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Viggo Stoltenberg-Hansen, born 1942, professor at Uppsala University, Department of Mathematics, is a Swedish mathematician/logician and expert on domain theory and recursion theory (also known as computability theory). Viggo received his PhD in Mathematics (titled "On Priority Arguments In Friedberg Theories") from University of Toronto in 1973.
Viggo Stoltenberg-Hansen and John Tucker developed in the early 1980s a general method of domain representations of topological algebras.
Viggo is the main author of the textbook "Mathematical Theory of Domains", Cambridge University Press, 1994 (coauthored by I. Lindström and E. Griffor), and also of a set of Marktoberdorf summer school lecture notes on domain theory.
Viggo Stoltenberg-Hansen and John Tucker made a thorough analysis of the computability associated to effective algebras and continuity of homomorphisms between such.
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In mathematics, computable numbers are the real numbers that can be computed to within any desired precision by a finite, terminating algorithm. They are also known as the recursive numbers, effective numbers or the computable reals or recursive reals.
Discrete mathematics is the study of mathematical structures that can be considered "discrete" rather than "continuous". Objects studied in discrete mathematics include integers, graphs, and statements in logic. By contrast, discrete mathematics excludes topics in "continuous mathematics" such as real numbers, calculus or Euclidean geometry. Discrete objects can often be enumerated by integers; more formally, discrete mathematics has been characterized as the branch of mathematics dealing with countable sets. However, there is no exact definition of the term "discrete mathematics".
Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal systems of logic such as their expressive or deductive power. However, it can also include uses of logic to characterize correct mathematical reasoning or to establish foundations of mathematics.
Domain theory is a branch of mathematics that studies special kinds of partially ordered sets (posets) commonly called domains. Consequently, domain theory can be considered as a branch of order theory. The field has major applications in computer science, where it is used to specify denotational semantics, especially for functional programming languages. Domain theory formalizes the intuitive ideas of approximation and convergence in a very general way and is closely related to topology.
In logic, a true/false decision problem is decidable if there exists an effective method for deriving the correct answer. Zeroth-order logic is decidable, whereas first-order and higher-order logic are not. Logical systems are decidable if membership in their set of logically valid formulas can be effectively determined. A theory in a fixed logical system is decidable if there is an effective method for determining whether arbitrary formulas are included in the theory. Many important problems are undecidable, that is, it has been proven that no effective method for determining membership can exist for them.
Categorical logic is the branch of mathematics in which tools and concepts from category theory are applied to the study of mathematical logic. It is also notable for its connections to theoretical computer science. In broad terms, categorical logic represents both syntax and semantics by a category, and an interpretation by a functor. The categorical framework provides a rich conceptual background for logical and type-theoretic constructions. The subject has been recognisable in these terms since around 1970.
Computational science, also known as scientific computing, technical computing or scientific computation (SC), is a field in mathematics that uses advanced computing capabilities to understand and solve complex problems. It is an area of science that spans many disciplines, but at its core, it involves the development of models and simulations to understand natural systems.
In the mathematical areas of order and lattice theory, the Kleene fixed-point theorem, named after American mathematician Stephen Cole Kleene, states the following:
Society for Industrial and Applied Mathematics (SIAM) is a professional society dedicated to applied mathematics, computational science, and data science through research, publications, and community. SIAM is the world's largest scientific society devoted to applied mathematics, and roughly two-thirds of its membership resides within the United States. Founded in 1951, the organization began holding annual national meetings in 1954, and now hosts conferences, publishes books and scholarly journals, and engages in advocacy in issues of interest to its membership. Members include engineers, scientists, and mathematicians, both those employed in academia and those working in industry. The society supports educational institutions promoting applied mathematics.
In computability theory, admissible numberings are enumerations (numberings) of the set of partial computable functions that can be converted to and from the standard numbering. These numberings are also called acceptable numberings and acceptable programming systems.
A computation tree is a representation for the computation steps of a non-deterministic Turing machine on a specified input. A computation tree is a rooted tree of nodes and edges. Each node in the tree represents a single computational state, while each edge represents a transition to the next possible computation. The number of nodes of the tree is the size of the tree and the length of the path from the root to a given node is the depth of the node. The largest depth of an output node is the depth of the tree. The leaves of the tree are called output nodes.
John Vivian Tucker is a British computer scientist and expert on computability theory, also known as recursion theory. Computability theory is about what can and cannot be computed by people and machines. His work has focused on generalising the classical theory to deal with all forms of discrete/digital and continuous/analogue data; and on using the generalisations as formal methods for system design; based on abstract data types and on the interface between algorithms and physical equipment.
Steve Vickers is a British mathematician and computer scientist. In the early 1980s, he wrote ROM firmware and manuals for three home computers, the ZX81, ZX Spectrum, and Jupiter Ace. The latter was produced by Jupiter Cantab, a short-lived company Vickers formed together with Richard Altwasser, after the two had left Sinclair Research. Since the late 1980s, Vickers has been an academic in the field of geometric logic, writing over 30 papers in scholarly journals on mathematical aspects of computer science. His book Topology via Logic has been influential over a range of fields. In October 2018, he retired as senior lecturer at the University of Birmingham. As announced on his university homepage, he continues to supervise PhD students at the university and focus on his research.
Johannes Aldert "Jan" Bergstra is a Dutch computer scientist. His work has focussed on logic and the theoretical foundations of software engineering, especially on formal methods for system design. He is best known as an expert on algebraic methods for the specification of data and computational processes in general.
The low basis theorem is one of several basis theorems in computability theory, each of which showing that, given an infinite subtree of the binary tree , it is possible to find an infinite path through the tree with particular computability properties. The low basis theorem, in particular, shows that there must be a path which is low; that is, the Turing jump of the path is Turing equivalent to the halting problem .
Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. Often, these applied methods are beyond simple geometry, and may include differential and integral calculus, difference and differential equations, matrix algebra, mathematical programming, or other computational methods. Proponents of this approach claim that it allows the formulation of theoretical relationships with rigor, generality, and simplicity.
In mathematical logic, true arithmetic is the set of all true first-order statements about the arithmetic of natural numbers. This is the theory associated with the standard model of the Peano axioms in the language of the first-order Peano axioms. True arithmetic is occasionally called Skolem arithmetic, though this term usually refers to the different theory of natural numbers with multiplication.
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models.
Validated numerics, or rigorous computation, verified computation, reliable computation, numerical verification is numerics including mathematically strict error evaluation, and it is one field of numerical analysis. For computation, interval arithmetic is used, and all results are represented by intervals. Validated numerics were used by Warwick Tucker in order to solve the 14th of Smale's problems, and today it is recognized as a powerful tool for the study of dynamical systems.
Applied category theory is an academic discipline in which methods from category theory are used to study other fields including but not limited to computer science, physics, natural language processing, control theory, probability theory and causality. The application of category theory in these domains can take different forms. In some cases the formalization of the domain into the language of category theory is the goal, the idea here being that this would elucidate the important structure and properties of the domain. In other cases the formalization is used to leverage the power of abstraction in order to prove new results about the field.
