by Cecil County with state aid. Work on both macadam roads was underway by 1911, and the Barksdale Road was completed to Elk Mills Road by 1915. The Barksdale Road was planned to be extended north through the village of Cowenton in 1917, but those plans were cancelled with the United States' entry into World War I. A disjoint segment of the Barksdale Road was built from the B&O; Railroad crossing through Barksdale in 1921 and 1922.
A map graph (top), the cocktail party graph K2,2,2,2, defined by corner adjacency of eight regions in the plane (lower left), or as the half-square of a planar bipartite graph (lower right, the graph of the rhombic dodecahedron) In graph theory, a branch of mathematics, a map graph is an undirected graph formed as the intersection graph of finitely many simply connected and internally disjoint regions of the Euclidean plane. The map graphs include the planar graphs, but are more general. Any number of regions can meet at a common corner (as in the Four Corners of the United States, where four states meet), and when they do the map graph will contain a clique connecting the corresponding vertices, unlike planar graphs in which the largest cliques have only four vertices.. Another example of a map graph is the king's graph, a map graph of the squares of the chessboard connecting pairs of squares between which the chess king can move.
Some classes of matroid have been defined from well-known families of graphs, by phrasing a characterization of these graphs in terms that make sense more generally for matroids. These include the bipartite matroids, in which every circuit is even, and the Eulerian matroids, which can be partitioned into disjoint circuits. A graphic matroid is bipartite if and only if it comes from a bipartite graph and a graphic matroid is Eulerian if and only if it comes from an Eulerian graph. Within the graphic matroids (and more generally within the binary matroids) these two classes are dual: a graphic matroid is bipartite if and only if its dual matroid is Eulerian, and a graphic matroid is Eulerian if and only if its dual matroid is bipartite.. Graphic matroids are one- dimensional rigidity matroids, matroids describing the degrees of freedom of structures of rigid beams that can rotate freely at the vertices where they meet.
The Burnside ring of a finite group G is constructed from the category of finite G-sets as a Grothendieck group. More precisely, let M(G) be the commutative monoid of isomorphism classes of finite G-sets, with addition the disjoint union of G-sets and identity element the empty set (which is a G-set in a unique way). Then A(G), the Grothendieck group of M(G), is an abelian group. It is in fact a free abelian group with basis elements represented by the G-sets G/H, where H varies over the subgroups of G. (Note that H is not assumed here to be a normal subgroup of G, for while G/H is not a group in this case, it is still a G-set.) The ring structure on A(G) is induced by the direct product of G-sets; the multiplicative identity is the (isomorphism class of any) one-point set, which becomes a G-set in a unique way.
In his later work, Tarski showed that, conversely, non-existence of paradoxical decompositions of this type implies the existence of a finitely-additive invariant measure. The heart of the proof of the "doubling the ball" form of the paradox presented below is the remarkable fact that by a Euclidean isometry (and renaming of elements), one can divide a certain set (essentially, the surface of a unit sphere) into four parts, then rotate one of them to become itself plus two of the other parts. This follows rather easily from a -paradoxical decomposition of , the free group with two generators. Banach and Tarski's proof relied on an analogous fact discovered by Hausdorff some years earlier: the surface of a unit sphere in space is a disjoint union of three sets and a countable set such that, on the one hand, are pairwise congruent, and on the other hand, is congruent with the union of and .
According to the Koebe–Andreev–Thurston circle-packing theorem, any planar graph may be represented by a packing of circular disks in the plane with disjoint interiors, such that two vertices in the graph are adjacent if and only if the corresponding pair of disks are mutually tangent. As show, for such a packing, there exists a circle that has at most 3n/4 disks touching or inside it, at most 3n/4 disks touching or outside it, and that crosses O(√n) disks. To prove this, Miller et al. use stereographic projection to map the packing onto the surface of a unit sphere in three dimensions. By choosing the projection carefully, the center of the sphere can be made into a centerpoint of the disk centers on its surface, so that any plane through the center of the sphere partitions it into two halfspaces that each contain or cross at most 3n/4 of the disks.
A perfect hash function for a specific set that can be evaluated in constant time, and with values in a small range, can be found by a randomized algorithm in a number of operations that is proportional to the size of S. The original construction of uses a two-level scheme to map a set of elements to a range of indices, and then map each index to a range of hash values. The first level of their construction chooses a large prime (larger than the size of the universe from which is drawn), and a parameter , and maps each element of to the index :g(x)=(kx\bmod p)\bmod n. If is chosen randomly, this step is likely to have collisions, but the number of elements that are simultaneously mapped to the same index is likely to be small. The second level of their construction assigns disjoint ranges of integers to each index .
The minimum weight cycle basis for a planar graph is not necessarily the same as the basis formed by its bounded faces: it can include cycles that are not faces, and some faces may not be included as cycles in the minimum weight cycle basis. However, there exists a minimum weight cycle basis in which no two cycles cross each other: for every two cycles in the basis, either the cycles enclose disjoint subsets of the bounded faces, or one of the two cycles encloses the other one. This set of cycles corresponds, in the dual graph of the given planar graph, to a set of cuts that form a Gomory–Hu tree of the dual graph, the minimum weight basis of its cut space.. Based on this duality, an implicit representation of the minimum weight cycle basis in a planar graph can be constructed in time O(n\log^3 n)..
Two graphs are also fractionally isomorphic if they have a common coarsest equitable partition. A partition of a graph is a collection of pairwise disjoint sets of vertices whose union is the vertex set of the graph. A partition is equitable if for any pair of vertices u and v in the same block of the partition and any block B of the partition, both u and v have the same number of neighbors in B. An equitable partition P is coarsest if each block in any other equitable partition is a subset of a block in P. Two coarsest equitable partitions P and Q are common if there is a bijection f from the blocks of P to the blocks of Q such for any blocks B and C in P, the number of neighbors in C of any vertex in B equals the number of neighbors in f(C) of any vertex in f(B).
Let G be a finite group (in fact everything will work verbatim for a profinite group). Then for any two finite G-sets X and Y we can define an equivalence relation among spans of G-sets of the form X\leftarrow U \rightarrow Y where two spans X\leftarrow U \rightarrow Y and X\leftarrow W \rightarrow Yare equivalent if and only if there is a G-equivariant bijection of U and W commuting with the projection maps to X and Y. This set of equivalence classes form naturally a monoid under disjoint union; we indicate with A(G)(X,Y) the group completion of that monoid. Taking pullbacks induces natural maps A(G)(X,Y)\times A(G)(Y,Z)\rightarrow A(G)(X,Z). Finally we can define the Burnside category A(G) of G as the category whose objects are finite G-sets and the morphisms spaces are the groups A(G)(X,Y).
The envy-free cake-cutting problem is to partition the cake to n disjoint pieces, one piece per agent, such for each agent, the value of his piece is weakly larger than the values of all other pieces (so no agent envies another agent's share). A corollary of the Dubins–Spanier convexity theorem (1961) is that there always exists a "consensus partition" - a partition of the cake to n pieces such that every agent values every piece as exactly 1/n. A consensus partition is of course EF, but it is not PE. Moreover, another corollary of the Dubins–Spanier convexity theorem is that, when at least two agents have different value measures, there exists a division that gives each agent strictly more than 1/n. This means that the consensus partition is not even weakly PE. Envy- freeness, as a criterion for fair allocation, were introduced into economics in the 1960s and studied intensively during the 1970s.
An (n+1)-dimensional cobordism is a quintuple (W; M, N, i, j) consisting of an (n+1)-dimensional compact differentiable manifold with boundary, W; closed n-manifolds M, N; and embeddings i\colon M \hookrightarrow \partial W, j\colon N \hookrightarrow\partial W with disjoint images such that :\partial W = i(M) \sqcup j(N)~. The terminology is usually abbreviated to (W; M, N).The notation "(n+1)-dimensional" is to clarify the dimension of all manifolds in question, otherwise it is unclear whether a "5-dimensional cobordism" refers to a 5-dimensional cobordism between 4-dimensional manifolds or a 6-dimensional cobordism between 5-dimensional manifolds. M and N are called cobordant if such a cobordism exists. All manifolds cobordant to a fixed given manifold M form the cobordism class of M. Every closed manifold M is the boundary of the non-compact manifold M × [0, 1); for this reason we require W to be compact in the definition of cobordism.
Two different pants decompositions for the surface of genus 2 The importance of the pairs of pants in the study of surfaces stems from the following property: define the complexity of a connected compact surface S of genus g with k boundary components to be \xi(S) = 3g - 3 + k, and for a non- connected surface take the sum over all components. Then the only surfaces with negative Euler characteristic and complexity zero are disjoint unions of pairs of pants. Furthermore, for any surface S and any simple closed curve c on S which is not homotopic to a boundary component, the compact surface obtained by cutting S along c has a complexity that is strictly less than S. In this sense, pairs of pants are the only "irreducible" surfaces among all surfaces of negative Euler characteristic. By a recursion argument, this implies that for any surface there is a system of simple closed curves which cut the surface into pairs of pants.
One says that a is a two-sided divisor of b if it is both a left divisor and a right divisor of b; in this case, it is not necessarily true that (using the previous notation) x=y, only that both some x and some y which each individually satisfy the previous equations in R exist in R. When R is commutative, a left divisor, a right divisor and a two-sided divisor coincide, so in this context one says that a is a divisor of b, or that b is a multiple of a, and one writes a \mid b . Elements a and b of an integral domain are associates if both a \mid b and b \mid a . The associate relationship is an equivalence relation on R, and hence divides R into disjoint equivalence classes. Notes: These definitions make sense in any magma R, but they are used primarily when this magma is the multiplicative monoid of a ring.
The German state was weak until 1933 and could not even protect itself under the terms of the Treaty of Versailles. The status of ethnic Germans, and the lack of contiguity of German majority lands resulted in numerous repatriation pacts whereby the German authorities would organize population transfers (especially the Nazi-Soviet population transfers arranged between Adolf Hitler and Joseph Stalin, and others with Benito Mussolini's Italy) so that both Germany and the other country would increase their homogeneity. However, these population transfers were considered but a drop in the pond, and the "Heim ins Reich" rhetoric over the continued disjoint status of enclaves such as Danzig and Königsberg was an agitating factor in the politics leading up to World War II, and is considered by many to be among the major causes of Nazi aggressiveness and thus the war. Adolf Hitler used these issues as a pretext for waging wars of aggression against Czechoslovakia and Poland.
The points and lines of the Fano plane that are disjoint from a non-incident point-line pair form a triangle, and the bitangents of a quartic have been considered as being in correspondence with the 28 triangles of the Fano plane.. The Levi graph of the Fano plane is the Heawood graph, in which the triangles of the Fano plane are represented by 6-cycles. The 28 6-cycles of the Heawood graph in turn correspond to the 28 vertices of the Coxeter graph.. The 28 bitangents of a quartic also correspond to pairs of the 56 lines on a degree-2 del Pezzo surface, and to the 28 odd theta characteristics. The 27 lines on the cubic and the 28 bitangents on a quartic, together with the 120 tritangent planes of a canonic sextic curve of genus 4, form a "trinity" in the sense of Vladimir Arnold, specifically a form of McKay correspondence,Arnold 1997, p.
In mathematics, the axiom of regularity (also known as the axiom of foundation) is an axiom of Zermelo–Fraenkel set theory that states that every non-empty set A contains an element that is disjoint from A. In first-order logic, the axiom reads: : \forall x\,(x eq \varnothing \rightarrow \exists y \in x\,(y \cap x = \varnothing)). The axiom of regularity together with the axiom of pairing implies that no set is an element of itself, and that there is no infinite sequence (an) such that ai+1 is an element of ai for all i. With the axiom of dependent choice (which is a weakened form of the axiom of choice), this result can be reversed: if there are no such infinite sequences, then the axiom of regularity is true. Hence, in this context the axiom of regularity is equivalent to the sentence that there are no downward infinite membership chains.
This analysis simultaneously gives a multi-scale grouping of the organisms of the present age, and aims to accurately reconstruct the branching process or evolutionary tree that in past ages produced these organisms.. The input to a clustering problem consists of a set of points. A cluster is any proper subset of the points, and a hierarchical clustering is a maximal family of clusters with the property that any two clusters in the family are either nested or disjoint. Alternatively, a hierarchical clustering may be represented as a binary tree with the points at its leaves; the clusters of the clustering are the sets of points in subtrees descending from each node of the tree.. In agglomerative clustering methods, the input also includes a distance function defined on the points, or a numerical measure of their dissimilarity. The distance or dissimilarity should be symmetric: the distance between two points does not depend on which of them is considered first.
The relation → is a partial order on those equivalence classes; it defines a poset. Let G < H denote that there is a homomorphism from G to H, but no homomorphism from H to G. The relation → is a dense order, meaning that for all (undirected) graphs G, H such that G < H, there is a graph K such that G < K < H (this holds except for the trivial cases G = K0 or K1). For example, between any two complete graphs (except K0, K1) there are infinitely many circular complete graphs, corresponding to rational numbers between natural numbers. The poset of equivalence classes of graphs under homomorphisms is a distributive lattice, with the join of [G] and [H] defined as (the equivalence class of) the disjoint union [G ∪ H], and the meet of [G] and [H] defined as the tensor product [G × H] (the choice of graphs G and H representing the equivalence classes [G] and [H] does not matter).
More than a dozen additional monitor-based languages had been created by 1990: Simone, Modula, CSP/k, CCNPascal, PLY, Pascal Plus, Mesa, SB-Mod, Concurrent Euclid, Pascalc, Concurrent C, Emerald, Real-time Euclid, Pascal-FC, Turing Plus, Predula. Concurrent Pascal was the first concurrent programming language: the first language developed specifically for concurrent programming, and more importantly, the first language to demonstrate that it was possible to incorporate secure, high-level facilities for concurrency, where the system could guarantee that processes access disjoint sets of variables only and do not interfere with each other in time dependent ways. Hoare described it as "an outstanding example of the best of academic research in this area." Source and portable code for Concurrent Pascal and the Solo operating system were distributed to at least 75 companies and 100 universities in 21 countries, resulting in its widespread adoption, porting and adaptation in both industry and academia.
To remove the condition of disjoint support, for each point P on C a local symbol :(f, g)P is defined, in such a way that the statement given is equivalent to saying that the product over all P of the local symbols is 1. When f and g both take the values 0 or ∞ at P, the definition is essentially in limiting or removable singularity terms, by considering (up to sign) :fagb with a and b such that the function has neither a zero nor a pole at P. This is achieved by taking a to be the multiplicity of g at P, and −b the multiplicity of f at P. The definition is then ::(f, g)P = (−1)ab fagb. See for example Jean-Pierre Serre, Groupes algébriques et corps de classes, pp. 44–46, for this as a special case of a theory on mapping algebraic curves into commutative groups.
100; , Corollary 24, p.10. Since outerplanar graphs, series- parallel graphs, and Halin graphs all have bounded treewidth, they all also have at most logarithmic pathwidth. As well as its relations to treewidth, pathwidth is also related to clique-width and cutwidth, via line graphs; the line graph L(G) of a graph G has a vertex for each edge of G and two vertices in L(G) are adjacent when the corresponding two edges of G share an endpoint. Any family of graphs has bounded pathwidth if and only if its line graphs have bounded linear clique-width, where linear clique-width replaces the disjoint union operation from clique-width with the operation of adjoining a single new vertex.. If a connected graph with three or more vertices has maximum degree three, then its cutwidth equals the vertex separation number of its line graph.. In any planar graph, the pathwidth is at most proportional to the square root of the number of vertices.
Open sets have a fundamental importance in topology. The concept is required to define and make sense of topological space and other topological structures that deal with the notions of closeness and convergence for spaces such as metric spaces and uniform spaces. Every subset A of a topological space X contains a (possibly empty) open set; the maximum (ordered under inclusion) such open set is called the interior of A. It can be constructed by taking the union of all the open sets contained in A. Given topological spaces X and Y, a function f from X to Y is continuous if the preimage of every open set in Y is open in X. The function f is called open if the image of every open set in X is open in Y. An open set on the real line has the characteristic property that it is a countable union of disjoint open intervals.
Elm Creek Boulevard was completed to County 81 in early 2008, but has yet to be signed as County 130. The original alignment of County 130 was 69th Avenue North from West River Road in Brooklyn Center to the east end of Eagle Lake in Maple Grove, then a road north of Eagle Lake and Cedar Island Lake to County 109, then Berkshire Lane (now Rice Lake Road) between County 109 and County 30. This original route was significantly altered due to the construction of Interstate 94 in the 1960s and the development of Maple Grove in the 1980s-90s, resulting in a period of two disjoint sections of County 130 between the mid-1970s and 2005. County Road 135 is a road in Orono that begins at County 15 and passes between Maxwell Bay, Stubbs Bay, and the north arm of Crystal Bay on Lake Minnetonka before ending at County 84.
An unrestricted grammar is a formal grammar G = (N, \Sigma, P, S), where N is a finite set of nonterminal symbols, \Sigma is a finite set of terminal symbols, N and \Sigma are disjoint,Actually, this is not strictly necessary since unrestricted grammars make no real distinction between the two. The designation exists purely so that one knows when to stop generating sentential forms of the grammar; more precisely, the language L(G) recognized by G is restricted to strings of terminal symbols P is a finite set of production rules of the form \alpha \to \beta where \alpha and \beta are strings of symbols in N \cup \Sigma and \alpha is not the empty string, and S \in N is a specially designated start symbol. As the name implies, there are no real restrictions on the types of production rules that unrestricted grammars can have.While Hopcroft and Ullman (1979) do not mention the cardinalities of N, \Sigma, P explicitly, the proof of their Theorem 9.3 (construction of an equivalent Turing machine from a given unrestricted grammar, p.
The "CIDOC object-oriented Conceptual Reference Model" (CRM) is a domain ontology, but includes its own version of an upper ontology. The core classes cover: ; Space-Time : includes title/identifier, place, era/period, time-span, and relationship to persistent items ; Events : includes title/identifier, beginning/ending of existence, participants (people, either individually or in groups), creation/modification of things (physical or conceptional), and relationship to persistent items ; Material Things : includes title/identifier, place, the information object the material thing carries, part-of relationships, and relationship to persistent items ; Immaterial Things : includes title/identifier, information objects (propositional or symbolic), conceptional things, and part-of relationships Examples of definitions: ; Persistent Item : a physical or conceptional item that has a persistent identity recognized within the duration of its existence by its identification rather than by its continuity or by observation. A Persistent Item is comparable to an endurant. ; Temporal Entity : includes events, eras/periods, and condition states which happen over a limited extent in time, and is disjoint with Persistent Item.
The first two sections of what would become the Patuxent Freeway opened in 1972. A short section of divided highway opened from just west of Oakland Mills Road in Guilford to a turnaround just west of Vollmerhausen Road that included MD 32's modern interchange with I-95. The other section was a two- lane road from MD 175 in Odenton northeast along Sappington Station Road and then east along the current alignments of MD 32 and I-97 to MD 178 just east of I-97 Exit 5 in Crownsville. The eastern section, which from 1972 to 1977 was one of the three disjoint sections of mainline MD 32 between Crownsville and Taneytown, included an interchange with MD 3 in Millersville. As of 1978, MD 32's interchange with the Baltimore-Washington Parkway was a partial cloverleaf interchange with four ramps on the north side of MD 32--due to the state highway closely paralleling the Baltimore & Ohio Railroad's spur to the military base--and a fifth ramp from the northbound Parkway to eastbound MD 32\.
1-planar graphs were first studied by , who showed that they can be colored with at most seven colors.. Later, the precise number of colors needed to color these graphs, in the worst case, was shown to be six.. The example of the complete graph K6, which is 1-planar, shows that 1-planar graphs may sometimes require six colors. However, the proof that six colors are always enough is more complicated. Coloring the vertices and faces of the triangular prism graph requires six colors Ringel's motivation was in trying to solve a variation of total coloring for planar graphs, in which one simultaneously colors the vertices and faces of a planar graph in such a way that no two adjacent vertices have the same color, no two adjacent faces have the same color, and no vertex and face that are adjacent to each other have the same color. This can obviously be done using eight colors by applying the four color theorem to the given graph and its dual graph separately, using two disjoint sets of four colors.
The coordinates (x_i,y_i,z_i) of the apexes of a solution to the tripod problem form a 2-comparable sets of triples, where two triples are defined as being 2-comparable if there are either at least two coordinates where one triple is smaller than the other, or at least two coordinates where one triple is larger than the other. This condition ensures that the tripods defined from these triples do not have intersecting rays. Another equivalent two-dimensional version of the question asks how many cells of an n\times n array of square cells (indexed from 1 to n) can be filled in by the numbers from 1 to n in such a way that the non-empty cells of each row and each column of the array form strictly increasing sequences of numbers, and the positions holding each value i form a monotone chain within the array. A collection of disjoint tripods with apexes (x_i,y_i,z_i) can be transformed into an array of this type by placing the number z_i in array cell (x_i,y_i) and vice versa.
Consider all possible relationships between the Subject (S) and the Predicate (P) represented using sets: Case 1: S = P (S and P perfectly overlap) Case 2: S is a subset of P Case 3: P is a subset of S Case 4: S and P are two overlapping sets Case 5: S and P are disjoint sets Case 6: S is the universe with P being a subset of P Case 7: P is the universe with S being a subset of S Validity of statements after Obversion: The obversion operation is performed by changing the quality of the statement and replacing the predicate with its complement. 1\. Statement: All S are P (Applicable for Case 1, 2, 6 and 7) Obverse: No S are non-P Validity: YES 2\. Statement: No S are P (Applicable for Case 5) Obverse: All S are non-P Validity: YES 3\. Statement: Some S are P (Applicable for Case 1, 2, 3, 4, 6 and 7) Obverse: Some S are not non-P Validity: YES 4\.
In coercive subtyping systems, subtypes are defined by implicit type conversion functions from subtype to supertype. For each subtyping relationship (S <: T), a coercion function coerce: S → T is provided, and any object s of type S is regarded as the object coerceS → T(s) of type T. A coercion function may be defined by composition: if S <: T and T <: U then s may be regarded as an object of type u under the compound coercion (coerceT → U ∘ coerceS → T). The type coercion from a type to itself coerceT → T is the identity function idT Coercion functions for records and disjoint union subtypes may be defined componentwise; in the case of width-extended records, type coercion simply discards any components which are not defined in the supertype. The type coercion for function types may be given by f'(s) = coerceS2 → T2(f(coerceT1 → S1(t))), reflecting the contravariance of function arguments and covariance of return values. The coercion function is uniquely determined given the subtype and supertype.
A 2-dimensional matching can be defined in a completely analogous manner. Let X and Y be finite, disjoint sets, and let T be a subset of X × Y. Now M ⊆ T is a 2-dimensional matching if the following holds: for any two distinct pairs (x1, y1) ∈ M and (x2, y2) ∈ M, we have x1 ≠ x2 and y1 ≠ y2. In the case of 2-dimensional matching, the set T can be interpreted as the set of edges in a bipartite graph G = (X, Y, T); each edge in T connects a vertex in X to a vertex in Y. A 2-dimensional matching is then a matching in the graph G, that is, a set of pairwise non-adjacent edges. Hence 3-dimensional matchings can be interpreted as a generalization of matchings to hypergraphs: the sets X, Y, and Z contain the vertices, each element of T is a hyperedge, and the set M consists of pairwise non-adjacent edges (edges that do not have a common vertex).
A clustered planar drawing In graph drawing, a clustered planar graph is a graph together with a hierarchical clustering on its vertices, such that the graph drawn together with a collection of simple closed curves surrounding each cluster, so that there are no crossings between graph edges or clusters.. A brief survey paper associated with an invited talk at SCG. The clustering can be described combinatorially by a collection of subsets of the vertices such that, for each two subsets, either both are disjoint or one is contained in the other. It is not required that the clustering be maximal nor that every vertex belong to a cluster. In a clustered planar drawing, no two edges may cross each other (that is, the graph must be planar), no two of the curves representing clusters may cross each other, an edge may cross a cluster boundary only if it connects a vertex inside the cluster to a vertex outside the cluster, and when an edge and cluster boundary cross they may cross only once.
In mathematics (differential geometry) by a ribbon (or strip) (X,U) is meant a smooth space curve X given by a three-dimensional vector X(s), depending continuously on the curve arc-length s (a\leq s \leq b), together with a smoothly varying unit vector U(s) perpendicular to X at each point (Blaschke 1950). The ribbon (X,U) is called simple and closed if X is simple (i.e. without self-intersections) and closed and if U and all its derivatives agree at a and b. For any simple closed ribbon the curves X+\varepsilon U given parametrically by X(s)+\varepsilon U(s) are, for all sufficiently small positive \varepsilon, simple closed curves disjoint from X. The ribbon concept plays an important role in the Călugăreanu-White-Fuller formula (Fuller 1971), that states that :Lk = Wr + Tw \;, where Lk is the asymptotic (Gauss) linking number (a topological quantity), Wr denotes the total writhing number (or simply writhe) and Tw is the total twist number (or simply twist).
Formally, a measurable cardinal is an uncountable cardinal number κ such that there exists a κ-additive, non-trivial, 0-1-valued measure on the power set of κ. (Here the term κ-additive means that, for any sequence Aα, α<λ of cardinality λ < κ, Aα being pairwise disjoint sets of ordinals less than κ, the measure of the union of the Aα equals the sum of the measures of the individual Aα.) Equivalently, κ is measurable means that it is the critical point of a non-trivial elementary embedding of the universe V into a transitive class M. This equivalence is due to Jerome Keisler and Dana Scott, and uses the ultrapower construction from model theory. Since V is a proper class, a technical problem that is not usually present when considering ultrapowers needs to be addressed, by what is now called Scott's trick. Equivalently, κ is a measurable cardinal if and only if it is an uncountable cardinal with a κ-complete, non-principal ultrafilter.
One structure that can be used for this purpose is the convex layers of the input point set, a family of nested convex polygons consisting of the convex hull of the point set and the recursively-constructed convex layers of the remaining points. Within a single layer, the points inside the query half-plane may be found by performing a binary search for the half-plane boundary line's slope among the sorted sequence of convex polygon edge slopes, leading to the polygon vertex that is inside the query half-plane and farthest from its boundary, and then sequentially searching along the polygon edges to find all other vertices inside the query half-plane. The whole half-plane range reporting problem may be solved by repeating this search procedure starting from the outermost layer and continuing inwards until reaching a layer that is disjoint from the query halfspace. Fractional cascading speeds up the successive binary searches among the sequences of polygon edge slopes in each layer, leading to a data structure for this problem with space O(n) and query time O(log n + h).
In a paper published in 1924, Stefan Banach and Alfred Tarski gave a construction of such a paradoxical decomposition, based on earlier work by Giuseppe Vitali concerning the unit interval and on the paradoxical decompositions of the sphere by Felix Hausdorff, and discussed a number of related questions concerning decompositions of subsets of Euclidean spaces in various dimensions. They proved the following more general statement, the strong form of the Banach–Tarski paradox: : Given any two bounded subsets and of a Euclidean space in at least three dimensions, both of which have a nonempty interior, there are partitions of and into a finite number of disjoint subsets, A=A_1 \cup \cdots\cup A_k, B=B_1 \cup \cdots\cup B_k (for some integer k), such that for each (integer) between and , the sets and are congruent. Now let be the original ball and be the union of two translated copies of the original ball. Then the proposition means that you can divide the original ball into a certain number of pieces and then rotate and translate these pieces in such a way that the result is the whole set , which contains two copies of .
A graph that has the complete graph K4 as a 1-shallow minor. Each of the four vertex subsets indicated by the dashed rectangles induces a connected subgraph with radius one, and there exists an edge between every pair of subsets. One way of defining a minor of an undirected graph G is by specifying a subgraph H of G, and a collection of disjoint subsets Si of the vertices of G, each of which forms a connected induced subgraph Hi of H. The minor has a vertex vi for each subset Si, and an edge vivj whenever there exists an edge from Si to Sj that belongs to H. In this formulation, a d-shallow minor (alternatively called a shallow minor of depth d) is a minor that can be defined in such a way that each of the subgraphs Hi has radius at most d, meaning that it contains a central vertex ci that is within distance d of all the other vertices of Hi. Note that this distance is measured by hop count in Hi, and because of that it may be larger than the distance in G., Section 4.2 "Shallow Minors", pp. 62–65.
Since these paths must leave the two vertices of the pair via disjoint edges, a k-vertex-connected graph must have degeneracy at least k. Concepts related to k-cores but based on vertex connectivity have been studied in social network theory under the name of structural cohesion.. If a graph has treewidth or pathwidth at most k, then it is a subgraph of a chordal graph which has a perfect elimination ordering in which each vertex has at most k earlier neighbors. Therefore, the degeneracy is at most equal to the treewidth and at most equal to the pathwidth. However, there exist graphs with bounded degeneracy and unbounded treewidth, such as the grid graphs.. The Burr–Erdős conjecture relates the degeneracy of a graph G to the Ramsey number of G, the least n such that any two-edge-coloring of an n-vertex complete graph must contain a monochromatic copy of G. Specifically, the conjecture is that for any fixed value of k, the Ramsey number of k-degenerate graphs grows linearly in the number of vertices of the graphs.. The conjecture was proven by .
For topological manifolds, there is a slightly stronger notion of triangulation: a piecewise-linear triangulation (sometimes just called a triangulation) is a triangulation with the extra property – defined for dimensions 0, 1, 2, . . . inductively – that the link of any simplex is a piecewise-linear sphere. The link of a simplex s in a simplicial complex K is a subcomplex of K consisting of the simplices t that are disjoint from s and such that both s and t are faces of some higher-dimensional simplex in K. For instance, in a two-dimensional piecewise-linear manifold formed by a set of vertices, edges, and triangles, the link of a vertex s consists of the cycle of vertices and edges surrounding s: if t is a vertex in this cycle, t and s are both endpoints of an edge of K, and if t is an edge in this cycle, it and s are both faces of a triangle of K. This cycle is homeomorphic to a circle, which is a 1-dimensional sphere. But in this article the word "triangulation" is just used to mean homeomorphic to a simplicial complex.
At each step one of the subsets and one of the input symbols of the automaton are chosen, and the subsets of states are refined into states for which a transition labeled would lead to , and states for which an -transition would lead somewhere else. When a set that has already been chosen is split by a refinement, only one of the two resulting sets (the smaller of the two) needs to be chosen again; in this way, each state participates in the sets for refinement steps and the overall algorithm takes time , where is the number of initial states and is the size of the alphabet.. Partition refinement was applied by in an efficient implementation of the Coffman–Graham algorithm for parallel scheduling. Sethi showed that it could be used to construct a lexicographically ordered topological sort of a given directed acyclic graph in linear time; this lexicographic topological ordering is one of the key steps of the Coffman–Graham algorithm. In this application, the elements of the disjoint sets are vertices of the input graph and the sets used to refine the partition are sets of neighbors of vertices.
The second edition describes its intended audience in an elaborate subtitle, a throwback to times when long subtitles were more common: "a study of Quasi-Convex, aplanar, tunneled orientable polyhedra of positive genus having regular faces with disjoint interiors, being an elaborate description and instructions for the construction of an enormous number or new and fascinating mathematical models of interest to students of euclidean geometry and topology, both secondary and collegiate, to designers, engineers and architects, to the scientific audience concerned with molecular and other structural problems, and to mathematicians, both professional and dilletante, with hundreds of exercises and search projects, many outlined for self-instruction". Reviewer H. S. M. Coxeter summarizes the book as "a remarkable combination of sound mathematics, art, instruction and humor", while Henry Crapo calls it "highly recommended" to others interested in polyhedra and their juxtapositions. Mathematician Joseph A. Troccolo calls a method of constructing physical models of polyhedra developed in the book, using cardboard and rubber bands, "of inestimable value in the classroom". One virtue of this technique is that it allows for the quick disassembly and reuse of its parts.
More generally, for any connected group G, there is a finite set T of finite places of k such that G satisfies weak approximation with respect to any set S that is disjoint with T . In particular, if k is an algebraic number field then any group G satisfies weak approximation with respect to the set S = S∞ of infinite places. The question asked in strong approximation is whether the embedding of G(k) in G(AS) has dense image, or equivalently whether the set :G(k)G(AS) is a dense subset in G(A). The main theorem of strong approximation states that a non-solvable linear algebraic group G over a global field k has strong approximation for the finite set S if and only if its radical N is unipotent, G/N is simply connected, and each almost simple component H of G/N has a non-compact component Hs for some s in S (depending on H). The proofs of strong approximation depended on the Hasse principle for algebraic groups, which for groups of type E8 was only proved several years later.
A Vitali set is a subset V of the interval [0, 1] of real numbers such that, for each real number r, there is exactly one number v \in V such that v-r is a rational number. Vitali sets exist because the rational numbers Q form a normal subgroup of the real numbers R under addition, and this allows the construction of the additive quotient group R/Q of these two groups which is the group formed by the cosets of the rational numbers as a subgroup of the real numbers under addition. This group R/Q consists of disjoint "shifted copies" of Q in the sense that each element of this quotient group is a set of the form for some r in R. The uncountably many elements of R/Q partition R, and each element is dense in R. Each element of R/Q intersects [0, 1], and the axiom of choice guarantees the existence of a subset of [0, 1] containing exactly one representative out of each element of R/Q. A set formed this way is called a Vitali set.
Maryland Route 7 (MD 7) is a collection of state highways in the U.S. state of Maryland. Known for much of their length as Philadelphia Road, there are five disjoint mainline sections of the highway totaling that parallel U.S. Route 40 (US 40) in Baltimore, Harford, and Cecil counties in northeastern Maryland. The longest section of MD 7 begins at US 40 just east of the city of Baltimore in Rosedale and extends through eastern Baltimore County and southern Harford County to US 40 in Aberdeen. The next segment of the state highway is a C-shaped route through Havre de Grace on the west bank of the Susquehanna River. The third mainline section of MD 7 begins in Perryville on the opposite east bank of the Susquehanna River and ends at US 40 a short distance west of the start of the fourth section, which passes through Charlestown and North East before ending at US 40, just west of Elkton. The fifth segment of the highway begins at South Street and passes through the eastern part of Elkton before reconnecting with US 40 just west of the Delaware state line.
A polyhedron realized from a circle packing. The circles representing the vertices of the polyhedron are their horizons on the sphere, and the circles representing the faces (dual vertices) are the intersections of the sphere with the face planes. According to one variant of the circle packing theorem, for every polyhedral graph and its dual graph, there exists a system of circles in the plane or on any sphere, representing the vertices of both graphs, so that two adjacent vertices in the same graph are represented by tangent circles, a primal and dual vertex that represent a vertex and face that touch each other are represented by orthogonal circles, and all remaining pairs of circles are disjoint.. From such a representation on a sphere, one can find a polyhedral realization of the given graph as the intersection of a collection of halfspaces, one for each circle that represents a dual vertex, with the boundary of the halfspace containing the circle. Alternatively and equivalently, one can find the same polyhedron as the convex hull of a collection of points (its vertices), such that the horizon seen when viewing the sphere from any vertex equals the circle that corresponds to that vertex.
The halved cube graph of order 4, obtained as the bipartite half of an order-4 hypercube graph In graph theory, the bipartite half or half-square of a bipartite graph G = (U,V,E) is a graph whose vertex set is one of the two sides of the bipartition (without loss of generality, U) and in which there is an edge uiuj for each two vertices ui and uj in U that are at distance two from each other in G.. That is, in a more compact notation, the bipartite half is G2[U] where the superscript 2 denotes the square of a graph and the square brackets denote an induced subgraph. For instance, the bipartite half of the complete bipartite graph Kn,n is the complete graph Kn and the bipartite half of the hypercube graph is the halved cube graph. When G is a distance-regular graph, its two bipartite halves are both distance-regular.. For instance, the halved Foster graph is one of finitely many degree-6 distance-regular locally linear graphs. The map graphs, that is, the intersection graphs of interior- disjoint simply-connected regions in the plane, are exactly the bipartite halves of bipartite planar graphs..
The state highway was widened with a pair of macadam shoulders from Westminster to New Windsor between 1936 and 1939, resulting in a wide roadway between the two towns. A disjoint section of MD 31 was added in 1939 along Water Tank Road east of Manchester. The state highway, which extended from the eastern town limit of Manchester at Millers Station Road to Carrs Road, was removed from the state highway system in 1956. The first section of MD 31 to be reconstructed was east from Libertytown between 1949 and 1951. The state highway was also widened and resurfaced within Westminster starting in 1952. Construction began on reconstructing and widening MD 31 from Manchester to Westminster in 1957. This project involved several relocations of the highway by the time it was completed in 1960; sections of old road became segments of MD 852, including MD 852G between Westminster and Mexico. MD 31 was relocated from New Windsor to southwest of Westminster between 1963 and 1965; the old highway, Old New Windsor Pike, became another segment of MD 852. The final relocation of MD 31 around Westminster occurred in 1967 when the highway was relocated between Old New Windsor Pike and what is now MD 140.
The second section runs between Minnehaha Avenue and the Ford Bridge over the Mississippi River towards Saint Paul, where it turns into Ramsey County Road 42. County Road 47 is 62nd Avenue North in Plymouth from Brockton Lane (County Road 101) to Northwest Boulevard (County Road 61). County Road 48 is Minnehaha Avenue in Minneapolis from East 46th Street (County 46) to East 28th Street, then 26th Avenue South from East 28th Street to Franklin Avenue (County 5). County Road 49 was a route that ran along South Diamond Lake Road in Dayton from County 12 to County 13, then ran concurrent with County 13 to 129th Avenue North, then followed 129th Avenue North into Rogers. When Interstate 94 was constructed in the early 1970s, County 49 was rerouted north of 129th Avenue North along the north frontage road of Interstate 94 and terminated at State Highway 101 (MN 101), but Hennepin County continued to maintain the segment of 129th Avenue North in Rogers between County 150 and the present County 81 as County 49, creating an anomaly of two disjoint sections of County 49 that had their western termini on the same highway (County 150 and Highway 101 form 1 continuous road).

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