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### Recursive functions of one variable

Julia Robinson
1968 Proceedings of the American Mathematical Society
Hence all primitive 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

Julia Robinson
1968 Proceedings of the American Mathematical Society
Hence all primitive 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]

A. Sergyeyev
2017 arXiv   pre-print
We present a novel construction of 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.  ...

### A new construction of recursion operators for systems of the hydrodynamic type

A. P. Fordy, T. B. Gürel
2000 Theoretical and mathematical physics
Therefore, without loss of 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).  ...

### 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

Jerzy S. Respondek
2020 Information
In this paper, we derive 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

M Gürses, A Karasu, R Turhan
2002 Journal of Nonlinear Mathematical Physics
A 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

Agustín Caparrós Quintero, Rafael Hernández Heredero
2018 Journal of Physics A: Mathematical and Theoretical
We derive the 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.  ...

### Symmetry group analysis and invariant solutions of hydrodynamic-type systems

M. B. Sheftel
2004 International Journal of Mathematics and Mathematical Sciences
Under additional restrictions on the systems, we obtain 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

Ahmet Satir
1998 Journal of Nonlinear Mathematical Physics
Differential constraints compatible with the linearized 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

S. C. Kleene
1943 Transactions of the American Mathematical Society
This paper contains a 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

S. C. Kleene
1943 Transactions of the American Mathematical Society
This paper contains a 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

Benno Fuchssteiner, Mauro Lo Schiavo
1993 Applied Mathematics Letters
These are 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.  ...

### Group analysis of hydrodynamic-type systems [article]

M.B. Sheftel
2001 arXiv   pre-print
The 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.  ...

### Generating conjugate directions for arbitrary matrices by matrix equations I

Cs.J. Hegedüs
1991 Computers and Mathematics with Applications
There are three basic techniques to achieve this aim: (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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