Acyclic Hypergraph
Acyclic may refer to: * In chemistry, a compound which is an open-chain compound, e.g. alkanes and acyclic aliphatic compounds * In mathematics: ** A graph without a cycle, especially *** A directed acyclic graph ** An acyclic complex is a chain complex all of whose homology groups are zero *** An acyclic space is a topological space all of whose homology groups are zero * In economics, an economic indicator An economic indicator is a statistic about an Economics, economic activity. Economic indicators allow analysis of economic performance and predictions of future performance. One application of economic indicators is the study of business cycles. ...

Open-chain Compound
In chemistry, an open-chain compound (or open chain compound) or acyclic compound (Greek prefix ''α'' 'without' and ''κύκλος'' 'cycle') is a compound with a linear structure, rather than a Cyclic compound, cyclic one. An open-chain compound having no side groups is called a straight-chain compound (also spelled as straight chain compound). Many of the simple molecules of organic chemistry, such as the alkanes and alkenes, have both linear and ring isomers, that is, both acyclic and cyclic compound, cyclic. For those with 4 or more carbons, the linear forms can have straight-chain or branched-chain isomers. The lowercase prefix ''n-'' denotes the straight-chain isomer; for example, ''n''-butane is straight-chain butane, whereas ''i''-butane is isobutane. Cycloalkanes are isomers of alkenes, not of alkanes, because the ring's closure involves a C-C bond. Having no rings (aromatic or otherwise), all open-chain compounds are aliphatic compound, aliphatic. Typically in biochemistr ...
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Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ...
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Cycle (graph Theory)
In graph theory, a cycle in a graph is a non-empty trail in which only the first and last vertices are equal. A directed cycle in a directed graph is a non-empty directed trail in which only the first and last vertices are equal. A graph without cycles is called an ''acyclic graph''. A directed graph without directed cycles is called a '' directed acyclic graph''. A connected graph without cycles is called a ''tree''. Definitions Circuit and cycle * A circuit is a non-empty trail in which the first and last vertices are equal (''closed trail''). : Let be a graph. A circuit is a non-empty trail with a vertex sequence . * A cycle or simple circuit is a circuit in which only the first and last vertices are equal. * ''n'' is called the length of the circuit resp. length of the cycle. Directed circuit and directed cycle * A directed circuit is a non-empty directed trail in which the first and last vertices are equal (''closed directed trail''). : Let be a directed grap ...
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Directed Acyclic Graph
In mathematics, particularly graph theory, and computer science, a directed acyclic graph (DAG) is a directed graph with no directed cycles. That is, it consists of vertices and edges (also called ''arcs''), with each edge directed from one vertex to another, such that following those directions will never form a closed loop. A directed graph is a DAG if and only if it can be topologically ordered, by arranging the vertices as a linear ordering that is consistent with all edge directions. DAGs have numerous scientific and computational applications, ranging from biology (evolution, family trees, epidemiology) to information science (citation networks) to computation (scheduling). Directed acyclic graphs are also called acyclic directed graphs or acyclic digraphs. Definitions A graph is formed by vertices and by edges connecting pairs of vertices, where the vertices can be any kind of object that is connected in pairs by edges. In the case of a directed graph, each edg ...
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Acyclic Complex
Homological algebra is the branch of mathematics that studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology (a precursor to algebraic topology) and abstract algebra (theory of modules and syzygies) at the end of the 19th century, chiefly by Henri Poincaré and David Hilbert. Homological algebra is the study of homological functors and the intricate algebraic structures that they entail; its development was closely intertwined with the emergence of category theory. A central concept is that of chain complexes, which can be studied through their homology and cohomology. Homological algebra affords the means to extract information contained in these complexes and present it in the form of homological invariants of rings, modules, topological spaces, and other "tangible" mathematical objects. A spectral sequence is a powerful tool for this. It has played an enormous role ...
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Acyclic Space
In mathematics, an acyclic space is a nonempty topological space ''X'' in which cycles are always boundaries, in the sense of homology theory. This implies that integral homology groups in all dimensions of ''X'' are isomorphic to the corresponding homology groups of a point. In other words, using the idea of reduced homology, :\tilde_i(X)=0, \quad \forall i\ge -1. It is common to consider such a space as a nonempty space without "holes"; for example, a circle or a sphere is not acyclic but a disc or a ball is acyclic. This condition however is weaker than asking that every closed loop in the space would bound a disc in the space, all we ask is that any closed loop—and higher dimensional analogue thereof—would bound something like a "two-dimensional surface." The condition of acyclicity on a space ''X'' implies, for example, for nice spaces—say, simplicial complexes—that any continuous map of ''X'' to the circle or to the higher spheres is null-homotopi ...
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