China
Africa
Explore chapters and articles related to this topic
The Landscape
Published in Alessio Plebe, Pietro Perconti, The Future of the Artificial Mind, 2021
Alessio Plebe, Pietro Perconti
Representations, however, are not concrete objects. They are like numbers and the basic units of arithmetic. More precisely, mental representations are abstract rules. Their task is to establish a relation between two domains They are comparable to the notion of mathematical function. A function in this context is a relation between two sets that can establish a correspondence between the elements of the first set, called domain, and the elements of the second set, called codomain. In the brain of an animal, a mental representation is like an abstract rule, even if it is physically realized in configurations that are present in the nervous system. The rule provides a link between behavioral types (domain) and environmental influences (codomain) Even if we are able to trace something in the brain that acts as a counterpart to a particular property of the world or object of the world, it is not a mental representation. It is a physical configuration that is connected to a particular class of environmental input via a mental representation. Thus understood, representation has the advantage of being a concept that can be used in a computational context while having a naturalistic counterpart in neural activations and in the class of environmental stimuli that can trigger them. For this reason, representation is the best theoretical construct available to a science of the mind that aspires to be both computational and naturalistic.
Preliminary Content
Published in John D. Ross, Kendall C. Richards, Introductory Analysis, 2020
John D. Ross, Kendall C. Richards
We next want to define the domain, codomain, and range of a function. Colloquially: the domain is the set of elements that can “be plugged in” to a function, the range is the set of elements that actually get produced by the function, and the codomain is a specified set that contains the range as a subset. We formalize this below: Let f be a function between S and T. The set of elements in S that appear in the function f (i.e., the set {a ∈ S | (a, b) ∈ f for some b ∈ T} is called the domain of the function f.Let f be a function between S and T. The set of elements in T that appear in the function f (i.e., the set {b ∈ T | (a, b) ∈ f for some a ∈ S} is called the range of the function f.Let f be a function between S and T. Then the set T is traditionally called the specified codomain of f.
Vector Spaces
Published in Jeff Suzuki, Linear Algebra, 2021
In a linear transformation that acts on a vector space Fn, the domain is easy to describe: it is the set of all vectors in Fn. Similarly, the codomain would be easy to describe. But the range itself might be “smaller” than the codomain. We'd like to be able to describe the range in some meaningful fashion. To that end, we'll need to introduce a few more ideas.
On totalisation of computable functions in a distributive environment
Published in International Journal of Parallel, Emergent and Distributed Systems, 2022
We remind the following substantial characteristics of functions: The domain of a function f is the set Dom f in which f can be (potentially) defined;the codomain of a function f is the set Codom f in which f can (potentially) take values;the definability domain of a function f is the largest set DDom f for all elements of which f is defined or formally, DDom f = {x; ∃ y ∈ Codom f (f(x) = y)};the range Rg f of a function f is the set of all elements from Codom f assigned by f to elements from DDom f, or formally, Rg f = {y; ∃ x ∈ Dom f (f(x) = y)}.
Cruxes for visual domain sonification in digital arts
Published in Digital Creativity, 2021
To further develop this potential, this paper introduces hierarchically structured groups of meaningful relations between core elements at the level of natural (physics-based) phenomena, continuing with their human perception (determined by physiology) and ending with artistic (‘intra-cognitive’) relations. The work builds on the mathematical concepts of domains, functions and mappings. In mathematics, a domain is a set of elements that a particular function links to another set of elements (also called a codomain). This linkage can also be viewed as a process of mapping elements from the domain to the elements in the codomain. And focused on artistic applications, the so-called cruxes are introduced. They represent the meaningful core mappings, which are divided into three groups mentioned above. In brief – the cruxes are about conceptualization, i.e. structured mappings between essential constituents of audio and visual domain in sonification explorations, while relying on digital technology. Although the basic intention of the apparatus deployment is in digital arts domain (with the emphasis on generative art), the anticipated applications of such mappings are much broader.