79,898 research outputs found
On bar recursion of types 0 and 1
For general information on bar recursion the reader should consult the papers of Spector [8], where it was introduced, Howard [2] and Tait [11]. In this note we shall prove that the terms of Godel's theory T (in its extensional version of Spector [8]) are closed under the rule BRo•1 of bar recursion of types 0 and 1. Our method of proof is based on the notion of an infinite term introduced by Tait [9]. The main tools of the proof are (i) the normalization theorem for (notations for) infinite terms and (ii) valuation functionals. Both are elaborated in [6]; for brevity some familiarity with this paper is assumed here. Using (i) and (ii) we reduce BRo.1 to ';-recursion with'; < co. From this the result follows by work of Tait [10], who gave a reduction of 2E-recursion to ';-recursion at a higher type. At the end of the paper we discuss a perhaps more natural variant of bar recursion introduced by Kreisel in [4]. Related results are due to rKeisel (in his appendix to [8]), who obtains results which imply, using the reduction given by Howard [2] of the constant of bar recursion of type '0 to the rule of bar recursion of type (0 ~ '0) ~ '0, that T is not closed under the rule of bar recursion of a type of level ~ 2, to Diller [1], who gave a reduction of BRo.1 to ';-recursion with'; bounded by the least (V-critical number, and to Howard [3], who gave an ordinal analysis of the constant of bar recursion of type O. I am grateful to H. Barendregt, W. Howard and G. Kreisel for many useful comments and discussions. Recall that a functional F of type 0 ~ (0 ~ '0) ~ (J is said to be defined by (the rule of) bar recursion of type '0 from Yand functionals G, H of the proper types i
Computation on abstract data types. The extensional approach, with an application to streams
AbstractIn this paper we specialize the notion of abstract computational procedure previously introduced for intensionally presented structures to those which are extensionally given. This is provided by a form of generalized recursion theory which uses schemata for explicit definition, conditional definition and least fixed point (LFP) recursion in functional of type level ⩽ 2 over any appropriate structure. It is applied here to the case of potentially infinite (and more general partial) streams as an abstract data type
Loop Equations and the Topological Phase of Multi-Cut Matrix Models
We study the double scaling limit of mKdV type, realized in the two-cut
Hermitian matrix model. Building on the work of Periwal and Shevitz and of
Nappi, we find an exact solution including all odd scaling operators, in terms
of a hierarchy of flows of matrices. We derive from it loop
equations which can be expressed as Virasoro constraints on the partition
function. We discover a ``pure topological" phase of the theory in which all
correlation functions are determined by recursion relations. We also examine
macroscopic loop amplitudes, which suggest a relation to 2D gravity coupled to
dense polymers.Comment: 24p
Field diffeomorphisms and the algebraic structure of perturbative expansions
We consider field diffeomorphisms in the context of real scalar field
theories. Starting from free field theories we apply non-linear field
diffeomorphisms to the fields and study the perturbative expansion for the
transformed theories. We find that tree level amplitudes for the transformed
fields must satisfy BCFW type recursion relations for the S-matrix to remain
trivial. For the massless field theory these relations continue to hold in loop
computations. In the massive field theory the situation is more subtle. A
necessary condition for the Feynman rules to respect the maximal ideal and
co-ideal defined by the core Hopf algebra of the transformed theory is that
upon renormalization all massive tadpole integrals (defined as all integrals
independent of the kinematics of external momenta) are mapped to zero.Comment: 8 pages, 2 figure
On Characterizing Spector Classes
We study in this paper characterizations of various interesting classes of relations arising in recursion theory. We first determine which Spector classes on the structure of arithmetic arise from recursion in normal type 2 objects, giving a partial answer to a problem raised by Moschovakis [8], where the notion of Spector class was first
essentially introduced. Our result here was independently discovered by S. G. Simpson (see [3]). We conclude our study of Spector classes by examining two simple relations between them and a natural hierarchy to which they give rise
Self-dual noncommutative \phi^4-theory in four dimensions is a non-perturbatively solvable and non-trivial quantum field theory
We study quartic matrix models with partition function Z[E,J]=\int dM
\exp(trace(JM-EM^2-(\lambda/4)M^4)). The integral is over the space of
Hermitean NxN-matrices, the external matrix E encodes the dynamics, \lambda>0
is a scalar coupling constant and the matrix J is used to generate correlation
functions. For E not a multiple of the identity matrix, we prove a universal
algebraic recursion formula which gives all higher correlation functions in
terms of the 2-point function and the distinct eigenvalues of E. The 2-point
function itself satisfies a closed non-linear equation which must be solved
case by case for given E. These results imply that if the 2-point function of a
quartic matrix model is renormalisable by mass and wavefunction
renormalisation, then the entire model is renormalisable and has vanishing
\beta-function.
As main application we prove that Euclidean \phi^4-quantum field theory on
four-dimensional Moyal space with harmonic propagation, taken at its
self-duality point and in the infinite volume limit, is exactly solvable and
non-trivial. This model is a quartic matrix model, where E has for N->\infty
the same spectrum as the Laplace operator in 4 dimensions. Using the theory of
singular integral equations of Carleman type we compute (for N->\infty and
after renormalisation of E,\lambda) the free energy density
(1/volume)\log(Z[E,J]/Z[E,0]) exactly in terms of the solution of a non-linear
integral equation. Existence of a solution is proved via the Schauder fixed
point theorem.
The derivation of the non-linear integral equation relies on an assumption
which we verified numerically for coupling constants 0<\lambda\leq (1/\pi).Comment: LaTeX, 64 pages, xypic figures. v4: We prove that recursion formulae
and vanishing of \beta-function hold for general quartic matrix models. v3:
We add the existence proof for a solution of the non-linear integral
equation. A rescaling of matrix indices was necessary. v2: We provide
Schwinger-Dyson equations for all correlation functions and prove an
algebraic recursion formula for their solutio
The algebro-geometric solutions for Degasperis-Procesi hierarchy
Though completely integrable Camassa-Holm (CH) equation and
Degasperis-Procesi (DP) equation are cast in the same peakon family, they
possess the second- and third-order Lax operators, respectively. From the
viewpoint of algebro-geometrical study, this difference lies in hyper-elliptic
and non-hyper-elliptic curves. The non-hyper-elliptic curves lead to great
difficulty in the construction of algebro-geometric solutions of the DP
equation. In this paper, we derive the DP hierarchy with the help of Lenard
recursion operators. Based on the characteristic polynomial of a Lax matrix for
the DP hierarchy, we introduce a third order algebraic curve
with genus , from which the associated Baker-Akhiezer
functions, meromorphic function and Dubrovin-type equations are established.
Furthermore, the theory of algebraic curve is applied to derive explicit
representations of the theta function for the Baker-Akhiezer functions and the
meromorphic function. In particular, the algebro-geometric solutions are
obtained for all equations in the whole DP hierarchy.Comment: 65 pages. arXiv admin note: text overlap with arXiv:solv-int/9809004
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