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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.<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
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ORC Resource Number #1470 Trout Pond: Using Algebra and Discrete Mathematics to Investigate Population Changes
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
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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
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

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

Encyclopaedia of Mathematics
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)
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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
    Index Main essay An arguable inconsistency in ZF Bhupinder Singh Anand A .pdf file of this essay before the current update is available at 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.
    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
    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
    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.
    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

    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
    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.
    Week Topics Monday
    (lecture) Wednesday
    (lecture) Wednesday
    (exercise) Extra
    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
    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.
    Subject areas covered:
    • Information technology Algorithms, logic General titles on the theory it
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    15. Grad Description
    Elements of set theory, functions and relations nondecimal numbers, Include Turing Machines, partial Recursive functions, Recursive and Recursively

    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
    Computability DCS 301 QMUL - Semester 6 - Winter 2007
    • 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:
    • 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,
    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.
    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.
    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
    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
    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

    21. The Set Of Primitive Recursive Functions
    Preliminaries; Sets of Compatible functions; Homogeneous relations; Primitive Recursiveness; The Set of Primitive Recursive functions; Examples
    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
    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.
    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

    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.
    FOM: Concepts of Recursion Theory
    Joseph Shoenfield jrs at
    Sat Aug 29 15:21:36 EDT 1998 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
    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
    var tcdacmd="dt";

    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
    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.
    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
    Bruce Ediger November 3, 2004
    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

    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.
    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.
    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
    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
    Physical Review Online Archive Physical Review Online Archive AMERICAN PHYSICAL SOCIETY
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    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)
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    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.
    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 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.
    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.
    Appropriate COMPASS placement score (College Algebra 53) or successful completion of MAT 115 Pre-Calc I is required.
    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://
    This thesis explores new ground regarding the composition of relations into this reachability question can be decided by primitive Recursive functions.
    Caltech Library System
    Browse Search Caltech Student Instructions
    Cook, Matthew M. (2005-05-27) Networks of relations.
    Type of Document Dissertation Author Cook, Matthew M. URN etd-06032005-140944 Persistent URL 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
    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

    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
    Discrete Structures (DS) DS1. Functions, relations, and sets [core]
    DS2. Basic logic
    DS3. Proof techniques
    DS4. Basics of counting
    DS5. Graphs and trees
    DS6. Discrete probability
    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.
    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
    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
    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
    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.
    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.

    42. Recursion
    Definition of recursion, possibly with links to more information and implementations. Write a function to compute GCD based on the following relations
    (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
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    Analytic Study of Periodic Chaos. II -Recursion Relations for the Power Spectra- Authors:

    Progress of Theoretical Physics, Vol. 73, No. 2, pp. 349-360 Publication Date:

    (c) 1985 Progress of Theoretical Physics Bibliographic Code:
    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
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    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
    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,
    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 @
    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
    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
    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
    Front for the arXiv Mon, 24 Dec 2007
    math CA math.CA/0509058 search register submit
    ... 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
    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
    var wgNotice = ""; var wgNoticeLocal = ""; var wgNoticeLang = "en"; var wgNoticeProject = "wikipedia";
    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.

    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
    var MyPageLoc = document.location; var MyPageTitle = document.title; G o o g ... e Web CTK Sites for teachers
    Sites for parents


    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, 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
    Browse: Collections Titles Authors By Topic Search Deep Blue: Advanced Search Login to
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    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:
    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. <> 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):

    51. Tutorial: SQL Server Recursion - Parent Child Relations - VBForums
    Tutorial SQL Server recursion Parent child relations UtilityBank - Tutorials.

    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

    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
    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
    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 hep-th
    Search or Article-id Help Advanced search All papers Titles Authors Abstracts Full text Help pages
    Full-text links: Download:
    Citations 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)
    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
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    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
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    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.

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