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1. JSTOR Recursive Number Theory (NorthHolland, Amsterdam) Recursive number theory is the study of the natural numbers in which only Recursive functions and relations may be defined. http://links.jstor.org/sici?sici=0025-5572(195812)2:42:342<326:RNT>2.0.CO;2-Q

2. Ohio Resource Center > Record > Trout Pond: Using Algebra And Discrete Mathemati use a variety of symbolic representations, including Recursive and parametric equations, for functions and relations;. Use mathematical models to represent http://www.ohiorc.org/record/1470.aspx

arOhioStandardTables[0]='ctl00_cpPageBody_tblStandardOhioProp31'; arOhioStandardImages[0]='imgStandardOhio31'; arNationalStandardTables[0]='ctl00_cpPageBody_tblStandardNationalProp202'; arNationalStandardImages[0]='imgStandardNational202'; Browse the Site Map About ORC Contact ORC Join ORC ... Go to Search Email Address: Password: Password Reminder Join ORC Mathematics Educators Science Educators ... Standards First ORC Resource Number #1470 Trout Pond: Using Algebra and Discrete Mathematics to Investigate Population Changes http://illuminations.nctm.org/index_o.aspx?id=142 PROFESSIONAL COMMENTARY This four-lesson unit uses iteration and recursion to model and analyze a changing fish population. Graphs, equations, tables, spreadsheets, and calculators are used to investigate the effect of varying parameters on the long-term population of fish in a trout pond. This four-lesson unit uses iteration and recursion to model and analyze a changing fish population. Graphs, equations, tables, spreadsheets, and calculators are used to investigate the effect of varying parameters on the long-term population of fish in a trout pond. Step-by-step instructions, activity sheets, questions for reflection, and suggestions for assessment are included. (author/sw) CAREER APPLICATION Maintaining an animal population at a desired level is important in agriculture and wildlife management, not to mention zoos. This lesson allows students to explore the combined effect of population decrease and restocking.

3. Recursive Function (mathematics) -- Britannica Online Encyclopedia The theory of Recursive functions was developed by the 20thcentury truth that all Recursive or computable functions and relations are representable in http://www.britannica.com/eb/topic-493975/article-9062919

Already a member? LOGIN Encyclopædia Britannica - the Online Encyclopedia Home Blog Advocacy Board ... Free Trial Britannica Online Content Related to this Topic Shopping Revised, updated, and still unrivaled. 2008 Britannica Ultimate DVD/CD-ROM The world's premier software reference source. Great Books of the Western World The greatest written works in one magnificent collection. Visit Britannica Store recursive function (mathematics) A selection of articles discussing this topic. Main article: recursive function in logic and mathematics, a type of function or expression predicating some concept or property of one or more variables, which is specified by a procedure that yields values or instances of that function by repeatedly applying a given relation or routine operation to known values of the function. The theory of recursive functions was developed by the 20th-century Norwegian Thoralf... application to formal systems application to formal systems (in metalogic: Discoveries about formal mathematical systems) application to formal systems (in metalogic: Decidability and undecidability) Alternatively, the above assumption can be avoided by resorting to a familiar lemma, or auxiliary truth: that all recursive or computable functions and relations are representable in the system (e.g., in N). Since truth in the language of a system is itself not representable (definable) in the system, it cannot, by the lemma, be recursive (i.e., decidable). Fibonacci sequence generation The resulting number sequence, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55 (Leonardo himself omitted the first term), in which each number is the sum of the two preceding numbers, is the first recursive number sequence (in which the relation between two or more successive terms can be expressed by a formula) known in Europe. Terms in the sequence were stated in a formula by the French-born mathematician...

4. Recursive Characterization Of Computable Real-valued Functions And Relations Recursive characterization of computable realvalued functions and relations. Source, Theoretical Computer Science archive http://portal.acm.org/citation.cfm?id=235897

5. Springer Online Reference Works Suppose that are place primitive Recursive functions, and let be primitive Recursive relations such that for any set of argument values at most one of them http://eom.springer.de/P/p074610.htm

Encyclopaedia of Mathematics P Article referred from Article refers to Primitive recursive function A function from natural numbers to natural numbers which can be obtained from the initial functions by a finite number of the operations of composition and primitive recursion Since the initial functions are computable and the operators of superposition and primitive recursion preserve computability, the set of all primitive recursive functions is a subclass of the class of all computable functions (cf. Computable function ). Every primitive recursive function is specified by a description of its construction from the initial functions (a primitive recursive description); hence the class of primitive recursive functions is countable. Practically all arithmetic functions used in mathematics for some concrete reason are primitive recursive functions; e.g. (the remainder from division of by (the prime number with index ), etc. A relation on natural numbers is called a primitive recursive relation if the function , equal to 1 if is true and if is false, is primitive recursive. One says that the relation

6. PlanetMath: Recursive Function AMS MSC, 03D20 (Mathematical logic and foundations Computability and recursion theory Recursive functions and relations, subRecursive hierarchies) http://planetmath.org/encyclopedia/PrimitiveRecursion.html

(more info) Math for the people, by the people. Encyclopedia Requests Forums Docs ... RSS Login create new user name: pass: forget your password? Main Menu sections Encyclop¦dia Papers Books Expositions meta Requests Orphanage Unclass'd Unproven ... Classification talkback Polls Forums Feedback Bug Reports downloads Snapshots PM Book information News Docs Wiki ChangeLog ... About recursive function (Definition) Intuitively, a recursive function is a positive integer valued function of one or more positive integer arguments which may be computed by a definite algorithm Recursive functions may be defined more rigorously as the largest class of partial functions from satisfying the following six criteria: The constant function defined by for all is a recursive function. The addition function and the multiplication function are recursive function. The projection functions with defined as are recursive functions. Closure under composition If is a recursive function and with are recursive functions, then , defined by is a recursive function. (Closure under primitive recursion) If and are recursive function, then

7. An Arguable Inconsistency In ZF We consequently note, in Metalemma 1, that we cannot introduce a finite number of arbitrary primitive Recursive functions and relations - as function http://alixcomsi.com/An_arguable_inconsistency_in_ZF.htm

Index Main essay An arguable inconsistency in ZF Bhupinder Singh Anand A .pdf file of this essay before the current update is available at http://arXiv.org/abs/math/0502503 Classical theory proves that every primitive recursive function is strongly representable in PA; that PA and PRA can both be interpreted in ZF; and that if ZF is consistent, then PA+PRA is consistent. We show that PA+PRA is inconsistent; it follows that ZF, too, is inconsistent. 1. Overview Classical theory proves that: a Peano Arithmetic PA , and primitive recursive arithmetic, PRA, can both be translated into Zermelo-Fraenkel Set Theory, ZF, under a non-standard interpretation b if ZF is consistent, then PA+PRA is consistent; c every primitive recursive function is strongly representable in PA. It also seems reasonable to suspect that: d although every primitive recursive function can be represented in PA, it cannot be defined in PA. In the appended Meta-theorem 1 and Meta-lemma 1 (cf. [ ]), we show that, if we assume that any finite segment of PRA is finitary (cf. [ ]), then:

8. KENT CITY SCHOOLS MATH COMMITTEE INPUT Algebra Standard (Grades 9-12) Use a variety of symbolic algebra representations, including Recursive, explicit, polar, and parametric equations, for functions and relations. http://kent.k12.oh.us/boe/Alg_Precalculus_Adv_Precalc_Senior_Math.htm

KENT CITY SCHOOLS MATH COMMITTEE INPUT Algebra Standard (Grades 9-12) NCTM STANDARD NCTM EXPECTATIONS Precalculus Adv. Precalculus College Math Fundamentals A. Understand patterns, relations, and functions Generalize patterns using explicitly defined and recursively defined functions Generalize patterns using explicitly and recursively defined functions. Generalize patterns using explicitly and recursively defined functions. Generalize patterns using explicitly and/or recursively defined functions. Understand relations and functions and select, convert flexibly among, and use various representations for them; Understand relations and functions and select, convert flexibly among, and use various representations for them. Understand relations and functions and select, convert flexibly among, and use various representations for them. Understand relations and functions and select, convert flexibly among, and use various representations for them. Analyze functions of one variable by investigating rates of change, intercepts, and zeros, asymptotes, and local and global behavior; Analyze functions of one variable by investigating rates of change, intercepts, zeros, asymptotes, limits, removable and essential discontinuities, and local and global behavior.

9. Languages And Machines 13.7 Turing Computability and MuRecursive functions 17.3 relations between Time and Space Complexity. 17.3 P-Space, NP-Space, and Savitch’s Theorem http://www.cs.wright.edu/~tsudkamp/book.htm

HOME COURSES RESEARCH PUBLICATIONS ... WSU Languages and Machines: An Introduction to the Theory of Computer Science THIRD EDITION Addison-Wesley Publishing Co. The primary objective of the book Languages and Machines is to give a mathematically sound presentation of the theory of computing at a level suitable for junior and senior level computer science majors. The topics covered include the theory of formal languages and automata, computability, computational complexity, and the deterministic parsing of context-free languages. To make these topics accessible to the undergraduate student, no special mathematical prerequisites are assumed. Rather, the mathematical tools of the theory of computing, naive set theory, recursive definitions, and proof by mathematical induction, are introduced in the course of the presentation. The presentation of formal language theory and automata develops the relationships between the grammars and abstract machines of the Chomsky hierarchy. Parsing context-free languages is introduced via standard graph-searching algorithms to make it accessible to students having taken a data structures course. Finite-state automata and Turing machines provide the framework for the study of effective computation. Topics covered include decidability, the Church-Turing thesis, and the equivalence of Turing computability and

10. Reasoning With Recursive Rules Note again, that ;; functions and relations must be defined before they can be This is an important feature, since recursion is a very ;; natural and http://www.isi.edu/isd/LOOM/PowerLoom/documentation/examples/recursion.html

Reasoning with Recursive Rules parent and ancestor relations, since those naturally lend themselves to recursive formulations. (demo 6) (in-package "STELLA") pause c (defmodule "/PL-KERNEL/PL-USER/RECURSION") (in-module "RECURSION") (clear-module "RECURSION") (reset-features) (in-dialect :KIF) (defconcept PERSON (?p) :documentation "The class of human beings.") (defrelation happy ((?p PERSON))) (defrelation has-parent ((?p PERSON) (?parent PERSON)) :documentation "True if `self' has `parent' as a parent.") (defrelation has-ancestor ((?p PERSON) (?ancestor PERSON)) :documentation "True if `self' has `ancestor' as an ancestor.") (assert (and (Person Abby) (Person Benny) (Person Carla) (Person Debbie) (Person Edward) (Person Fred))) (assert (has-parent Abby Benny)) (assert (has-ancestor Benny Carla)) (assert (has-parent Carla Debbie)) (assert (has-ancestor Debbie Edward)) (assert (has-parent Edward Fred)) (retrieve (?z PERSON) (has-ancestor Abby ?z)) (retrieve 2) (retrieve) (retrieve) (retrieve) (retrieve all (?z PERSON) (has-ancestor Benny ?z))

11. Bibliographie Real Recursive functions and real extentions of Recursive functions. for understanding the relations between several analog computational models. http://www.loria.fr/~hainry/publi/Hainry.html

Publications computability, computation over reals, elementary functions, real computable functions bib http Olivier Bournez and Emmanuel Hainry. An analog characterization of elementarily computable functions over the real numbers. In 2nd APPSEM II Workshop - APPSEM'2004, Tallinn, Estonia , Apr 2004. analog models, complexity, computability bib http 31st International Colloqiuim on Automata, Languages and Programming - ICALP'2004, Turku, Finland , volume 3142 of Lecture Notes in Computer Science , pages 269280. Springer, Jul 2004. analog models, complexity, computability bib http analog models, computability bib http Olivier Bournez and Emmanuel Hainry. Real recursive functions and real extentions of recursive functions. In Maurice Margenstern, editor, Machines, Computations, and Universality - MCU 2004, St Petersburg, Russia , volume 3354 of Lecture Notes in Computer Science , pages 116127. Springer, 2005. Recently, functions over the reals that extend elementarily computable functions over the integers have been proved to correspond to the smallest class of real functions containing some basic functions and closed by composition and linear integration. We extend this result to all computable functions: functions over the reals that extend total recursive functions over the integers are proved to correspond to the smallest class of real functions containing some basic functions and closed by composition, linear integration and a very natural unique minimization schema. bib http Olivier Bournez and Emmanuel Hainry. Elementary computable functions over the real numbers and R-sub-recursive functions.

12. Recursive Characterization Of Computable Real-Valued Functions And Relations Recursive Characterization of Computable RealValued functions and relations. Vasco Brattka. Journal Title Theoretical Computer Science. Date 1996 http://wotan.liu.edu/docis/show?doc=dbl/tcstcs/1996_162_1_45_RCOCRF.htm&query=si

13. Theory Of Computing 2007/2008 (FUB MSc In Computer Science) - Lectures the basic definitions regarding functions, relations, and their properties showing computability of primitive Recursive functions; bounded operators http://www.inf.unibz.it/~calvanese/teaching/07-08-tc/lectures.html

Free University of Bolzano/Bozen Faculty of Computer Science Master of Science in Computer Science Theory of Computing Lectures A.Y. 2007/2008 Prof. Diego Calvanese Teaching material Introduction to Automata Theory, Languages, and Computation (3rd edition). J.E. Hopcroft, R. Motwani, J.D. Ullman. Addison Wesley, 2007. Lecture Notes for Theory of Computing . Diego Calvanese. 2007. Made available as scanned pages in pdf. Exercises on Theory of Computing . Will be made available as scanned pages in pdf. Lectures Summary Week Topics Monday (lecture) Wednesday (lecture) Wednesday (exercise) Extra (lecture) Oct. 8 Course introduction Course introduction, basic notions about sets Lec 1,2 Basic notions about relations, functions, languages Lec 3,4 Formal proofs Exer 1 Oct. 15 Undecidability Undecidable problems Lec 5,6 The Turing Machine Lec 7,8 Turing Machines Exer 2 Oct. 22 (Extended) Turing Machines Programming techniques for TMs Lec 9,10 Multitape and nondeterministic TMs Lec 11,12 Nondeterministic TMs and extensions of TMs Exer 3 Oct. 29 Decidability and undecidability Church-Turing Thesis Lec 13,14

14. Book Languages And Machines, An Introduction To The Theory Of Computer Science ( relations, and functions 1.3 Equivalence relations 1.4 Countable and Chapter 13 MuRecursive functions 13.1 Primitive Recursive functions 13.2 Some http://www.lavoisier.fr/notice/gbQZOR333E2RB32O.html

Search on All Book CD-Rom eBook Software The french leading professional bookseller Description Approximate price Languages and machines, an introduction to the theory of computer science (3rd ed ) Author(s) : SUDKAMP Thomas Publication date : 03-2005 Language : ENGLISH 672p. Hardback Status : In Print (Delivery time : 10 days) Description 'Languages and Machines' covers key concepts and theorems of the theory of computation. The third edition provides a mathematically sound presentation augmented with the theory of computer science. It incorporates step-by-step, unhurried proofs, worked-out examples and illustrations providing students the needed aid in understanding concepts. Summary Subject areas covered: Information technology Algorithms, logic General titles on the theory it New search Your basket Information New titles BiblioAlerts E-books Customer services Open an account Ordering non-listed items Order tracking Help Lavoisier.fr Back to the home page Company information Terms and conditions Partner's sites ... basket Special Offer www.Lavoisier.fr

15. Grad Description Elements of set theory, functions and relations nondecimal numbers, Include Turing Machines, partial Recursive functions, Recursive and Recursively http://www.jsums.edu/~sst/cset/ComputerScience/computerscience-gradcurriculum.ht

16. Computability DCS301 Primitive Recursive functions and relations. Ackermann function. Read Chapters 6 and 7 of textbook. Look at Handout 3 and at Cohn, Algebra, Vol 1, Chapter 2 http://www.dcs.qmul.ac.uk/~bellin/coursepage.htm

Computability DCS 301 QMUL - Semester 6 - Winter 2007 Information: Two lectures each week: Tuesday 12:00-13:00 Room CS 338 Wednesday 11:00-13:00 Room FB 328 Homework discussion: Wednesday 12pm, FB 328 Office hours: Tuesday after 15pm, Room CS 331 Teaching Assistant: Corrado Biasi TA's Office hours: by appointment TA's Email contact: biasiman@dcs.qmul.ac.uk You need to do a certain number of homework assignments Midterm on Wednesday 21st February at noon, Room FB 328 The midterm is worth 15% of the final mark, the remaining 15% is determined by the best marks in half of the homework assignments. You must obtain a mark of at least 40/100 in the exam in order to pass the course. If you reach at least 40/100 in the exam, then the final mark will be determined as follows: Exam will be worth 70% of the final mark. Coursework+Midterm will be worth 30% of the final mark, determined either by the average of the six best homework assignments; or 15% by the Midterm and 15% by the average of the three best homework assignments Textbooks: G.S.Boolos, J.P.Burgess and R.C.Jeffrey

17. Overview Of Topics Operations on Sets of Objects; functions and relations; Domain and Range; Recursive functions; Composition of functions; Inverse functions; Bijection, http://www.4dsolutions.net/ocn/mainoutline.html

Linear Overview: A Sequential Outline of Topics Topic Headings** Description Bits, Bytes and Bases Positional Notation Roman Numerals From the Abacus to Computers Powers of 10 Powers of 2 Binary Numbers Hexadecimal Numbers Learn some of the terminology surrounding number systems to the base N, with emphasis on those used in computing. Discuss positional notation systems with Roman Numerals for contrast. Research the history of computing devices, from the abacus until the present time. Preview exponential notation. Cardinality vs. Ordinality Numbers for Naming Addressing Schemes Relational Databases ASCII and Unicode Permutations Ordering a Set The Number Line Develop cardinal sense by looking at symbols paired with unique binary identifiers as per ASCII and Unicode mappings, contrast with ordinal sense. Consider alphanumeric addressing schemes such as URIs for web-based resources. Disucuss the omnipresence of relational data bases in contemporary society. Review hexadecimal numbers. Look at greater-than, less-than and equals operators in the context of the number line as an organizing heuristic. Algorithms and Operations Basic Operations Modulo Arithmetic Groups, Rings, and Fields

18. Big-Oh For Recursive Functions: Recurrence Relations This web page gives an introduction to how recurrence relations can be used to help determine the bigOh running time of Recursive functions. http://www.cs.duke.edu/~ola/ap/recurrence.html

Big-Oh for Recursive Functions: Recurrence Relations It's not easy trying to determine the asymptotic complexity (using big-Oh) of recursive functions without an easy-to-use but underutilized tool. This web page gives an introduction to how recurrence relations can be used to help determine the big-Oh running time of recursive functions. This material is taken from what we present in our courses at Duke University and was given at a College Board AP workshop in August of 1998 at Berkeley. Motivation The problem below appeared as AB Problem 3 on the 1996 AP exam, for which a C++ translation has been made. Given a binary tree, is it a search tree? In part A students are asked to write the function ValsLess In Part B, students are asked to write IsBST using ValsLess and assuming that a similar function ValsGreater exists. The solution is shown below: Before continuing you should try to determine/guess/reason about what the complexity of IsBST is for an n -node tree. Assume that ValsLess and ValsGreater both run in O(n) time for an n-node tree.

19. Recursive Function Theory The relation between Recursive functions and the description of flow control by flow charts is described in Reference 7. An ALGOL program can be described http://www-formal.stanford.edu/jmc/basis1/node13.html

Next: On the Relations between Up: Relation to Other Formalisms Previous: Relation to Other Formalisms Recursive function theory Our characterization of as the set of functions computable in terms of the base functions in cannot be independently verified in general since there is no other concept with which it can be compared. However, it is not hard to show that all partial recursive functions in the sense of Church and Kleene are in In order to prove this we shall use the definition of partial recursive functions given by Davis [3]. If we modify definition 1.1 of page 41 of Davis [3] to omit reference to oracles we have the following: A function is partial recursive if it can be obtained by a finite number of applications of composition and minimalization beginning with the functions on the following list: All the above functions are in Any is closed under composition so all that remains is to show that is closed under the minimalization operation. This operation is defined as follows: The operation of minimalization associates with each total function the function whose value for given is the least y for which and which is undefined if no such y exists. We have to show that if

20. Call Graphs Of Nestedly Recursive Functions - Wolfram Demonstration Project Nestedly Recursive functions nestedly call previous instances of themselves. Even very simple recursion relations can lead to a complex sequence of values http://demonstrations.wolfram.com/CallGraphsOfNestedlyRecursiveFunctions/

Call Graphs of Nestedly Recursive Functions loadFlash(630, 581, 'CallGraphsOfNestedlyRecursiveFunctions'); Nestedly recursive functions nestedly "call" previous instances of themselves. Even very simple recursion relations can lead to a complex sequence of values for nestedly recursive functions. The recursion relations are set up so that whenever they sample below n=1, the f[n] is taken to have value 1. f[n]=3 f[n - f[n - 1]] is the simplest example that seems to yield complex behavior. Functions like these were mentioned in A New Kind of Science , but first studied in detail in Stephen Wolfram's Live Experiment at the opening of the first NKS Summer School, in June 2003. "Indirect calls" are instances of the "inner f" in the recursion relation. " Call Graphs of Nestedly Recursive Functions " from The Wolfram Demonstrations Project http://demonstrations.wolfram.com/CallGraphsOfNestedlyRecursiveFunctions/ Contributed by: Stephen Wolfram Based on: A live experiment at the 2003 NKS Summer School Computer Systems Computers Mathematical Logic ... Atom

21. The Set Of Primitive Recursive Functions Preliminaries; Sets of Compatible functions; Homogeneous relations; Primitive Recursiveness; The Set of Primitive Recursive functions; Examples http://mizar.org/JFM/Vol13/comput_1.html

Journal of Formalized Mathematics Volume 13, 2001 University of Bialystok Association of Mizar Users The Set of Primitive Recursive Functions Grzegorz Bancerek University of Bialystok, Shinshu University, Nagano Piotr Rudnicki University of Alberta, Edmonton Summary. We follow [ ] in defining the set of primitive recursive functions. The important helper notion is the homogeneous function from finite sequences of natural numbers into natural numbers where homogeneous means that all the sequences in the domain are of the same length. The set of all such functions is then used to define the notion of a set closed under composition of functions and under primitive recursion. We call a set primitively recursively closed iff it contains the initial functions (nullary constant function returning 0, unary successor and projection functions for all arities) and is closed under composition and primitive recursion. The set of primitive recursive functions is then defined as the smallest set of functions which is primitive recursively closed. We show that this set can be obtained by primitive recursive approximation. We finish with showing that some simple and well known functions are primitive recursive. This work has been supported by NSERC Grant OGP9207, NATO CRG 951368 and TYPES grant IST-1999-29001.

22. A PVS Formalization Of Well Founded Recursion Our theory applies to the definition of total Recursive relations the definition of total Recursive functions is a special case. http://uuu.enseirb.fr/~gloess/enseignement/spp/now/well_founded_recursion.html

A PVS formalization of well founded recursion created november 22, 2002, from 20012002 SPP , last edition 24 October 2006 back to SPP Although PVS language provides a powerful construction for defining recursive functions (that can be proved "to terminate" using a measure and well founded relation), it turns out that a theory of recursive functions may be developed from PVS typed higher order logic primitive concepts! We don't even need inductive sets: these can be reconstructed from scratch, see Bruno Dutertre "mu calculus" theory. We actually use it in our reconstruction of recursive functions. All theories can be restored from three library dumps ( to be restored in the same order exit PVS after proving each library ~gloess/pvs/2.4/nasa/fixpoints/p24_fixedpoints.dmp this library of fixpoint induction (for sets) has been developed by Bruno Dutertre (SRI International) and is available at NASA; it builds upon PVS mucalculus prelude theories, by essentially proving that mu and nu operators respectively compute the least and greatest fixpoint of a monotonic functional; ~gloess/pvs/2.4/scott/2.4.scott.6feb2002.induction.dump

23. FOM: Concepts Of Recursion Theory This was not surprising; Church s Thesis shows that any result about Recursive functions or relations is in soie sense a result about computability. http://cs.nyu.edu/pipermail/fom/1998-August/002024.html

FOM: Concepts of Recursion Theory Joseph Shoenfield jrs at math.duke.edu Sat Aug 29 15:21:36 EDT 1998 Previous message: FOM: On "grasping" Con(ZFC) Next message: FOM: Concepts of Recursion Theory (reply to Shoenfield) Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] Previous message: FOM: On "grasping" Con(ZFC) Next message: FOM: Concepts of Recursion Theory (reply to Shoenfield) Messages sorted by: [ date ] [ thread ] [ subject ] [ author ] ... More information about the FOM mailing list

24. Finite Sets: Counting, Recursion, And Logic, Next Steps Iteration and Recursion; Sets defined by Propositions; The Art of Counting; relations and functions; Digraphs as relations; The Art of Searching http://www.mathwright.com/book_pgs/book651.html

This Microworld is the second half of a 12-week course in Discrete Mathematics. The term "Discrete Mathematics" in this Microworld will refer loosely to the collection of techniques, ideas, and constructions that have evolved over the years to describe artificial systems, and in particular, those systems from which the modern theories of computing have evolved. So, for example, an excellent model for our subject is the Turing Machine, or, equivalently, the lambda-calculus of Alonzo Church. In fact, the lambda-calculus is the conceptual basis for the computer language, LISP , which is the language in which this Microworld is written. We recommend that you follow the 9 readings of this Microworld in the order in which they appear below, beginning with "Iteration and Recursion." You may read the 72-page collection of lectures either within the Microworld, or as a Word 2000 document. To download the Lectures as Word Document, click here , then extract the file to your disk. Also, to download the Laboratory Instructions as Word Document, click

25. IngentaConnect Recursive Characterization Of Computable Real-valued Functions An Recursive functions we introduce a class of Recursive relations in metric spaces such that each relation is generated from a class of basic relations by a http://www.ingentaconnect.com/content/els/03043975/1996/00000162/00000001/art002

var tcdacmd="dt"; Home About Ingenta Ingenta Labs Ingenta Blog ... Check our FAQs Or contact us to report problems with: Subscription access Article delivery Registration Library administrator tasks ... Other problems For Publishers Why go online? Why choose IngentaConnect? Beyond Print: enhancing your service Access and authentication ... Contact us For Researchers For Authors About IngentaConnect Search and browse Publications available ... Register For Librarians Resource Zone Why choose IngentaConnect? Why choose IngentaConnect Complete? Activating Subscriptions ... Register CM8ShowAd("HorizontalBanner"); Recursive characterization of computable real-valued functions and relations Author: Brattka V. Source: Theoretical Computer Science , Volume 162, Number 1, 5 August 1996 , pp. 45-77(33) Publisher: Elsevier view table of contents Key: - Free Content - New Content - Subscribed Content - Free Trial Content CM8ShowAd("Skyscraper"); Language: English Document Type: Research article DOI: Affiliations: Theoretische Informatik I, FernUniversitat Hagen, D-58084 Hagen, Germany

26. Logic, 8 We define relations, functions and so on in the natural way RUP f1 . .. Recursion Theory – compare this to Sipser’s Computation, Part II http://www.media.mit.edu/physics/pedagogy/babbage/texts/rt.html

Introduction to Logic and Recursion Theory This is a transcription of relevant notes from the class 18.511 taught by Prof. Sacks in the Spring of 1998, organized and reinterpreted. Homework problems starting with problem 9 are solved in vitro. Notation is indecipherable. Propositional Calculus Propositional calculus is an example of a formal system . One must specify atomic symbols , which consist of letters A n , or symbols, and connectives expression is a finite sequence of atomic symbols. The set of well-formed formulas (WFFs) is defined recursively as follows: (A n g). This lets up build up new propositions from old ones. They are associative, etc. in the commonly held sense of these notions. A truth valuation c to the set of all WFFs, simply given by defining it recursively in the obvious fashion. Two WFFs are semantically equivalent disjunctive normal form semantically complete . It is obvious that we cannot discard the ! symbol, but one can combine the two to make the NAND operator, which is, all by itself, semantically complete, the Schaeffer stroke . In quantum logic, the CNOT is semantically complete, combined with an arbitrary unitary operator.

27. Recursive Functions A less trivial one we now introduce is defined by two relations. Immediately we run into the problems of such Recursive functions. http://www.geocities.com/CapeCanaveral/Lab/3550/meaning.htm

Recursive Functions and the true meaning of Zimaths by Tawanda Gwena Functions? Again! Sorry, but that's the main ingredient of mathematics. But today, instead of our standard plain functions, we do the recursive functions, definitely a lot more flavour in them. First the word recursive. In general usage it means going back to oneself, a bit like a dog chasing its own tail, or like the story on our cover is a story about itself. Now in mathematics it means a function defined in terms of itself. Let's try doing that. The easiest case is f(x)=f(x). This leads us nowhere, as every possible function satisfies that relation, e.g. sin(x)=sin(x). A less trivial one we now introduce is defined by two relations. They are f(0)=0 and f(n)=n+f(n-1). The first relation is there as a blocker, otherwise we might go off to negative infinity, because we do not know when to stop. As a first exercise we work out f(4): f(4) = 4+f(3) = 4+3+f(2) = 4+3+2+f(1) = 4+3+2+1+f(0) = 4+3+2+1+0 = 10 This turns out to be just the function f(n)=n(n+1)/2. Not too interesting because there is a simple formula to give you the value easily.

28. SRF srf (Simple Recursive functions) interprets a very simple programming .. The file examples/relations in the source distribution has functions that match http://www.inlumineconsulting.com:8080/website/srf.html

SRF Bruce Ediger November 3, 2004 Synopsis srf S imple R ecursive F unctions) interprets a very simple programming language similar to Stephen Kleene's recursive functions. You can use srf to help understand recursive functions, or Peano arithmetic. Command line srf [-L filename [-L otherfile ...]] [-t timeout] Once you've built and installed srf , you can start it at a shell prompt: 9:54PM 3 % ./srf For better or for worse, srf doesn't provide a prompt. You just enter things after starting it. You can give it command line options. The -L option causes srf to read in a file or files before reading interactive input: 9:55PM 4 % ./srf -L examples/numbers load file named "examples/numbers" 1 2 3 ... 10 You can use more than one -L option - srf reads them in order before allowing interactive input. The -t option allows you to get srf to interrupt execution of long-running calculations after a certain number of seconds: 10:02PM 12 % ./srf -t 5 -L examples/functions ... x=pow(2,10); 1024 y=pow(x,10); Timeout Unset value srf reads from stdin, so you can also use it with shell redirection, or, bless your heart, as a filter.

29. Vectorization Of Recursion Relations. Techniques for efficient vectorization of the simple Recursive relation (x sub j) = (a sub j)(x sub (j1)) + (d sub j) and the tridiagonal series of http://stinet.dtic.mil/oai/oai?verb=getRecord&metadataPrefix=html&identifier=ADA

30. Discrete Mathematics/Recursion - Wikibooks, Collection Of Open-content Textbooks In mathematics, we can create Recursive functions, which depend on its previous values to create new ones. We often call these recurrence relations. http://en.wikibooks.org/wiki/Discrete_mathematics/Recursion

var wgNotice = ""; var wgNoticeLocal = ""; var wgNoticeLang = "en"; var wgNoticeProject = "wikibooks"; Discrete mathematics/Recursion From Wikibooks, the open-content textbooks collection Discrete mathematics Jump to: navigation search In this section we will look at certain mathematical processes which deal with the fundamental property of recursion at its core. Contents What is recursion? Exponents Recursive definitions Recurrence relations ... edit What is recursion? Recursion, to put it simply, is the process of describing an action in terms of itself. This may seem a bit strange to understand, but once it "clicks" it can be an extremely powerful way of expressing certain ideas. Let's look at some examples to make things clearer. edit Exponents When we calculate an exponent, say x , we multiply x by itself three times. If we have x , we multiply x by itself five times. However, if we want a recursive definition of exponents, we need to define the action of taking exponents in terms of itself. So we note that x for example, is the same as x x . But what is x x is the same as x x . We can continue in this fashion up to x =1. What can we say in general then? Recursively

31. Recurrence Relations/recursion & Induction Recurrence relations and Recursive definitions have a lot in common with induction the value of a function at a higher value is defined in terms of its http://mathcircle.berkeley.edu/BMC4/Handouts/induct/node9.html

Next: Double Induction, etc. Up: Related Topics Previous: Related Topics Recurrence relations and recursive definitions have a lot in common with induction - the value of a function at a higher value is defined in terms of its values at a smaller value. Often a recursive construction will require an inductive proof of its correctness. Example: , what is a closed form expression for Then, show that the best solution to the towers of Hanoi puzzle requires steps. A classic example of a recursive definition is that of the Fibonnaci numbers: they are defined as: and, for Prove the following identity for Fibonacci numbers: for all Prove that consecutive Fibonacci numbers are always relatively prime. (Hint: Try finite descent!) For the Fibonacci numbers, show: Show that every positive integer can be expressed uniquely as the sum of distinct, non-consecutive Fibonacci numbers (here, non-consecutive means that no two of the Fibonacci number in the sum are consecutive Fibonacci numbers). Next: Double Induction, etc. Up: Related Topics Previous: Related Topics Zvezdelina Stankova-Frenkel 2001-11-18

32. Phys. Rev. A 37 (1988): C. K. Lutrus And S. H. Suck Salk - Recursion Relations F Recursion relations for the overlap of a Morse continuum state with a Lanczos In their study, recursion relations for Green’s functions in the Lanczos http://link.aps.org/doi/10.1103/PhysRevA.37.3151

Physical Review Online Archive Physical Review Online Archive AMERICAN PHYSICAL SOCIETY Home Browse Search Members ... Help Abstract/title Author: Full Record: Full Text: Title: Abstract: Cited Author: Collaboration: Affiliation: PACS: Phys. Rev. Lett. Phys. Rev. A Phys. Rev. B Phys. Rev. C Phys. Rev. D Phys. Rev. E Phys. Rev. ST AB Phys. Rev. ST PER Rev. Mod. Phys. Phys. Rev. (Series I) Phys. Rev. Volume: Page/Article: MyArticles: View Collection Help (Click on the to add an article.) Phys. Rev. A 37, 3151 - 3153 (1988) Previous article Next article Issue 8 View Page Images PDF (351 kB), or Buy this Article Use Article Pack Export Citation: BibTeX EndNote (RIS) Recursion relations for the overlap of a Morse continuum state with a Lanczos basis state C. K. Lutrus and S. H. Suck Salk Department of Physics and Graduate Center for Cloud Physics Research, University of Missouri Rolla, Rolla, Missouri 65401-0249 Received 18 May 1987 In the resonant reactive scattering theory of Mundel, Berman, and Domcke [Phys. Rev. A , 181 (1985)], the overlap of a Morse continuum state and a Lanczos basis state appears in the expression of transition amplitude. In their study, recursion relations for Green’s functions in the Lanczos basis were used for computational efficiency. In this paper we derive new recursion relations specifically for the evaluation of overlap between the Morse continuum wave and Lanczos basis state that appears in the transition amplitude of resonant scattering. They are found to be simple to use with great accuracy.

33. Math 210 Topics include counting and combinations, laws of logic, methods of proof, set theory, cardinality, proof by induction, recursion, and relations/functions. http://www.cptc.ctc.edu/library/div4/math210sweerus4.htm

4500 Steilacoom Blvd. SW Lakewood WA Discrete Mathematics Math 2 Core Academics/Related Instruction 5 Credits Instructor: Dr. Neil Sweerus Office Hours: by appointment Telephone: Location: Building 15, Room 107 Email Address neil.sweerus@cptc.edu Revision Dates: Discrete Mathematics Math 2 COURSE DESCRIPTION: Develop tools for reasoning about discrete mathematical objects. Topics include counting and combinations, laws of logic, methods of proof, set theory, cardinality, proof by induction, recursion, and relations/functions. COURSE RATIONALE: This course will provide the student with critical thinking and problem solving skills through mathematical reasoning and knowledge of topics in discrete mathematics that are frequently encountered in computer applications, computer programming, and computer science. PREREQUISITES: Appropriate COMPASS placement score (College Algebra 53) or successful completion of MAT 115 Pre-Calc I is required. COLLEGE-WIDE CORE ABILITIES: Core abilities are transferable skills that are essential to an individual’s success, regardless of occupation or community setting. These skills: Complement specific occupational skills Broaden one’s ability to function outside a given occupation, and

34. Cook, Matthew M. (2005-05-27) Networks Of Relations. Http://resolver.caltech.edu This thesis explores new ground regarding the composition of relations into this reachability question can be decided by primitive Recursive functions. http://etd.caltech.edu/etd/available/etd-06032005-140944/

Caltech Library System About Browse Search Caltech Student Instructions Cook, Matthew M. (2005-05-27) Networks of relations. http://resolver.caltech.edu/CaltechETD:etd-06032005-140944 Type of Document Dissertation Author Cook, Matthew M. URN etd-06032005-140944 Persistent URL http://resolver.caltech.edu/CaltechETD:etd-06032005-140944 Title Networks of relations Degree PhD Option Computation and Neural Systems Advisory Committee Advisor Name Title Erik Winfree Committee Chair Chris Umans Committee Member Jehoshua Bruck Committee Member Leonard J. Schulman Committee Member Yaser S. Abu-Mustafa Committee Member Keywords fan-out primitive recursive undecidability finite chemical reaction networks lattice relations predicates Date of Defense Availability unrestricted Abstract The canonical form for linear segments can be represented as a matrix, leading us to matrix networks. We study the question of how we can perform a change of basis in matrix networks, which brings us to a new understanding of Valiant's recent holographic algorithms, a new source of polynomial time algorithms for counting problems on graphs that would otherwise appear to take exponential time. We show how the holographic transformation can be understood as a collection of changes of basis on individual edges of the graph, thus providing a new level of freedom to the method, as each edge may now independently choose a basis so as to transform the matrices into the required form. Consideration of zipper networks makes it clear that "fan-out," i.e., the ability to duplicate information (for example allowing a variable to be used in many places), is most naturally itself represented as a relation along with everything else. This is a notable departure from the traditional lack of representation for this ability. This deconstruction of fan-out provides a more general model for combining relations than was provided by previous models, since we can examine both the traditional case where fan-out (the equality relation on three variables) is available and the more interesting case where its availability is sub ject to the same limitations as the availability of other relations. As we investigate the composition of relations in this model where fan-out is explicit, what we find is very different from what has been found in the past.

35. Well-Foundedness And Recursion This is a proof of the recursion theorem for wellfounded Recursive definitions. It is modelled on Tobias Nipkow s proof for Isabelle HOL and uses relations http://www.rbjones.com/rbjpub/pp/x004.html

Well-Foundedness and Recursion (for noframe browsers)

36. Libra: Recursive Characterization Of Computable Real-Valued Functions And Relati Real Recursive functions and Real Extensions of Recursive functions(2004) (citation2). Olivier Bournez Emmanuel Hainry. Abstract Recently, functions over http://libra.msra.cn/papercited.aspx?id=759674

37. Discrete Structures (DS) DS1. Functions, Relations, And Sets [core Explain with examples the basic terminology of functions, relations, and sets. Relate the ideas of mathematical induction to recursion and recursively http://www.sigcse.org/cc2001/DS.html

Discrete Structures (DS) DS1. Functions, relations, and sets [core] DS2. Basic logic [core] DS3. Proof techniques [core] DS4. Basics of counting [core] DS5. Graphs and trees [core] DS6. Discrete probability [core] Discrete structures is foundational material for computer science. By foundational we mean that relatively few computer scientists will be working primarily on discrete structures, but that many other areas of computer science require the ability to work with concepts from discrete structures. Discrete structures includes important material from such areas as set theory, logic, graph theory, and combinatorics. The material in discrete structures is pervasive in the areas of data structures and algorithms but appears elsewhere in computer science as well. For example, an ability to create and understand a formal proof is essential in formal specification, in verification, and in cryptography. Graph theory concepts are used in networks, operating systems, and compilers. Set theory concepts are used in software engineering and in databases. As the field of computer science matures, more and more sophisticated analysis techniques are being brought to bear on practical problems. To understand the computational techniques of the future, today's students will need a strong background in discrete structures.

38. ACL2 Version 2.3 Such wellfounded relations are used in the admissibility test for Recursive functions, in particular, to show that the recursion terminates. http://www.cs.utexas.edu/users/moore/acl2/v2-3/acl2-doc-56.html

REFINEMENT record that one equivalence relation refines another Major Section: RULE-CLASSES See rule-classes for a general discussion of rule classes and how they are used to build rules from formulas. An example corollary formula from which a :refinement rule might be built is: Example: (implies (bag-equal x y) (set-equal y x)). Also see defrefinement General Form: (implies (equiv1 x y) (equiv2 x y)) and must be known equivalence relations. The effect of such a rule is to record that is a refinement of . This means that rewrite rules may be used while trying to maintain . See equivalence for a general discussion of the issues. The macro form (defrefinement equiv1 equiv2) is an abbreviation for a defthm of rule-class :refinement that establishes that is a refinement of . See defrefinement Suppose we have the rewrite rule (bag-equal (append a b) (append b a)) which states that append is commutative modulo bag-equality. Suppose further we have established that bag-equality refines set-equality. Then when we are simplifying append expressions while maintaining set-equality we use append 's commutativity property, even though it was proved for bag-equality.

39. ScienceDirect - Physics Letters A : The Recursion Relations For The N-dimensiona Two recursion relations in terms of raising and lowering operators are derived for only the ‘principal’ and ‘angularmomentum’ quantum numbers through http://linkinghub.elsevier.com/retrieve/pii/S0375960104008321

Athens/Institution Login Not Registered? User Name: Password: Remember me on this computer Forgotten password? Home Browse My Settings ... Help Quick Search Title, abstract, keywords Author e.g. j s smith Journal/book title Volume Issue Page Physics Letters A Volume 328, Issues 2-3 , 26 July 2004, Pages 123-126 Abstract Full Text + Links PDF (144 K) Related Articles in ScienceDirect Modeling quantum harmonic oscillator in complex domain Modeling quantum harmonic oscillator in complex domain Volume 30, Issue 2 October 2006 Pages 342-362 Ciann-Dong Yang Abstract Abstract Full Text + Links PDF (377 K) Moment method and the Schrodinger equation in the large... ... Physics Letters A N limit Physics Letters A Volume 97, Issue 5 29 August 1983 Pages 178-182 J. P. Ader Abstract N expansion. Based on recursion relations satisfied by moments of the coordinate operator, this method which allows to compute energy levels and wavefunctions is applied to four examples: the harmonic oscillator, the rotating harmonic oscillator, a linear plus Coulomb potential and a logarithmic one. Abstract Abstract + References PDF (310 K) The exact solutions of the Schrodinger equation with th...

40. 22c:245 Advanced Artificial Intelligence The Y combinator for the definition of Recursive functions. Confluence of the reduction relation. -equivalence of -terms. Reduction strategies. http://www.cs.uiowa.edu/~tinelli/classes/185/Fall06/logs.shtml

The University of Iowa 22c:185 Programming Language Foundations Fall 2006 Course Info Announcements Staff and Hours Syllabus Course Work Class Logs Exercises Exams WebCT Resources Readings OCaml Learning Research Class Logs and Required Readings The cited references can be found in the Readings section. Date Topics Readings Introduction and syllabus overview. Main aspects of programming languages: syntax and semantics. Examples of syntax and semantics in natural languages. The case for formal semantics of PLs. Examples of how different semantics affect the equivalence or inequivalence of two program fragments. Perceived disadvantages of formal semantics. Advantages of formal semantics in implementation, verification and design of PLs. Brief overview of three major styles of formal semantics: operational, axiomatic and denotational. - Chap. 1 of [Win] - Chap. 11-16 of [Nis] (optional) Concrete and abstract syntax. Examples of concrete grammars and parse trees. Examples of abstract grammars and abstract syntax trees. IMP, a simple imperative language.

41. Recurrence Relations And Recursion - Maple Application Center - Maplesoft In this module, we ll examine recursion and solving recurrence relations in various forms and from symbolic, numeric, and geometric points of view. http://www.maplesoft.com/Applications/app_center_view.aspx?AID=853

42. Recursion Definition of recursion, possibly with links to more information and implementations. Write a function to compute GCD based on the following relations http://www.nist.gov/dads/HTML/recursion.html

recursion (algorithmic technique) Definition: An algorithmic technique where a function, in order to accomplish a task, calls itself with some part of the task. Specialization (... is a kind of me.) tail recursion collective recursion See also iteration divide and conquer divide and marriage before conquest recursive ... recurrence relation Note: Every recursive solution involves two major parts or cases, the second part having three components. base case(s) , in which the problem is simple enough to be solved directly, and recursive case(s) . A recursive case has three components: divide the problem into one or more simpler or smaller parts of the problem, call the function (recursively) on each part, and combine the solutions of the parts into a solution for the problem. Depending on the problem, any of these may be trivial or complex. Here are some exercises to help you learn recursion. Although recursion may not be the best way to write some of these functions, it is good practice. Write a function to compute the sum of all numbers from 1 to n. Write a function to compute 2 to the power of a non-negative integer.

43. Analytic Study Of Periodic Chaos. II ---Recursion Relations For The Power Spectr On the basis of the analytic formulas for the correlation functions of nonperiodic orbits, the recursion relations are obtained which fully characterize the http://adsabs.harvard.edu/abs/1985PThPh..73..349Y

Sign on Smithsonian/NASA ADS Physics Abstract Service Find Similar Abstracts (with default settings below Electronic Refereed Journal Article (HTML) References in the article Citations to the Article (1) ... Translate Abstract Title: Analytic Study of Periodic Chaos. II -Recursion Relations for the Power Spectra- Authors: Publication: Progress of Theoretical Physics, Vol. 73, No. 2, pp. 349-360 Publication Date: Origin: PTP (c) 1985 Progress of Theoretical Physics Bibliographic Code: Abstract and -alpha, where alpha is Feigenbaum's rescaling factor. Universal aspects of the recursion relations are also discussed. Bibtex entry for this abstract Preferred format for this abstract (see Preferences Find Similar Abstracts: Use: Authors Title Abstract Text Return: Query Results Return items starting with number Query Form Database: Astronomy Physics arXiv e-prints Smithsonian/NASA ADS Homepage ADS Sitemap Query Form Basic Search ... FAQ

44. Recursion Relations In Component Form Recursion relations in Component Form. We will now present formulae for calculating , and . We do this by expanding ( gif ) and ( gif http://www.hpcoders.com.au/docs/thesis/node45.html

Next: Consistency with Biorthonormality Up: Perturbation of the Previous: Perturbation of the Recursion relations in Component Form We will now present formulae for calculating and . We do this by expanding ( ) and ( ) in the and basis (zeroth order basis) with the following definition: We now define an operator so that where is the projection onto the orthogonal complement of . In other words, if the domain of is restricted to make one to one, then is the inverse of By applying on equation ( ), the multipole functions can be generated according to Or expressed in the zeroth order basis for where are the matrix elements of the operator , and are the eigenvalues of These relations will generate the multipole functions from the seeds . If then and may be chosen arbitrarily subject to To complete the prescription, we need a formula for calculating the , and we also need to check that the biorthonormality condition ( is satisfied in the case . For convenience of notation, we will define for every n less than zero. This means that all the power series expansions hold for every integer

45. Venanzio Capretta's Home Page For each language, we can define recursion and induction principles that work the use of wellfounded relations, implementation of operational semantics, http://www.cs.ru.nl/~venanzio/

Venanzio Capretta Postdoctoral researcher Foundation Group Computer Science Institute (iCIS) Radboud University Nijmegen P.O. Box 9010, NL-6500 GL Nijmegen The Netherlands e-mail: venanzio @ cs.ru.nl telephone: +31-24-3652631 fax: +31-24-3652525 room: 02.528 We know nothing, not even whether we know or do not know, or what it is to know or not to know, or in general whether anything exists or not. Metrodorus of Chios Work in progress These are some articles that I and some coauthors are working on. Click on the title to get the PDF file (they are also available in PostScript format). An Introduction to CoRecursive Algebras with Tarmo Uustalu and Varmo Vene The Foundations Group / Brouwer Institute Seminar, Nijmegen, Tuesday 4 December 2007. Higher Order Abstract Syntax in Type Theory Coq formalization and example application A polymorphic representation of induction-recursion Also in PostScript format Publications Combining de Bruijn Indices and Higher-Order Abstract Syntax in Coq abstract Coauthor: Amy Felty to appear in Proceedings of TYPES 2006 bibtex entry Recursive Coalgebras from Comonads (long version) ( abstract Coauthors: Tarmo Uustalu and Varmo Vene Information and Computation , volume 204, issue 4 (2006), pages 437-468. (

46. Question About Indicator Function Of Recursive And R.e. Sets - Object Mix relation and Recursive relation . In the first case, they say the characteristic function has to be effectively computable (ie. partial http://objectmix.com/theory/307390-question-about-indicator-function-recursive-r

Object Mix Question about indicator function of recursive and r.e. sets This is a discussion on Question about indicator function of recursive and r.e. sets within the Theory forums, part of the Theory and Concepts category; Hi all, at our lecture from Theory of Computation, we've been told that - the set M is recursive, if ... Object Mix Theory and Concepts Theory Question about indicator function of recursive and r.e. sets User Name Remember Me? Password Home Register FAQ Calendar ... Display Modes 11-27-2007, 03:16 PM Question about indicator function of recursive and r.e. sets Hi all, at our lecture from Theory of Computation, we've been told that - the set M is recursive, if its indicator function is total recursive function (or partial recursive function, it doesn't make a difference here, since indicator function is always EVERYWHERE defined). Ok, I thought I understood this. But today, I found this quotation from some book on Wikipedia: "The characteristic function of a k-place relation is the k-argument function that takes the value 1 for a k-tuple if the relation holds of the k-tuple, and the value if it does not; and a relation is

47. Front: [math.CA/0509058] Differential Recursion Relations For Laguerre Functions Title Differential Recursion relations for Laguerre functions on Symmetric we considered the family of generalized Laguerre functions on $\Omega$ that http://front.math.ucdavis.edu/0509.5058

Front for the arXiv Mon, 24 Dec 2007 Front math CA math.CA/0509058 search register submit journals ... iFAQ math.CA/0509058 Title: Differential Recursion Relations for Laguerre Functions on Symmetric Cones Authors: Michael Aristidou , Mark Davidson , Gestur Olafsson Categories: math.CA Classical Analysis and ODEs MSC: Abstract: Owner: Gestur Olafsson Version 1: Fri, 2 Sep 2005 22:23:59 GMT - for questions or comments about the Front arXiv contact page - for questions about downloading and submitting e-prints

48. Recursion - Wikipedia, The Free Encyclopedia Some specific kinds of recurrence relation can be solved to obtain a nonRecursive definition. A classic example of recursion is the definition of the http://en.wikipedia.org/wiki/Recursion

var wgNotice = ""; var wgNoticeLocal = ""; var wgNoticeLang = "en"; var wgNoticeProject = "wikipedia"; Recursion From Wikipedia, the free encyclopedia Jump to: navigation search This article is about the concept of recursion. For the novel, see Recursion (novel) . For computer applications, see Recursion (computer science) . For other uses, see recursive A visual form of recursion known as the Droste effect Recursion , in mathematics and computer science , is a method of defining functions in which the function being defined is applied within its own definition. The term is also used more generally to describe a process of repeating objects in a self-similar way. For instance, when the surfaces of two mirrors are almost parallel with each other the nested images that occur are a form of recursion. Contents Formal definitions of recursion Recursion in language Recursion in plain English Recursive humor ... edit Formal definitions of recursion In mathematics and computer science , recursion specifies (or constructs) a class of objects or methods (or an object from a certain class) by defining a few very simple base cases or methods (often just one), and defining rules to break down complex cases into simpler cases. For example, the following is a recursive definition of person's ancestors:

49. Tower Of Hanoi The above expression is known as a recurrence relation which, as you might have noticed, is but a Recursive function. TN is defined in terms of only one of http://www.cut-the-knot.org/recurrence/hanoi.shtml

var MyPageLoc = document.location; var MyPageTitle = document.title; G o o g ... e Web CTK Sites for teachers Sites for parents Awards Interactive Activities ... Sites for parents Tower of Hanoi The Tower of Hanoi puzzle was invented by the French mathematician Edouard Lucas in 1883. We are given a tower of eight disks (initially four in the applet below), initially stacked in increasing size on one of three pegs. The objective is to transfer the entire tower to one of the other pegs (the rightmost one in the applet below), moving only one disk at a time and never a larger one onto a smaller. The puzzle is well known to students of Computer Science since it appears in virtually any introductory text on data structures or algorithms. Its solution touches on two important topics discussed later on: recursive functions and stacks recurrence relations The applet has several controls that allow one to select the number of disks and observe the solution in a Fast or Slow manner. To solve the puzzle drag disks from one peg to another following the rules. You can drop a disk on to a peg when its center is sufficiently close to the center of the peg. The applet expects you to move disks from the leftmost peg to the rightmost peg. This applet requires Sun's Java VM 2 which your browser may perceive as a popup. Which it is not. If you want to see the applet work, visit Sun's website at http://www.java.com/en/download/index.jsp, download and install Java VM and enjoy the applet.

50. Deep Blue At The University Of Michigan: QCD Recursion Relations From The Larges Moreover, the proof of the gluon recursion relations hinges on an identity in momentum space which we show to be nothing but the Fourier transform of the http://deepblue.lib.umich.edu/handle/2027.42/49174

Browse: Collections Titles Authors By Topic Search Deep Blue: Advanced Search Login to My Deep Blue to Deposit/Edit Items authorized users Receive email updates Edit Your Profile Get more info About Deep Blue Help Contact Us Deep Blue at the University of Michigan ... Interdisciplinary and Peer-Reviewed Please use this persistent URL to cite or link to this item: http://hdl.handle.net/2027.42/49174 Title: QCD recursion relations from the largest time equation Author(s): Vaman, Diana Yao, York-Peng Issue Date: 1-Apr-2006 Publisher: IOP Publishing Ltd Citation: Vaman, Diana; Yao, York-Peng (2006). " QCD recursion relations from the largest time equation." Journal of High Energy Physics. 2006(04): 030. <http://hdl.handle.net/2027.42/49174> Abstract: We show how by reassembling the tree level gluon Feynman diagrams in a convenient gauge, space-cone, we can explicitly derive the BCFW recursion relations. Moreover, the proof of the gluon recursion relations hinges on an identity in momentum space which we show to be nothing but the Fourier transform of the largest time equation. Our approach lends itself to natural generalizations to include massive scalars and even fermions. Persistent URL (URI): http://hdl.handle.net/2027.42/49174

51. Tutorial: SQL Server Recursion - Parent Child Relations - VBForums Tutorial SQL Server recursion Parent child relations UtilityBank - Tutorials. http://www.vbforums.com/showthread.php?t=366078

52. Citebase - Mirzakhani's Recursion Relations, Virasoro Constraints And The KdV Hi We present in this paper a differential version of Mirzakhani s recursion relation for the WeilPetersson volumes of the moduli spaces of bordered Riemann http://www.citebase.org/abstract?id=oai:arXiv.org:math/0601194

53. TFL An Environment For Terminating Functional Programs Proving termination of a Recursively defined function divides into two tasks (1) finding a wellfounded relation R; (2) showing that the Recursive calls http://www.cl.cam.ac.uk/~ks121/tfl.html

TFL: An Environment for Terminating Functional Programs TFL is an environment for defining and reasoning about terminating programs written in a purely functional manner. It has the following significant features: Higher order. Programs are represented by the native functions of higher order logic. This light representation means that the proof tools already provided in the logic can be immediately applied to programs. Since functions in the logic can be higher order and (ML-style) polymorphic, the class of programs that can be represented can also be higher order and polymorphic. Therefore a wide class of programs from popular functional languages like ML and Haskell can be quickly formalized and reasoned about. Total. All programs in TFL terminate; therefore, a program defined in TFL will have the same behaviour regardless of which compiler or reduction strategy it is executed under. Thus TFL captures a wide class of programs that are inter-language portable Fully formal.

54. Shodor Interactivate: Recursion The Recursion uses JAVA technology. We are trying to determine whether you have this technology installed on your computer. This should only take a few http://www.shodor.org/interactivate/activities/Recursion/

Shodor Interactivate: Recursion Activity The activity will start momentarily Activity Description Graph recursive functions by defining f(0)=C and defining f(n) based on f(n-1). var data = "One moment please. The Recursion uses JAVA technology. We are trying to determine whether you have this technology installed on your computer. This should only take a few moments." data = data + "Taking too long? You may also choose to skip this detection process."; document.write(data); JavaScript is Disabled Shodor Interactivate utilizes JavaScript technology, you must enable it to effectively use our site. If you wish to leave JavaScript disabled, you may attempt to view the activity . Here is some additional information about this activity that may not be accessible without JavaScript. Recursion Place in mathematics curriculum: This activity can be used to: Teach students about recursive equations Show up to three recursive equations on a graph Trace individual values of a recursive equation using a graph This activity allows the user to explore different recursive equations. This activity would work well in small groups of 2-3 for about 40 minutes if you use the

55. [hep-th/0501052] Direct Proof Of Tree-Level Recursion Relation In Yang-Mills The Recently, by using the known structure of oneloop scattering amplitudes for gluons in Yang-Mills theory, a recursion relation for tree-level scattering http://arxiv.org/abs/hep-th/0501052

arXiv.org hep-th Search or Article-id Help Advanced search All papers Titles Authors Abstracts Full text Help pages Full-text links: Download: PostScript PDF Other formats Citations SLAC-SPIRES HEP (refers to , cited by , arXiv reformatted) CiteBase p revious n ... ext High Energy Physics - Theory Title: Direct Proof Of Tree-Level Recursion Relation In Yang-Mills Theory Authors: Ruth Britto Freddy Cachazo Bo Feng Edward Witten (Submitted on 7 Jan 2005 ( ), last revised 8 Feb 2005 (this version, v2)) Abstract: Recently, by using the known structure of one-loop scattering amplitudes for gluons in Yang-Mills theory, a recursion relation for tree-level scattering amplitudes has been deduced. Here, we give a short and direct proof of this recursion relation based on properties of tree-level amplitudes only. Comments: 10 pp. Added section 4: Proof of MHV Recursion Relations Subjects: High Energy Physics - Theory (hep-th) Journal reference: Phys.Rev.Lett. 94 (2005) 181602 Cite as: arXiv:hep-th/0501052v2 Submission history From: Freddy Cachazo [ view email Fri, 7 Jan 2005 20:04:03 GMT (15kb)

56. Recursion In E-R Relationships Recursion is extremely powerful in modelling. A good example is its use in modelling a family tree. The simplest model stores data about one entity (Person) http://blogs.ittoolbox.com/eai/implementation/archives/recursion-in-er-relations

dojo.setModulePrefix("ITtoolbox", "../widgets");dojo.require("ITtoolbox.UserBadge");dojo.require("ITtoolbox.InviteBadge"); ITtoolbox Blogs 370,128 blog subscriptions Welcome, Guest Sign in Create user What is ITtoolbox? A community where peers share knowledge about information technology. Take the tour Ignore this text box. It is used to detect spammers.If you enter anything into this text box, no search results will be displayed. Sign in to ITtoolbox E-mail or User ID Password or Sign up! Forgot password? Help Find Members ... Blogs Home Browse All Blogs Recent Entries Recent Comments Popular Entries Popular Blogs ... ITtoolbox Blogs Entry Blog Main Blog Archive Author Bio Connect to this blog ... Next Entry Recursion in E-R Relationships Craig Borysowich (Chief Technology Tactician) Posted 1/28/2007 Comments Trackbacks (0) Some relationships involve only one entity. For example: Employee reports to Employee. This type of relationship is called a recursive relationship. A loop is used to show that an occurrence of an entity may be associated with other occurrences of the same entity. Recursion is extremely powerful in modelling.