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Recursive functions of one variable

1968
*
Proceedings of the American Mathematical Society
*

Hence all primitive

doi:10.1090/s0002-9939-1968-0230618-2
fatcat:hjmgv2nh7vdgjdnqera7vq4x2m
*recursive*functions of one variable are obtained from 0,*I*, S, K, and L by compositions, pairings, and*general**recursions*of the*form*given in the lemma. ... Hence every primitive*recursive*function is obtained from the usual initial functions together with J, K, and L by*general**recursion*of the*form*in the lemma and substitution. ...##
###
Recursive Functions of One Variable

1968
*
Proceedings of the American Mathematical Society
*

Hence all primitive

doi:10.2307/2035317
fatcat:326uymhf35gmhlsqtzaltqfs74
*recursive*functions of one variable are obtained from 0,*I*, S, K, and L by compositions, pairings, and*general**recursions*of the*form*given in the lemma. ... Hence every primitive*recursive*function is obtained from the usual initial functions together with J, K, and L by*general**recursion*of the*form*in the lemma and substitution. ...##
###
Recursion Operators for Multidimensional Integrable PDEs
[article]

2017
*
arXiv
*
pre-print

We present a novel construction of

arXiv:1710.05907v1
fatcat:nrzqyddy55dwzljpi22hrloxju
*recursion*operators for scalar second-order integrable multidimensional PDEs with isospectral Lax pairs written in terms of first-order scalar differential operators. ... Imposing the requirements in question gives rise to a (in*general*overdetermined) system of*equations*for f j*i*, and any solution of this system gives a*recursion*operator for (1). ... (1) but also to more*general**equations*of the*form*F ( x, u, (k) u ) = 0, (12) where (k) u denotes the derivatives of u up to order k, and k 2. ...##
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A new construction of recursion operators for systems of the hydrodynamic type

2000
*
Theoretical and mathematical physics
*

Therefore, without loss of

doi:10.1007/bf02551167
fatcat:kicyltyxsjf3pmbolfsrfqbde4
*generality*, we start with the diagonal system q~=v*i*(q) q;,*i*=1,2. (1) ... We use this to give a simple construction of*recursion*operators for these systems, not always coinciding with those of Sheftel and Teshukov. ... We choose ~*i*in the*form*1 ~*i*(qi) _ ai(qi) '~' ai E ~. (8) 2. We require the*form*of wave*equation*(7) to be invariant under the transformation V(ql,q2) = V(ql,q2)(ql)C~2(q2)a~, qi = fi(qi). ...##
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Chapter VIII: Recursively defined functions
[chapter]

1977
*
Lecture Notes in Computer Science
*

*Recursive*

*equations*provide a convenient language for defining computable functions. Recent studies by Cadiou [Cad72], Vuillemin [Vu74], Downey and Sexthi [DS? ... 6] and Berry and L~vy [BL77] separate computational steps involving

*recursively*defined functions from those involving given primitive functions, and are based on continuous interpretations of primitive ... Consider the SRS <Z,Z#,+,A> where A is a set of

*recursive*

*equations*. Theorem 21 Let A be a set of

*recursive*

*equations*containing exactly one

*equation*of the

*form*f(X 1 ..... Xn)=A for each fcFn, n~p. ...

##
###
Recursive Matrix Calculation Paradigm by the Example of Structured Matrix

2020
*
Information
*

In this paper, we derive

doi:10.3390/info11010042
fatcat:ispkczu4uban3nfaicbxpmeaii
*recursive*algorithms for calculating the determinant and inverse of the*generalized*Vandermonde matrix. ... methods, developed for the*general*matrices. ... Acknowledgments:*I*would like to thank my university colleagues for stimulating discussions and reviewers for apt remarks which significantly improved the paper. ...##
###
Time-Dependent Recursion Operators and Symmetries

2002
*
Journal of Nonlinear Mathematical Physics
*

A

doi:10.2991/jnmp.2002.9.2.5
fatcat:6jqvpm6jwbccfa5hrijsrcrp3e
*general*formula is given for the missing term of the*recursion*operators. Apart from the*recursion*operators a method is introduced to calculate the correct symmetries. ... It has been previously observed that the symmetries of the integrable evolution*equations*obtained through their*recursion*operators do not satisfy the symmetry*equations*. ... Time-Dependent*Recursion*Operators and Symmetries ...##
###
Formal recursion operators of integrable nonevolutionary equations and Lagrangian systems

2018
*
Journal of Physics A: Mathematical and Theoretical
*

We derive the

doi:10.1088/1751-8121/aad63e
fatcat:6ew7oicfpjemlc5jjhhu3ej6qy
*general*structure of the space of formal*recursion*operators of nonevolutionary*equations*~$q_{tt}=f(q,q_{x},q_t,q_{xx},q_{xt},q_{xxx},q_{xxxx})$. ... The key technique relays on exploiting a homogeneity of the determining*equations*of formal*recursion*operators. ... This extension can be further applied to more*general**equations*as follows. An integrable*equation*(1.6) admits two types of formal*recursion*operators, L*i*and M j ,*i*, j ∈ Z. ...##
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Symmetry group analysis and invariant solutions of hydrodynamic-type systems

2004
*
International Journal of Mathematics and Mathematical Sciences
*

Under additional restrictions on the systems, we obtain

doi:10.1155/s0161171204206147
fatcat:73li7pg43bgsrajyx3koh34uw4
*recursion*operators for symmetries and use them to construct infinite discrete sets of exact solutions of the studied*equations*. ... We find the interrelation between higher symmetries and*recursion*operators. Two-component systems are studied in more detail thann-component systems. ... , Formula (3.14) gives us a*general**form*of the first-order*recursion*operator for a*generic*function α(ρ). ...##
###
Differential Constraints Compatible with Linearized Equations

1998
*
Journal of Nonlinear Mathematical Physics
*

Differential constraints compatible with the linearized

doi:10.2991/jnmp.1998.5.4.3
fatcat:wme6pfr7zfdmxbwsz4gx73gwxm
*equations*of partial differential*equations*are examined.*Recursion*operators are obtained by integrating the differential constraints. ... Acknowledgments*I*would like to thank Norbert Euler and the referee for their constructive comments. ... Next, we consider differential*equations*of the following*general**form*, which includes the KdV*equation*, q t = P (q, q x , q xx , q xxx ). (24) The linearization of (24) takes the*form*Ψ t = αΨ xxx + βΨ ...##
###
Recursive predicates and quantifiers

1943
*
Transactions of the American Mathematical Society
*

This paper contains a

doi:10.1090/s0002-9947-1943-0007371-8
fatcat:eth5qmjytzhizobcrew5u5qezm
*general*theorem on the quantification of*recursive*predicates, with applications to the foundations of mathematics. ... The*general*theorem asserts that to each of an enumeration of predicate*forms*, there is a predicate not expressible in that*form*. The predicates considered belong to elementary number theory. ...*I*. The*general*theorem on*recursive*predicates and quantifiers 1 . Primitive*recursive*functions. ...##
###
Recursive Predicates and Quantifiers

1943
*
Transactions of the American Mathematical Society
*

This paper contains a

doi:10.2307/1990131
fatcat:wjmsir4qdzaevoloouubrbvjli
*general*theorem on the quantification of*recursive*predicates, with applications to the foundations of mathematics. ... The*general*theorem asserts that to each of an enumeration of predicate*forms*, there is a predicate not expressible in that*form*. The predicates considered belong to elementary number theory. ...*I*. The*general*theorem on*recursive*predicates and quantifiers 1 . Primitive*recursive*functions. ...##
###
Nonlinear PDE's and recursive flows: applications

1993
*
Applied Mathematics Letters
*

These are

doi:10.1016/0893-9659(93)90158-j
fatcat:oyjvz64jfvhczfqdeaspaiz72y
*equations*having the property that a Wronskian determinant,*formed*by the*I*-and t-derivatives, is equal to zero on the solution manifold of the*equation*. ... The theory developed in a preeeeding paper [l] is applied for the solution of the Cauehy problem for the se-called Wronskian partial differential*equations*. ... Observe that this*equation*may be easily rewritten as o(h;'f-*i*(st/s)) = h2, a*form*which will be called the associated PDE. ...##
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Group analysis of hydrodynamic-type systems
[article]

2001
*
arXiv
*
pre-print

The

arXiv:math-ph/0104014v1
fatcat:5445juyitvhghckoygrkql3zhq
*recursion*operators for symmetries are obtained and used for constructing infinite series of exact solutions of the studied*equations*. ... We study point and higher symmetries for the hydrodynamic-type systems with two independent variables t and x with and without explicit dependence of the*equations*on t,x. ...*Recursion*operator R of the*form*(2.76)*generates*an infinite set of higher symmetries (2.3) of any order N. ...##
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Generating conjugate directions for arbitrary matrices by matrix equations I

1991
*
Computers and Mathematics with Applications
*

There are three basic techniques to achieve this aim: (

doi:10.1016/0898-1221(91)90232-s
fatcat:nsqvkm5it5httdqtad2aforbfu
*i*) minimizing a quadratic*form*, (ii)*generation*by projections, and (iii) use of matrix*equations*. ... Among matrix*equation**forms*Hestenes -Stiefel type*recursions*and L~nczos type*recursions*are mentioned, where the*recursion*matrices are bidiagonal matrices in the simple case. ... THE DIRECT*RECURSION*SCHEME OF HESTENES -STIEFEL TYPE ILl. The matrix*equations*The*general*matrix*recursion**equations*of Hestenes -Stiefel type are given by (1.26) -(1.28). ...
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