Mike Healy. Our Tutorial at IJCNN 2009 We gave a tutorial at the International Joint Conference on Neural Networks 2009. The title is Tutorial A1_T11: Fundamentals of Categorical Neural Semantic Theory, and it was given jointly with my colleague Tom Caudell during the tutorial sessions. It consisted of three slide presentations: An Introduction and Background; Fundamentals of the Theory; and Category Theory Applied to Improving the Performance of the Neocognitron and ART networks. Handouts were provided except for Fundamentals of the Theory.
Research My research is in the mathematical semantics of biological and computational systems. Two major areas of application for mathematical semantics are knowledge representation and the technological evolution of the philosophical notion of ontology. Currently, we are developing a categorical mathematical model of faceted ontologies. Technical questions and comments are welcome. Publications in this line of research M. R. M. M. M. M. M. Shawn E. Dulany B. M. Mathematical and theoretical biology. Mathematical and theoretical biology is an interdisciplinary scientific research field with a range of applications in biology, biotechnology, and medicine.[1] The field may be referred to as mathematical biology or biomathematics to stress the mathematical side, or as theoretical biology to stress the biological side.[2] It includes at least four major subfields: biological mathematical modeling, relational biology/complex systems biology (CSB), bioinformatics and computational biomodeling/biocomputing.
Mathematical biology aims at the mathematical representation, treatment and modeling of biological processes, using a variety of applied mathematical techniques and tools. It has both theoretical and practical applications in biological, biomedical and biotechnology research. For example, in cell biology, protein interactions are often represented as "cartoon" models, which, although easy to visualize, do not accurately describe the systems studied. Importance[edit] Category theory.
A category with objects X, Y, Z and morphisms f, g, g ∘ f, and three identity morphisms (not shown) 1X, 1Y and 1Z. Several terms used in category theory, including the term "morphism", differ from their uses within mathematics itself. In category theory, a "morphism" obeys a set of conditions specific to category theory itself. Thus, care must be taken to understand the context in which statements are made. An abstraction of other mathematical concepts[edit] The most accessible example of a category is the category of sets, where the objects are sets and the arrows are functions from one set to another.
However, the objects of a category need not be sets, and the arrows need not be functions; any way of formalising a mathematical concept such that it meets the basic conditions on the behaviour of objects and arrows is a valid category, and all the results of category theory will apply to it. Category theory has several faces known not just to specialists, but to other mathematicians. The n. nLab. Physics, Topology, Logic and Computation: a Rosetta Stone | The n. Category Theory. [ Jocelyn Ireson-Paine's Home Page | Publications | Dobbs Code Talk Index | Dobbs Blog Version ] Here is a dialogue I read recently about implementing Windows XP: Daniel de França: Hi John! If I am very patient, may I be able to write Windows XP using manifolds and cobordisms? […] John Baez: If string theory is correct, Windows XP already runs using manifolds and cobordisms.
Manifold and cobordism are things used in topology: manifolds represent spaces, and cobordisms represent mappings between them. My own work on spreadsheets, which I blogged in Spreadsheet Components, Google Spreadsheets, and Code Reuse and Which Spreadsheet Components Would You Like to See? One day in Oxford, I walked into a room called KB7. However, it applied to much more. (Feedback is an example of a concept at this level. Goguen's work was at the same level of abstraction, and you can see how general he thought it was from the abstract he wrote: To be continued... Category Theory and Biology | The n. Category Theory Demonstrations.
[ Jocelyn Ireson-Paine's Home Page | Publications | What might Categories do for AI and Cognitive Science? | n-Category Café thread about Graphical Category Theory Demonstrations ] I've lost a few mails from people who asked about improvements to, or problems with, the demonstrations. One of them was about viewing angle brackets in Windows. If you sent me something and haven't had a reply, could you resend?
Thanks. This is an interactive page, with buttons you press to get examples of basic constructions in category theory. To run the demonstrations, press one of the buttons in the form below these instructions. The demonstrations do not use Java or JavaScript, and so should work in any browser. The demonstrations generate diagrams as well as text, and I'm happy for you to save and use these. Note that the demonstration handles random sets crudely. If you like these demonstrations, please consider sponsoring them. The algorithms are coded in Jan Wielemaker's free open-source SWI-Prolog.
Saunders Mac Lane. Saunders Mac Lane (4 August 1909 – 14 April 2005) was an American mathematician who co-founded category theory with Samuel Eilenberg. Early life and education[edit] Mac Lane was born in Norwich, Connecticut, near where his family lived in Taftville.[1] He was christened "Leslie Saunders MacLane", but "Leslie" fell into disuse because his parents, Donald MacLane and Winifred Saunders, came to dislike it. He began inserting a space into his surname because his first wife found it difficult to type the name without a space.[2] He was the oldest of three brothers; one of his brothers, Gerald MacLane, also became a mathematics professor at Rice University and Purdue University.
Another sister died as a baby. His father and grandfather were both ministers; his grandfather had been a Presbyterian, but was kicked out of the church for believing in evolution, and his father was a Congregationalist. In high school, Mac Lane's favorite subject was chemistry. Career[edit] Contributions[edit] Edsko de Vries - FMG - Research - Computer Science Department - Computer Science - Trinity College Dublin. TheCatsters's Channel Eugenia Cheng. Simon Willerton. Nicolas Rashevsky. Nicolas Rashevsky (November 9, 1899 – January 16, 1972) was an American theoretical physicist who pioneered mathematical biology, and is also considered the father of mathematical biophysics and theoretical biology.[1][2][3][4][5] Academic career[edit] He studied theoretical physics at the University of Kiev in Ukraine (then Russian Empire) before 1917, and immigrated first to Turkey, then to Poland, France, and finally to the US in 1924 because of the October revolution.
[citation needed] He was awarded a Rockefeller Fellowship in 1934 and went to the University of Chicago to take up the appointment of assistant professor in the Department of Physiology. Major scientific contributions[edit] In 1938 he published the first book on mathematical biology and mathematical biophysics entitled: "Mathematical Biophysics: Physico-Mathematical Foundations of Biology. " In the same year he established the World' s first[citation needed] PhD program in Mathematical Biology at the University of Chicago. Robert Rosen (theoretical biologist) Robert Rosen (June 27, 1934 – December 28, 1998) was an American theoretical biologist and Professor of Biophysics at Dalhousie University.[1] Rosen was born on June 27, 1934 in Brownsville (a section of Brooklyn), in New York City.
He studied biology, mathematics, physics, philosophy, and history; particularly, the history of science. In 1959 he obtained a PhD in relational biology, a specialization within the broader field of Mathematical Biology, under the guidance of Professor Nicolas Rashevsky at the University of Chicago. He remained at the University of Chicago until 1964,[2] later moving to the University of Buffalo (now known as the State University of New York (SUNY)) at Buffalo on a full associate professorship, while holding a joint appointment at the Center for Theoretical Biology. He served as president of the Society for General Systems Research, (now known as ISSS), in 1980-81. stands for the metabolic and -systems in terms of enzymes ( -mapping). Panmere. The Mathematics Genealogy Project - Aloisius Louie. More Than Life Itself: A Synthetic Continuation in Relational Biology (Categories) (Volume 1) (9783868380446): A. Louie. MEMORY EVOLUTIVE SYSTEM HomePage.
Untitled Document. Innovaxiom - Conférence mathématiques innovantes 2010 - Andrée Ehresmann. Memory Evolutive Systems; Hierarchy, Emergence, Cognition, Volume 4 (Studies in Multidisciplinarity) (9780444522443): A C Ehresmann, J.P. Vanbremeersch. Functor in nLab. Graph in nLab. HaskellWiki. Category theory. Haskell/Category theory. This article attempts to give an overview of category theory, in so far as it applies to Haskell. To this end, Haskell code will be given alongside the mathematical definitions. Absolute rigour is not followed; in its place, we seek to give the reader an intuitive feel for what the concepts of category theory are and how they relate to Haskell. Introduction to categories[edit] A simple category, with three objects A, B and C, three identity morphisms and , and two other morphisms .
A category is, in essence, a simple collection. A collection of objects.A collection of morphisms, each of which ties two objects (a source object and a target object) together. Lots of things form categories. Is a morphism in Grp iff: It may seem that morphisms are always functions, but this needn't be the case. ) defines a category where the objects are the elements of P, and there is a morphism between any two objects A and B iff . Are both functions with source object and target object Category laws[edit] . ?
. . Learning Haskell through Category Theory, and Adventuring in Category Land: Like Flatterland, Only About Categories « Monadically Speaking: Adventures in Programming Language Theory. Two days ago, there was an interesting post by Andrzej Jaworski, entitled “[Haskell] Teach theory then Haskell as example,” dated “Wed, 14 Jan 2009 04:37:33 +0100,” on the Haskell mailing list on the Haskell programming language, recommending that Haskell be taught “on most abstract terms in a framework of higher order logic, types and CT right from the start.” While I agreed with the gist of his post, I hadn’t found an appropriate publication on category theory that addressed the subject at the proper pace, level, and perspective. Most publications did not explain enough detail, assumed too many topics not covered, or did not explain the concepts in a manner which would allow me to form a visual framework of reference in my mind.
In my response, I listed the following publications on category theory, a branch of mathematics which forms a theoretical framework for Haskell. Category Theory Books:Conceptual Mathematics: A First Introduction to Categories (Paperback) by F. I added: Yes! CatTheory and Func Prog Michiexile/MATH198. Computational category theory.
Welcome to the Computational Category Theory Project Computational Category Theory is an implementation of concepts and constructions from category theory in the functional programming language Standard ML. The original ideas are due to R.M. Burstall, and it was developed by D. Rydeheard, with help from D.T. Sannella and others in the University of Edinburgh theoretical computer science community. The Manual For full details of the project, there is a copy of the manual (in PDF or Postscript) available here. The Programs To access the Computational Category Theory programs go to this directory. Any comments? The manual is available for personal use only and not to be distributed, made multiple copies, made available on other websites, or sold in any format.
Standard ML. SML is a modern descendant of the ML programming language used in the Logic for Computable Functions (LCF) theorem-proving project. It is distinctive among widely used languages in that it has a formal specification, given as typing rules and operational semantics in The Definition of Standard ML (1990, revised and simplified as The Definition of Standard ML (Revised) in 1997).[1] Language[edit] fun factorial n = if n = 0 then 1 else n * factorial (n-1) A Standard ML compiler is required to infer the static type int -> int of this function without user-supplied type annotations.
I.e., it has to deduce that n is only used with integer expressions, and must therefore itself be an integer, and that all value-producing expressions within the function return integers. fun factorial 0 = 1 | factorial n = n * factorial (n - 1) This can be rewritten using a case statement like this: val rec factorial = fn n => case n of 0 => 1 | n => n * factorial (n - 1) or as a lambda function: Type synonyms[edit] Functional Programming Course. Overview The purpose of this course is to introduce the theory and practice of functional programming (FP).
The characteristic feature of FP is the emphasis on computation as evaluation. The traditional distinction between program and data characteristic of imperative programming (IP) is replaced by an emphasis on classifying expressions by types that specify their applicative behavior. Types include familiar (fixed and arbitrary precision) numeric types, tuples and records (structs), classified values (objects), inductive types such as trees, functions with specified inputs and outputs, and commands such as input and output.
The advantages of FP are significant: Verification: There is a close correspondence between the mathematical reasoning that justifies the correctness of a program and the program itself. Moreover, FP generalizes IP by treating commands as forms of data that may be executed for their effects. Prerequisites: 15-151 or 21-127. Past Instances.