Civil Rights
Movements, leaders, victories and the continuing fight for equality.
q2K BHP
Black History Portal
THE BHP WIRE —
HIDDEN TRUTHS
What's New!
Explore the people, places, events, achievements, struggles and stories that shaped our journey.
Movements, leaders, victories and the continuing fight for equality.
Innovation, patents, science, technology and world-changing contributions.
Pioneers, champions, Negro Leagues, records, activism and excellence.
Meet the people whose lives, choices and achievements shaped the journey.
Black towns, communities, institutions and places where history happened.
Moments that changed communities, movements, institutions and the nation.
Katherine Johnson’s mathematical calculations helped guide some of America’s most important early space missions while she confronted the racial and gender barriers faced by Black women in twentieth-century America.
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 mathematics, given two partially ordered sets P and Q, a function f: P → Q between them is Scott-continuous (named after the mathematician Dana Scott) if it preserves all directed suprema. That is, for every directed subset D of P with supremum in P, its image has a supremum in Q, and that supremum is the image of the supremum of D, i.e. ∐f[D] = f(∐D), where ∐ is the directed join.[1] When Q is the poset of truth values, i.e. Sierpiński space, then Scott-continuous functions are characteristic functions of open sets, and thus Sierpiński space is the classifying space for open sets.[2]
A subset U of a partially ordered set P is called Scott-open if it is an upper set and if it is inaccessible by directed joins, i.e. if all directed sets D with supremum in U have non-empty intersection with U. The Scott-open subsets of a partially ordered set P form a topology on P, the Scott topology. A function between partially ordered sets is Scott-continuous if and only if it is continuous with respect to the Scott topology.[1]
The Scott topology was first defined by Dana Scott for complete lattices and later defined for arbitrary partially ordered sets.[3]
Scott-continuous functions are used in the study of models for lambda calculi[3] and the denotational semantics of computer programs.
A Scott-continuous function is always monotonic, meaning that if A ≤P B for A, B ∈ P, then f(A) ≤Q f(B).
A subset of a directed complete partial order is closed with respect to the Scott topology induced by the partial order if and only if it is a lower set and closed under suprema of directed subsets.[4]
A directed complete partial order (dcpo) with the Scott topology is always a Kolmogorov space (i.e., it satisfies the T0 separation axiom).[4] However, a dcpo with the Scott topology is a Hausdorff space if and only if the order is trivial.[4] The Scott-open sets form a complete lattice when ordered by inclusion.[5]
For any Kolmogorov space, the topology induces an order relation on that space, the specialization order: x ≤ y if and only if every open neighbourhood of x is also an open neighbourhood of y. The order relation of a dcpo D can be reconstructed from the Scott-open sets as the specialization order induced by the Scott topology. However, a dcpo equipped with the Scott topology need not be sober: the specialization order induced by the topology of a sober space makes that space into a dcpo, but the Scott topology derived from this order is finer than the original topology.[4]
The open sets in a given topological space when ordered by inclusion form a lattice on which the Scott topology can be defined. A subset X of a topological space T is compact with respect to the topology on T (in the sense that every open cover of X contains a finite subcover of X) if and only if the set of open neighbourhoods of X is open with respect to the Scott topology.[5]
For CPO, the cartesian closed category of dcpo's, two particularly notable examples of Scott-continuous functions are curry and apply.[6]
Nuel Belnap used Scott continuity to extend logical connectives to a four-valued logic.[7]
{{cite journal}}: Cite uses deprecated parameter |citeseerx= (help)
Source: Wikipedia. Article content is retrieved live through the MediaWiki API.
In mathematics, given two partially ordered sets P and Q, a function f: P → Q between them is Scott-continuous (named after the mathematician Dana Scott) if it preserves all directed suprema. That is, for every directed subset D of P with supremum in P, its image has a supremum in Q, and that supremum is the image of the supremum of D, i.e. ∐f[D] = f(∐D), where ∐ is the directed join. When Q is the poset of truth values, i.e. Sierpiński space, then Scott-continuous functions are characteristic functions of open sets, and thus Sierpiński space is the classifying space for open sets. A subset U of a partially ordered set P is called Scott-open if it is an upper set and if it is inaccessible by directed joins, i.e. if all directed sets D with supremum in U have non-empty intersection with U. The Scott-open subsets of a partially ordered set P form a topology on P, the Scott topology. A function between partially ordered sets is Scott-continuous if and only if it is continuous with respect to the Scott topology. The Scott topology was first defined by Dana Scott for complete lattices and later defined for arbitrary partially ordered sets. Scott-continuous functions are used in the study of models for lambda calculi and the denotational semantics of computer programs.
In the mathematical areas of order and lattice theory, the Kleene fixed-point theorem, named after American mathematician Stephen Cole Kleene, states the following: Kleene Fixed-Point Theorem. Suppose ( L , ⊑ ) {\displaystyle (L,\sqsubseteq )} is a directed-complete partial order (dcpo) with a least element, and let f : L → L {\displaystyle f:L\to L} be a Scott-continuous (and therefore monotone) function. Then f {\displaystyle f} has a least fixed point, which is the supremum of the ascending Kleene chain of f . {\displaystyle f.} The ascending Kleene chain of f is the chain ⊥ ⊑ f ( ⊥ ) ⊑ f ( f ( ⊥ ) ) ⊑ ⋯ ⊑ f n ( ⊥ ) ⊑ ⋯ {\displaystyle \bot \sqsubseteq f(\bot )\sqsubseteq f(f(\bot ))\sqsubseteq \cdots \sqsubseteq f^{n}(\bot )\sqsubseteq \cdots } obtained by iterating f on the least element ⊥ of L. Expressed in a formula, the theorem states that lfp ( f ) = sup ( { f n ( ⊥ ) ∣ n ∈ N } ) {\displaystyle {\textrm {lfp}}(f)=\sup \left(\left\{f^{n}(\bot )\mid n\in \mathbb {N} \right\}\right)} where lfp {\displaystyle {\textrm {lfp}}} denotes the least fixed point. Although Tarski's fixed point theorem does not consider how fixed points can be computed by iterating f from some seed (also, it pertains to monotone functions on complete lattices), this result is often attributed to Alfred Tarski who proves it for additive functions. Moreover, the Kleene fixed-point theorem can be extended to monotone functions using transfinite iterations.
Scott Ian (born Scott Ian Rosenfeld, December 31, 1963) is an American musician, best known as the rhythm guitarist, lyricist and co-founder of the thrash metal band Anthrax, of which he is the sole continuous member. Ian is also the guitarist, lyricist, and a founding member of the crossover thrash band Stormtroopers of Death, and is the rhythm guitarist for the metal bands the Damned Things and Mr. Bungle. He has hosted The Rock Show on VH1 and has appeared on VH1's I Love the... series, Heavy: The Story of Metal, and SuperGroup.
A continuous truss bridge is a truss bridge that extends without hinges or joints across three or more supports. A continuous truss bridge may use less material than a series of simple trusses because a continuous truss distributes live loads across all the spans; in a series of simple trusses, each truss must be capable of supporting the entire load. Although some continuous truss bridges resemble cantilever bridges and may be constructed using cantilever techniques, there are essential differences between the two forms. Cantilever bridges need not connect rigidly mid-span, as the cantilever arms are self-supporting. Although some cantilever bridges appear continuous due to decorative trusswork at the joints, these bridges will remain standing if the connections between the cantilevers are broken or the suspended span (if any) is removed. Conversely, continuous truss bridges rely on rigid truss connections throughout the structure for stability. Severing a continuous truss mid-span endangers the structure, as exemplified by the collapse of Baltimore's Francis Scott Key Bridge in March 2024. However, continuous truss bridges do not experience the tipping forces that a cantilever bridge must resist because the main span of a continuous truss bridge is supported at both ends. It is possible to convert a series of simple truss spans into a continuous truss. For example, the northern approach to the Golden Gate Bridge was initially constructed as a series of five simple truss spans. In 2001, a seismic retrofit project connected the five spans into a single continuous truss bridge.
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 →The Greenwood District.