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In 1968 Shirley Chisholm became the first Black woman elected to the United States Congress. In 1972 she launched a campaign for the Democratic presidential nomination, breaking another political barrier.
MORE →Reflects the personal views, recollections, and perspective of the author, Mike Davis.
This is a personal recollection on the Move fire on May 13, 1985
In computational complexity theory, the complexity class FNP is the function problem extension of the decision problem class NP. The name is somewhat of a misnomer, since technically it is a class of binary relations, not functions, as the following formal definition explains:
This definition does not involve nondeterminism and is analogous to the verifier definition of NP.
There is an NP language directly corresponding to every FNP relation, sometimes called the decision problem induced by or corresponding to said FNP relation. It is the language formed by taking all the x for which there exists some y such that P(x,y) holds; however, there may be more than one FNP relation that induces a particular decision problem.
Many problems in NP, including many NP-complete problems, can be specified by asking whether a particular object exists, such as a satisfying assignment, a graph coloring, or a clique of a certain size. These problems often correspond to relations in FNP that ask not only whether an object exists but what value or values such an object can have. When an FNP relation corresponds in this way to an NP-complete problem, the relation is self-reducible. Bellare and Goldwasser showed in 1994 using some standard complexity-theoretic assumptions that there exist problems in NP such that none of their FNP versions are self-reducible, implying that they are harder than their corresponding decision problem.[2]
For each P in FNP, the search problem associated with P is: given x, find a y such that P(x,y) holds, or state that no such y exists. The search problem for every relation in FNP can be solved deterministically in polynomial time if and only if P = NP. This result is usually stated as "FP = FNP if and only if P = NP"; however, for this statement to be true, it is necessary to redefine FP and FNP so that the members of FP and FNP are not relations, but are instead search problems associated with relations.
Let P1 and P2 be two problems in FNP, with associated verification algorithms A1, A2. A reduction P1 and P2 is defined as two polynomial-time computable functions, f and g, such that[3]
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In computational complexity theory, the complexity class FNP is the function problem extension of the decision problem class NP. The name is somewhat of a misnomer, since technically it is a class of binary relations, not functions, as the following formal definition explains: A binary relation P(x,y), where y is at most polynomially longer than x, is in FNP if and only if there is a deterministic polynomial-time algorithm that can determine whether P(x,y) holds given both x and y. This definition does not involve nondeterminism and is analogous to the verifier definition of NP. There is an NP language directly corresponding to every FNP relation, sometimes called the decision problem induced by or corresponding to said FNP relation. It is the language formed by taking all the x for which there exists some y such that P(x,y) holds; however, there may be more than one FNP relation that induces a particular decision problem. Many problems in NP, including many NP-complete problems, can be specified by asking whether a particular object exists, such as a satisfying assignment, a graph coloring, or a clique of a certain size. These problems often correspond to relations in FNP that ask not only whether an object exists but what value or values such an object can have. When an FNP relation corresponds in this way to an NP-complete problem, the relation is self-reducible. Bellare and Goldwasser showed in 1994 using some standard complexity-theoretic assumptions that there exist problems in NP such that none of their FNP versions are self-reducible, implying that they are harder than their corresponding decision problem. For each P in FNP, the search problem associated with P is: given x, find a y such that P(x,y) holds, or state that no such y exists. The search problem for every relation in FNP can be solved deterministically in polynomial time if and only if P = NP. This result is usually stated as "FP = FNP if and only if P = NP"; however, for this statement to be true, it is necessary to redefine FP and FNP so that the members of FP and FNP are not relations, but are instead search problems associated with relations.
In the mathematical field of infinite group theory, the Nottingham group is the group J(Fp) or N(Fp) consisting of formal power series t + a2t2+... with coefficients in Fp. The group multiplication is given by formal composition also called substitution. That is, if f = t + ∑ n = 2 ∞ a n t n {\displaystyle f=t+\sum _{n=2}^{\infty }a_{n}t^{n}} and if g {\displaystyle g} is another element, then g f = f ( g ) = g + ∑ n = 2 ∞ a n g n {\displaystyle gf=f(g)=g+\sum _{n=2}^{\infty }a_{n}g^{n}} . The group multiplication is not abelian. The group was studied by number theorists as the group of wild automorphisms of the local field Fp((t)) and by group theorists including D. Johnson (1988) and the name "Nottingham group" refers to his former domicile. This group is a finitely generated pro-p-group, of finite width. For every finite group of order a power of p there is a closed subgroup of the Nottingham group isomorphic to that finite group.
In computational complexity theory, the complexity class FP is the set of function problems that can be solved by a deterministic Turing machine in polynomial time (and for which the function problem also represents a predicate decidable in polynomial time). It is the function problem version of the decision problem class P. Roughly speaking, it is the class of functions that can be efficiently computed on classical computers without randomization. The difference between FP and P is that problems in P have one-bit, yes/no answers, while problems in FP can have any output that can be computed in polynomial time. For example, adding two numbers is an FP problem, while determining if their sum is odd is in P. So any decision problem in P can be thought of as a function that outputs 0 or 1, so it’s trivially in FP. In short: P ⊆ F P {\displaystyle {\mathsf {P}}\subseteq {\mathsf {FP}}} . Polynomial-time function problems are fundamental in defining polynomial-time reductions, which are used in turn to define the class of NP-complete problems.
FP (for functional programming) is a programming language created by John Backus to support the function-level programming paradigm. It allows building programs from a set of generally useful language primitives and avoiding named variables (a style also called tacit programming or "point free"). It was heavily influenced by APL developed by Kenneth E. Iverson in the early 1960s. The FP language was introduced in Backus's 1977 Turing Award paper, "Can programming be liberated from the von Neumann style?: a functional style and its algebra of programs". The paper sparked interest in functional programming research, eventually leading to modern functional languages, which are largely founded on the lambda calculus paradigm, and not the function-level paradigm Backus had hoped. In his Turing award paper, Backus described how the FP style is different: An FP system is based on the use of a fixed set of combining forms called functional forms. These, plus simple definitions, are the only means of building new functions from existing ones; they use no variables or substitutions rules, and they become the operations of an associated algebra of programs. All the functions of an FP system are of one type: they map objects onto objects and always take a single argument. FP was used little beyond academia. In the 1980s, Backus created a successor language, FL, as an internal project at IBM Research.
Before the 1921 destruction of Tulsa’s Greenwood District, Black residents had created a remarkable center of business and community life. The district included stores, professional offices, entertainment venues and homes owned by Black citizens. Understanding Greenwood means learning what was built—not only what was burned.
MORE →Joe Louis.