Set Of Nonnegative Integers Research Articles

Counterexamples in the Theory of ω-functions

Let ϵ stand for the set of nonnegative integers (numbers), V for the class of all subcollections of ϵ (sets), Λ for the set of isols, and ΛR for the set of regressive isols. A function, f, is a mapping from a subset of ϵ into ϵ and δf and ρf denote the domain and range of f respectively. The relation of inclusion is denoted by ⊂ and that of proper inclusion by ⊊. The sets α and β are recursively equivalent (written α ≃ β), if δf = α and ρf = β for some function f with a one-to-one partial recursive extension.

Characterization of recursively enumerable sets

AbstractLet N, O and S denote the set of nonnegative integers, the graph of the constant 0 function and the graph of the successor function respectively. For sets P, Q, R ⊆ N2 operations of transposition, composition, and bracketing are defined as follows: P∪ = {〈x, y〉 ∣ 〈y, x〉 ∈ P}, PQ = {〈x, z〉 ∣ ∃y〈x, y〉 ∈ P & 〈y, z〉 ∈ Q}, and [P, Q, R] = ⋃n ∈ M(Pn Q Rn).Theorem. The class of recursively enumerable subsets of N2 is the smallest class of sets with O and S as members and closed under transposition, composition, and bracketing.This result is derived from a characterization by Julia Robinson of the class of general recursive functions of one variable in terms of function composition and “definition by general recursion.” A key step in the proof is to show that if a function F is defined by general recursion from functions A, M, P and R then F = [P∪, A∪M, R].The above definitions of the transposition, composition, and bracketing operations on subsets of N2 can be generalized to subsets of X2 for an arbitrary set X. In this abstract setting it is possible to show that the bracket operation can be defined in terms of K, L, transposition, composition, intersection, and reflexive transitive closure where K: X → X and L: X → X are functions for decoding pairs.

Interpolation and Embedding in the Recursively Enumerable Degrees

By a degree is meant a degree of recursive unsolvability. A degree is recursively enumerable (r.e.) just if it contains a recursively enumerable subset of N, the set of non-negative integers. Two infinite injury priority arguments are presented, in ? 2 and ? 3, which are generalizations of parts of Sacks' proof that the r.e. degrees are dense [9]. These results are in a form sufficiently flexible to admit a variety of applications. It is found, for example, that a degree which is a minimal r.e.-uniform upper bound for a sequence of degrees must be the join of a finite subsequence. By way of contrast, any non-recursive r.e. degree is a minimal upper bound for some strictly ascending sequence of r.e. degrees. It is also found that if a, b are r.e. and a < b, then any countable partial ordering is embeddable in the r.e. degrees between a and b with joins preserved whenever they exist. In ? 4 it is shown that if a' = 0' then there is always such an embedding which also preserves greatest and least elements when they exist. The proof of the latter result is a finite injury priority argument which is more closely related to two splitting theorems of Sacks [8, Th. 2 of ? 5 and Th. 2 of ? 61 than to the preceding results. Our basic notation follows Kleene [4], Kleene and Post [5], and Sacks [8]. A convenient property of Kleene's T-predicates is that if Tn(e, x1, * * * x ,nq y) or Tn(p e xi, ...,xny) holds then U(y) <y and xi<y for 1<? i n. As does Sacks we denote this fact by GND. It is convenient to have U(O) = 2, an inessential departure from Kleene. Another slight variation from Kleene's notation is that we put

ω-homomorphisms and ω-groups

Let ε stand for the set of nonnegative integers (numbers), V for the class of all subcollections of ε (sets), Λ for the set of isols, and for the set of mappings from a subset of ε into ε (functions). I f is a function we write δf and ρf for its domain and range respectively. The relation of inclusion is denoted by ⊂ and that of proper inclusion by ⊊. The sets α and β are recursively equivalent [written: α ≃ β], if δf = α and ρf = β for some function f with a one-to-one partial recursive extension f. We denote the recursive equivalence type of α, {σ ∈ V ∣ ≃ α}, by Req(α). Also R stands for Req(ε), while ΛR denotes the collection of all regressive isols. The reader is assumed to be familiar with the contents of [1] and [6].

Commutativity and common fixed points in recursion theory

Let N be the set of nonnegative integers and, for eEN, let 45e be the partial recursive function of one argument having index e. In 1938 [1, The Recursion Theorem] Kleene showed that if f is any recursive function then, for some number c, /,c-qfCf(,). It follows that if We (the recursively enumerable (r.e.) set with index e) is defined as the domain of 0e, then W= Wf(,). Call a number-theoretic function h well-defined on the r.e. sets if, for all m, nEN, Wm Wn*Wh(m) -Wh(n). In this paper we show that if f, g are recursive functions which are well-defined on the r.e. sets and which commute as maps of the r.e. sets (i.e., for all n&VN, Wf(g(n)) = Wg(f(n))), then they have a common fixed point (i.e., for some e=N, We = Wf(e) = Wg()) We also give an example which shows that the assumption of well-definedness cannot be eliminated. First we prove a lemma related to the Myhill-Sheplherdson Theorem [2, p. 359, Theorem XXIX (6) ]. From now on, wheneverf is well defined on the r.e. sets and W is an r.e. set, we shall write f(W) for Wf(e) where e is any number such that W We.

On Some Special Factorizations of (1-xn)/(1-x)

Let A: a1 &lt; a2 &lt; … &lt; ak be a set of non-negative integers We call the corresponding polynomial the characteristic polynomial, or briefly, the c-polynomial of A. Any polynomial of such a form we call a c-polynomial and any factorization of a c-polynomial into others of the kind we call a c-factorization. If a c-polynomial cannot be factored in this way we call it c-irreducible. In this note we will determine all c-factorizations of the polynomial 1 + x + x2 + … + xn-1, and will find under what circumstances the c-irreducible factors are also irreducible in the usual sense, i.e., irreducible over the field of rationals.

On Complementing Sets of Nonnegative Integers

On Complementing Sets of Nonnegative Integers

On Complementing Sets of Nonnegative Integers

On Complementing Sets of Nonnegative Integers

Theory of Vavilov-Cherenkov radiation in an isotrophic optically active medium: B. M. Bolotovskii and O. S. Mergelyan, Optics & Spectrosc., 14 (3), March 1963, 204

A necessary and sufficient condition is obtained for the linear span of a system of monomials {zλ:λ∈Λ} to be dense in the space of all continuous functions defined on the line segments emerging from the origin, where Λ is a set of nonnegative integers. The result is a generalization of the Müntz theorem to the segments emerging from the origin and an extension of the Mergelyan theorem to lacunary polynomials.

On the Multiplicity of Representations of Numbers by Sums

Let A: {a1, ≤ a2, ≤ ߪ ≤ at} be a set of non-negative integers not exceeding n, and let rn(A;m) = rr(m) denote the numbers of representations of m as the sum of k a's, order taken into account, i. e.1Since 0 ≤ ai1 + ai2 + … + aik ≤ kn, we have2

Power Series Representing Certain Rational Functions

Let denote the set of functions of a complex variable z, regular at z = 0, and let I denote the set of non-negative integers. For f ∈ putFor a given subset 0 of there arises the problem of characterizing the admissible gap sets If of functions f in 0. When 0 is the set R of rational functions a complete solution in given by the following theorem:(A) Let f ∈ R and let If be infinite. Then there exist integers L, L1, L2… , L3 such that 0 ≤ L1 &lt; L2 … &lt; Ls &lt; L, and If = {n|n ∈ I, n ≡ Lj (mod L), j =1, … , s} U I\ where V is a Unite exceptional set.

On Sums of Sets of Integers

Small italics denote integers. Let A, B, … be sets of non-negative integers. Let A (h) be the number of positive integers in A that are not greater than h. Finally let A + B denote the set of all integers of the form a + b where a ⊂ A, b ⊂ B. The following result is implicitly contained in Mann's Proposition 11 (4):Theorem 1. Let n &gt; 0 and(1.1) 0⊂4, 0⊂B, n⊄C = A + B.

On intervals of prescribed lengths

The present paper is an outgrowth of a conversation with Maurice Sion. In this conversation he asked whether the closed unit interval always contains a set of Lebesgue measure zero which cannot be embraced by a sequence of open intervals of prescribed lengths adding up to 1. I posed the corresponding question for the unit open interval. Answers to these questions are given by Theorems 3 and 6 below. Let us agree: X is the set of non-negative integers; a sequence is a function whose domain is c; S embraces A if and only if S is such a sequence of sets that

On the Sum of Two Sets of Integers

In his beautiful paper: proof of the fundamental theorem on the density of sums of sets of positive integers' Mr. Mann succeeded in proving the (a, A)hypothesis and a generalization of it that had been conjectured for more than ten years. We found that his method can be simplified considerably and even yields some stronger results. Let A, B respectively be sets of nonnegative integers a, b. Let C = A + B be the set of all integers of the form a + b. Let A(x), B(x), C(x) denote the number of positive integers of the sets <x.2 Mr. Mann proved the following theorem: If 0 C A and O C B, and if C(n) < n, then