Random recursive construction of Salem sets

We introduce a random recursive method for constructing random Salem sets inR ^d. The method is inspired by Salem's construction [13] of certain signular monotonic functions. This work contains parts of the author's forthcoming doctoral thesis [2] which were presented at theConference on Harmonic Analysis from the Pichorides Viewpoint in Anogia Academic Village on Crete in July 1995.     

Random recursive construction of self-similar fractal measures. The noncompact case

Summary The self-similarity of sets (measures) is often defined in a constructive way. In the present paper it will be shown that the random recursive construction model of Falconer, Graf and Mauldin/Williams for (statistically) self-similar sets may be generalized to the noncompact case. We define a sequence of random finite measures, which converges almost surely to a self-similar random limit measure. Under certain conditions on the generating Lipschitz maps we determine the carrying dimension of the limit measure.     

Wave recursive interpolation

In this paper it is studied that the generated theory of wave recursive interpolation of uniform T-subdivi-ston scheme include wave parameter.The paper analyses the convergence of sequences of control polygons produced by wave recursive interpolation T-subdivision scheme of the formj=l,2,…,T-1;m=O,l,…,nTk;k=0,l,2,…,and differentiability of the limit curve.     

Extension of partial recursive functions and functions with a recursive graph

It is proved that every degree of complexity of mass problems, containing the decision problem of a recursive enumerable set, contains also the problem of extension of a partial recursive function, the graph of which is recursive. Some properties of functions with a recursive graph are considered.Translated from Matematicheskie Zametki, Vol. 5, No. 2, pp. 261–267, February, 1969.The author is deeply indebted to V. A. Uspenskii for discussion of the results.     

The first example of a recursive function which is not primitive recursive

The first example of a recursive function which is not primitive recursive is usually attributed to W. Ackermann. The authors of the present paper show that such an example can also be found in a paper by G. Sudan, published concomitantly with Ackermann's paper.     

Invariantly recursive credibility estimation

Necessary and sufficient conditions are given for recursivity of credibility estimators to be invariant against changes in term-length. A stationary invariant risk process is analysed, and a concept of asymptotic efficiency is developed.     

Intuitionistically provable recursive well-orderings

We consider intuitionistic number theory with recursive infinitary rules (HA^*). Any primitive recursive binary relation for which transfinite induction schema is provable is in fact well founded. Its ordinal is less than ε⁰ if the transfinite induction schema is intuitionistically provable in elementary number theory. These results are provable intuitionistically. In fact, it suffices to consider transfinite induction with respect to one particular number-theoretic property.     

Nilpotent and recursive flows

In this paper we introduce a class of nonlinear vector fields on infinite dimensional manifolds such that the corresponding evolution equations can be solved with the same method one uses to solve ordinary differential equations with constant coeficients. Mostly, these equations are nonlinear partial differential equations. It is shown that these flows are characterized by a generalization of the ‘method of variation of constants’ which is widely used for second order problems to find general solutions out of particular ones. Invariant densities are constructed for these flows in a natural way. These invariant densities are providing an essential tool for solving initial value and boundary value problems for the equations under consideration. Many applications are presented     

Maximum-likelihood recursive nonlinear filtering

The basic model for the general nonlinear filtering problem consists of a nonlinear plant driven by noise followed by nonlinear observation with additive noise. The object is to estimate, at each instant, the current state of the plant, given thea priori information and the history of the observations up to the current time. The estimation procedure studied here is that of computing, at each instant, the most probable trajectory given the data at that time, and taking its final value. The purpose of the present paper is to clarify some earlier studies of this procedure.This research was partially supported by the National Science Foundation, Grant No. GK-806.     

Iterated relative recursive enumerability

A result of Soare and Stob asserts that for any non-recursive r.e. setC, there exists a r.e.[C] setA such thatAC is not of r.e. degree. A setY is called [of]m-REA (m-REA[C] [degree] iff it is [Turing equivalent to] the result of applyingm-many iterated hops to the empty set (toC), where a hop is any function of the formXX W _e ^X . The cited result is the special casem=0,n=1 of our Theorem. Form=0,1, and any (m+1)-REA setC, ifC is not ofm-REA degree, then for alln there exists an-r.e.[C] setA such thatA C is not of (m+n)-REA degree. We conjecture that this holds also form2.German speakers should not be unduely influenced by the acronym for this titlePartially supported by an NSF Postdoctoral Fellowship and the US Army Research Office through the Mathematical Sciences Institute of Cornell University     

On recursive decompositions of measures

Recursive decompositions of a complex measure relative to a nonnegative measure are considered and properties of the decompositions and conditions for convergence are established.     

Canonical recursive functions and operations

A series of properties of canonical recursive functions and operations are established, allowing the possibility of extrapolating to these functions and operations the method proposed by R. L. Goodstein for constructing equational calculi.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 88, pp. 218–235, 1979.     

Nonstandard arithmetic and recursive comprehension

First order reasoning about hyperintegers can prove things about sets of integers. In the author’s paper Nonstandard Arithmetic and Reverse Mathematics, Bulletin of Symbolic Logic 12 (2006) 100-125, it was shown that each of the “big five” theories in reverse mathematics, including the base theory , has a natural nonstandard counterpart. But the counterpart of has a defect: it does not imply the Standard Part Principle that a set exists if and only if it is coded by a hyperinteger. In this paper we find another nonstandard counterpart, , that does imply the Standard Part Principle.     

A new recursive theorem onn-extendibility

A graphG having a 1-factor is calledn-extendible if every matching of sizen extends to a 1-factor. LetG be a 2-connected graph of order 2p. Letr0 andn>0 be integers such thatp–rn+1. It is shown that ifG/S isn-extendible for every connected subgraphS of order 2r for whichG/S is connected, thenG isn-extendible.