scholarly journals RELATIVE DEFINABILITY OF n-GENERICS

2018 ◽  
Vol 83 (04) ◽  
pp. 1345-1362
Author(s):  
WEI WANG
Keyword(s):  
Positive Integer ◽  
Initial Segment ◽  
Cohen Forcing ◽  
Image Position ◽  
Generic Set

AbstractA set $G \subseteq \omega$ is n-generic for a positive integer n if and only if every ${\rm{\Sigma }}_n^0$ formula of G is decided by a finite initial segment of G in the sense of Cohen forcing. It is shown here that every n-generic set G is properly ${\rm{\Sigma }}_n^0$ in some G-recursive X. As a corollary, we also prove that for every $n > 1$ and every n-generic set G there exists a G-recursive X which is generalized ${\rm{lo}}{{\rm{w}}_n}$ but not generalized ${\rm{lo}}{{\rm{w}}_{n - 1}}$. Thus we confirm two conjectures of Jockusch [4].

scholarly journals The Divisibility of Divisor Functions

1961 ◽  
Vol 5 (1) ◽  
pp. 35-40 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Positive Integer ◽  
Divisor Functions ◽  
Image Position ◽  
Positive Integers

For any positive integers n and v letwhere d runs through all the positive divisors of n. For each positive integer k and real x > 1, denote by N(v, k; x) the number of positive integers n ≦ x for which σv(n) is not divisible by k. Then Watson [6] has shown that, when v is odd,as x → ∞; it is assumed here and throughout that v and k are fixed and independent of x. It follows, in particular, that σ (n) is almost always divisible by k. A brief account of the ideas used by Watson will be found in § 10.6 of Hardy's book on Ramanujan [2].

The Distribution Of Totatives

1955 ◽  
Vol 7 ◽  
pp. 347-357 ◽  
Author(s):  
D. H. Lehmer
Keyword(s):  
Positive Integer ◽  
Prime Factors ◽  
Image Position

This paper is concerned with the numbers which are relatively prime to a given positive integerwhere the p's are the distinct prime factors of n. Since these numbers recur periodically with period n, it suffices to study the ϕ(n) numbers ≤n and relatively prime to n.

scholarly journals An algebraically closed field

1968 ◽  
Vol 9 (2) ◽  
pp. 146-151 ◽  
Author(s):  
F. J. Rayner
Keyword(s):  
Positive Integer ◽  
Power Series ◽  
Formal Power Series ◽  
Characteristic Zero ◽  
Closed Field ◽  
Algebraically Closed Field ◽  
Zero Characteristic ◽  
Image Position ◽  
Formal Power ◽  
Puiseux Expansions

Letkbe any algebraically closed field, and denote byk((t)) the field of formal power series in one indeterminatetoverk. Letso thatKis the field of Puiseux expansions with coefficients ink(each element ofKis a formal power series intl/rfor some positive integerr). It is well-known thatKis algebraically closed if and only ifkis of characteristic zero [1, p. 61]. For examples relating to ramified extensions of fields with valuation [9, §6] it is useful to have a field analogous toKwhich is algebraically closed whenkhas non-zero characteristicp. In this paper, I prove that the setLof all formal power series of the form Σaitei(where (ei) is well-ordered,ei=mi|nprt,n∈ Ζ,mi∈ Ζ,ai∈k,ri∈ Ν) forms an algebraically closed field.

scholarly journals An Integral involving a Modified Bessel Function of the Second Kind and an E-function

1953 ◽  
Vol 1 (3) ◽  
pp. 119-120 ◽  
Author(s):  
Fouad M. Ragab
Keyword(s):  
Positive Integer ◽  
Bessel Function ◽  
Modified Bessel Function ◽  
Image Position

§ 1. Introductory. The formula to be established iswhere m is a positive integer,and the constants are such that the integral converges.

scholarly journals Integrals involving products of modified Bessel functions of the second kind

1963 ◽  
Vol 6 (2) ◽  
pp. 70-74 ◽  
Author(s):  
F. M. Ragab
Keyword(s):  
Positive Integer ◽  
Bessel Functions ◽  
Modified Bessel Functions ◽  
Image Position

It is proposed to establish the two following integrals.where n is a positive integer, x is real and positive, μi and ν are complex, and Δ (n; a) represents the set of parameterswhere n is a positive integer and x is real and positive.

Sublattices and Initial Segments of the Degrees of Unsolvability

1970 ◽  
Vol 22 (3) ◽  
pp. 569-581 ◽  
Author(s):  
S. K. Thomason
Keyword(s):  
Lower Bound ◽  
Upper Bound ◽  
Lattice Theory ◽  
Initial Segment ◽  
Finite Lattice ◽  
Equivalence Relations ◽  
Finite Lattices ◽  
Image Position ◽  
Degrees Of Unsolvability

In this paper we shall prove that every finite lattice is isomorphic to a sublattice of the degrees of unsolvability, and that every one of a certain class of finite lattices is isomorphic to an initial segment of degrees.Acknowledgment. I am grateful to Ralph McKenzie for his assistance in matters of lattice theory.1. Representation of lattices. The equivalence lattice of the set S consists of all equivalence relations on S, ordered by setting θ ≦ θ’ if for all a and b in S, a θ b ⇒ a θ’ b. The least upper bound and greatest lower bound in are given by the ⋃ and ⋂ operations:

Determining a Set from the Cardinalities of its Intersections with Other Sets

1964 ◽  
Vol 16 ◽  
pp. 94-97 ◽  
Author(s):  
David G. Cantor
Keyword(s):  
Positive Integer ◽  
Image Position

Let n be a positive integer and put N = {1, 2, . . . , n}. A collection {S1, S2, . . . , St} of subsets of N is called determining if, for any T ⊂ N, the cardinalities of the t intersections T ∩ Sj determine T uniquely. Let €1, €2, . . . , €n be n variables with range {0, 1}. It is clear that a determining collection {Sj) has the property that the sums

Note Upon The Generalized Cayleyan Operator

1949 ◽  
Vol 1 (1) ◽  
pp. 48-56 ◽  
Author(s):  
H. W. Turnbull
Keyword(s):  
Positive Integer ◽  
Original Proof ◽  
Image Position

The following note which deals with the effect of a certain determinantal operator when it acts upon a product of determinants was suggested by the original proof which Dr. Alfred Young gave of the propertysubsisting between the positive P and the negative N substitutional operators, θ being a positive integer. This result which establishes the idempotency of the expression θ−1NP within an appropriate algebra is fundamental in the Quantitative Substitutional Analysis that Young developed.

On the Integral Part of a Linear form with Prime Variables

1966 ◽  
Vol 18 ◽  
pp. 621-628 ◽  
Author(s):  
I. Danicic
Keyword(s):  
Positive Integer ◽  
Linear Form ◽  
Image Position

The object of this paper is to prove the following:Theorem. Suppose that λ, μ are real non-zero numbers, not both negative, λ is irrational, and k is a positive integer. Then there exist infinitely many primes p and pairs of primes p1, p2 such thatIn particular [λp1 + μp2] represents infinitely many primes.Here [x] denotes the greatest integer not exceeding x.

Irreducibility of Bernoulli Polynomials of Higher Order

1962 ◽  
Vol 14 ◽  
pp. 565-567 ◽  
Author(s):  
P. J. McCarthy
Keyword(s):  
Positive Integer ◽  
Bernoulli Number ◽  
Bernoulli Polynomials ◽  
Constant Term ◽  
Rational Numbers ◽  
Higher Order ◽  
Image Position

The Bernoulli polynomials of order k, where k is a positive integer, are defined byBm(k)(x) is a polynomial of degree m with rational coefficients, and the constant term of Bm(k)(x) is the mth Bernoulli number of order k, Bm(k). In a previous paper (3) we obtained some conditions, in terms of k and m, which imply that Bm(k)(x) is irreducible (all references to irreducibility will be with respect to the field of rational numbers). In particular, we obtained the following two results.

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