Domain Theory - School of Computer Science, University of

... How can we give a non-circular account of its meaning? Suppose we are working inside some mathematical structure D. We want to find an element d ∈ D such that substituting d for x in (1) yields a valid equation. The right-hand-side of (1) can be read as a function of X, semantically as f : D → D. We ...

... How can we give a non-circular account of its meaning? Suppose we are working inside some mathematical structure D. We want to find an element d ∈ D such that substituting d for x in (1) yields a valid equation. The right-hand-side of (1) can be read as a function of X, semantically as f : D → D. We ...

Domain Theory

... How can we give a non-circular account of its meaning? Suppose we are working inside some mathematical structure D. We want to find an element d ∈ D such that substituting d for x in (1) yields a valid equation. The right-hand-side of (1) can be read as a function of X, semantically as f : D → D. We ...

... How can we give a non-circular account of its meaning? Suppose we are working inside some mathematical structure D. We want to find an element d ∈ D such that substituting d for x in (1) yields a valid equation. The right-hand-side of (1) can be read as a function of X, semantically as f : D → D. We ...

Annals of Pure and Applied Logic Dynamic topological S5

... open sets ∅, {0}, {1} and X ; f (0) = 1 and f (1) = 0; and V (p) = {0}. It is easy to check that M 6 (2p ⊃ ◦2p). Fact 2.4.2. (2◦p ⊃ ◦2p) ∈ S5H – S5Ct. Proof. Given the soundness of S5Ct for trivial spaces, it suffices to find a dynamic topological model M = hX , f , V i, where X is trivial and M ...

... open sets ∅, {0}, {1} and X ; f (0) = 1 and f (1) = 0; and V (p) = {0}. It is easy to check that M 6 (2p ⊃ ◦2p). Fact 2.4.2. (2◦p ⊃ ◦2p) ∈ S5H – S5Ct. Proof. Given the soundness of S5Ct for trivial spaces, it suffices to find a dynamic topological model M = hX , f , V i, where X is trivial and M ...

FINITE TOPOLOGIES AND DIGRAPHS

... The one to one correspondence between finite preorder relations and finite topologies with the same underlying set of points, and also between finite posets and finite T0 topologies is well known. Then, the one to one correspondence between finite digraphs and topologies is easily deducible. In fact ...

... The one to one correspondence between finite preorder relations and finite topologies with the same underlying set of points, and also between finite posets and finite T0 topologies is well known. Then, the one to one correspondence between finite digraphs and topologies is easily deducible. In fact ...

Partial Continuous Functions and Admissible Domain Representations

... This suggests representing maps from X to Y by partial continuous functions from D to E: We say that f : D * E represents f : X → Y if the diagram f ...

... This suggests representing maps from X to Y by partial continuous functions from D to E: We say that f : D * E represents f : X → Y if the diagram f ...

Extension of continuous functions in digital spaces with the

... Let us agree that Zn is equipped with the Khalimsky topology from now on, unless otherwise stated. This makes it meaningful, for example, to talk about continuous functions Z → Z. What properties then, does such a function have? First of all, it is necessarily Lipschitz with Lipschitz constant 1. We ...

... Let us agree that Zn is equipped with the Khalimsky topology from now on, unless otherwise stated. This makes it meaningful, for example, to talk about continuous functions Z → Z. What properties then, does such a function have? First of all, it is necessarily Lipschitz with Lipschitz constant 1. We ...

THURSTON OBSTRUCTIONS FOR CUBIC BRANCHED

... any obstruction for F is a Levy cycle. This note is part of the work in [Sha15] studying the mating of pairs of cubic polynomials which each have a fixed critical point. ...

... any obstruction for F is a Levy cycle. This note is part of the work in [Sha15] studying the mating of pairs of cubic polynomials which each have a fixed critical point. ...

The Filter Dichotomy and medial limits

... Talagrand [24] proved that a filter on ω is meager if and only if there exists a finite-to-one function h : ω → ω such that {h[x] | x ∈ F } is the set of cofinite subsets of ω. The Filter Dichotomy can then be restated as: for every uniform filter F on ω there is a finite-to-one function h : ω → ω s ...

... Talagrand [24] proved that a filter on ω is meager if and only if there exists a finite-to-one function h : ω → ω such that {h[x] | x ∈ F } is the set of cofinite subsets of ω. The Filter Dichotomy can then be restated as: for every uniform filter F on ω there is a finite-to-one function h : ω → ω s ...

2 Continuous maps. 3 Induced topologies.

... If f : X → Y is a function and V is a topology on Y , the topology on X induced by f is the topology whose open sets are exactly the sets of the form f −1 (V ) for V ∈ V. This topology is denoted f ∗ (U). The formula below defines the pull-back topology. U ∈ f ∗ (V) ⇔ ∃(V ∈ V) U = f −1 (V ) ...

... If f : X → Y is a function and V is a topology on Y , the topology on X induced by f is the topology whose open sets are exactly the sets of the form f −1 (V ) for V ∈ V. This topology is denoted f ∗ (U). The formula below defines the pull-back topology. U ∈ f ∗ (V) ⇔ ∃(V ∈ V) U = f −1 (V ) ...

Functions 1 2.2 Definition of a Function

... a unique element of the second set. The first set is called the domain of the function. For each element x of the domain, the corresponding element y of the second set is called the image of x under the function. The set of all images of the elements of the domain is called the range of the function ...

... a unique element of the second set. The first set is called the domain of the function. For each element x of the domain, the corresponding element y of the second set is called the image of x under the function. The set of all images of the elements of the domain is called the range of the function ...

2.5

... We can think of a function as a machine that takes some input x and turns it into some output f (x). The set of numbers that we put into the machine is the domain of the function, and the set of numbers that comes out is the range. ...

... We can think of a function as a machine that takes some input x and turns it into some output f (x). The set of numbers that we put into the machine is the domain of the function, and the set of numbers that comes out is the range. ...

PDF

... that for a domain D being connected is equivalent to being path-connected. Since we have that every component of a region D will be a domain, we have that every region has at most countably many components. This definition has no particular relationship to the notion of an integral domain, used in a ...

... that for a domain D being connected is equivalent to being path-connected. Since we have that every component of a region D will be a domain, we have that every region has at most countably many components. This definition has no particular relationship to the notion of an integral domain, used in a ...

# Julia set

In the context of complex dynamics, a topic of mathematics, the Julia set and the Fatou set are two complementary sets (Julia 'laces' and Fatou 'dusts') defined from a function. Informally, the Fatou set of the function consists of values with the property that all nearby values behave similarly under repeated iteration of the function, and the Julia set consists of values such that an arbitrarily small perturbation can cause drastic changes in the sequence of iterated function values.Thus the behavior of the function on the Fatou set is 'regular', while on the Julia set its behavior is 'chaotic'.The Julia set of a function f is commonly denoted J(f), and the Fatou set is denoted F(f). These sets are named after the French mathematicians Gaston Julia and Pierre Fatou whose work began the study of complex dynamics during the early 20th century.