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In mathematics, a partial cyclic order is a ternary relation that generalizes a cyclic order in the same way that a partial order generalizes a linear order.
Over a given set, a partial cyclic order is a ternary relation that is:
linear extension, Szpilrajn extension theorem
standard example
The relationship between partial and total cyclic orders is more complex than the relationship between partial and total linear orders. To begin with, not every partial cyclic order can be extended to a total cyclic order. An example is the following relation on the first thirteen letters of the alphabet: {acd, bde, cef, dfg, egh, fha, gac, hcb} ∪ {abi, cij, bjk, ikl, jlm, kma, lab, mbc}. This relation is a partial cyclic order, but it cannot be extended with either abc or cba; either attempt would result in a contradiction. [4]
The above was a relatively mild example. One can also construct partial cyclic orders with higher-order obstructions such that, for example, any 15 triples can be added but the 16th cannot. In fact, cyclic ordering is NP-complete, since it solves 3SAT. This is in stark contrast with the recognition problem for linear orders, which can be solved in linear time. [5] [6]
In mathematics and computer science, currying is the technique of converting a function that takes multiple arguments into a sequence of functions that each take a single argument. For example, currying a function that takes three arguments creates three functions:
In mathematics, especially order theory, a partially ordered set formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order." The word partial in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize total orders, in which every pair is comparable.
In mathematics, a cyclic order is a way to arrange a set of objects in a circle. Unlike most structures in order theory, a cyclic order is not modeled as a binary relation, such as "a < b". One does not say that east is "more clockwise" than west. Instead, a cyclic order is defined as a ternary relation [a, b, c], meaning "after a, one reaches b before c". For example, [June, October, February]. A ternary relation is called a cyclic order if it is cyclic, asymmetric, transitive, and total. Dropping the "total" requirement results in a partial cyclic order.
In mathematics, the Weierstrass function is an example of a real-valued function that is continuous everywhere but differentiable nowhere. It is an example of a fractal curve. It is named after its discoverer Karl Weierstrass.
Noncommutative logic is an extension of linear logic which combines the commutative connectives of linear logic with the noncommutative multiplicative connectives of the Lambek calculus. Its sequent calculus relies on the structure of order varieties, and the correctness criterion for its proof nets is given in terms of partial permutations. It also has a denotational semantics in which formulas are interpreted by modules over some specific Hopf algebras.
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 Robert P. Dilworth (1950).
In mathematics, a stiff equation is a differential equation for which certain numerical methods for solving the equation are numerically unstable, unless the step size is taken to be extremely small. It has proven difficult to formulate a precise definition of stiffness, but the main idea is that the equation includes some terms that can lead to rapid variation in the solution.
Hilbert's nineteenth problem is one of the 23 Hilbert problems, set out in a list compiled in 1900 by David Hilbert. It asks whether the solutions of regular problems in the calculus of variations are always analytic. Informally, and perhaps less directly, since Hilbert's concept of a "regular variational problem" identifies precisely a variational problem whose Euler–Lagrange equation is an elliptic partial differential equation with analytic coefficients, Hilbert's nineteenth problem, despite its seemingly technical statement, simply asks whether, in this class of partial differential equations, any solution function inherits the relatively simple and well understood structure from the solved equation. Hilbert's nineteenth problem was solved independently in the late 1950s by Ennio De Giorgi and John Forbes Nash, Jr.
In mathematics, the dimension of a partially ordered set (poset) is the smallest number of total orders the intersection of which gives rise to the partial order. This concept is also sometimes called the order dimension or the Dushnik–Miller dimension of the partial order. Dushnik & Miller (1941) first studied order dimension; for a more detailed treatment of this subject than provided here, see Trotter (1992).
In convex analysis and the calculus of variations, both branches of mathematics, a pseudoconvex function is a function that behaves like a convex function with respect to finding its local minima, but need not actually be convex. Informally, a differentiable function is pseudoconvex if it is increasing in any direction where it has a positive directional derivative. The property must hold in all of the function domain, and not only for nearby points.
In mathematics, a ternary relation or triadic relation is a finitary relation in which the number of places in the relation is three. Ternary relations may also be referred to as 3-adic, 3-ary, 3-dimensional, or 3-place.
In graph theory, a comparability graph is an undirected graph that connects pairs of elements that are comparable to each other in a partial order. Comparability graphs have also been called transitively orientable graphs, partially orderable graphs, containment graphs, and divisor graphs. An incomparability graph is an undirected graph that connects pairs of elements that are not comparable to each other in a partial order.
Stochastic partial differential equations (SPDEs) generalize partial differential equations via random force terms and coefficients, in the same way ordinary stochastic differential equations generalize ordinary differential equations.
In numerical linear algebra, the alternating-direction implicit (ADI) method is an iterative method used to solve Sylvester matrix equations. It is a popular method for solving the large matrix equations that arise in systems theory and control, and can be formulated to construct solutions in a memory-efficient, factored form. It is also used to numerically solve parabolic and elliptic partial differential equations, and is a classic method used for modeling heat conduction and solving the diffusion equation in two or more dimensions. It is an example of an operator splitting method.
In mathematics, quaternionic analysis is the study of functions with quaternions as the domain and/or range. Such functions can be called functions of a quaternion variable just as functions of a real variable or a complex variable are called.
In functional analysis, a branch of mathematics, a selection theorem is a theorem that guarantees the existence of a single-valued selection function from a given multi-valued map. There are various selection theorems, and they are important in the theories of differential inclusions, optimal control, and mathematical economics.
In mathematics, Nambooripad order is a certain natural partial order on a regular semigroup discovered by K S S Nambooripad in late seventies. Since the same partial order was also independently discovered by Robert E Hartwig, some authors refer to it as Hartwig–Nambooripad order. "Natural" here means that the order is defined in terms of the operation on the semigroup.
In mathematics, a cyclically ordered group is a set with both a group structure and a cyclic order, such that left and right multiplication both preserve the cyclic order.
Vector optimization is a subarea of mathematical optimization where optimization problems with a vector-valued objective functions are optimized with respect to a given partial ordering and subject to certain constraints. A multi-objective optimization problem is a special case of a vector optimization problem: The objective space is the finite dimensional Euclidean space partially ordered by the component-wise "less than or equal to" ordering.
In graph theory, an orientation of an undirected graph is an assignment of a direction to each edge, turning the initial graph into a directed graph.
