The þ-function in λ-K-conversion

1937 ◽  
Vol 2 (4) ◽  
pp. 164-164 ◽  
Author(s):  
A. M. Turing
Keyword(s):  
Positive Integer ◽  
Natural Number ◽  
Definition Of ◽  
Image Position

In the theory of conversion it is important to have a formally defined function which assigns to any positive integer n the least integer not less than n which has a given property. The definition of such a formula is somewhat involved: I propose to give the corresponding formula in λ-K-conversion, which will (naturally) be much simpler. I shall in fact find a formula þ such that if T be a formula for which T(n) is convertible to a formula representing a natural number, whenever n represents a natural number, then þ(T, r) is convertible to the formula q representing the least natural number q, not less than r, for which T(q) conv 0.2 The method depends on finding a formula Θ with the property that Θ conv λu·u(Θ(u)), and consequently if M→Θ(V) then M conv V(M). A formula with this property is,The formula þ will have the required property if þ(T, r) conv r when T(r) conv 0, and þ(T, r) conv þ(T, S(r)) otherwise. These conditions will be satisfied if þ(T, r) conv T(r, λx·þ(T, S(r)), r), i.e. if þ conv {λptr·t(r, λx·p(t, S(r)), r)}(þ). We therefore put,This enables us to define also a formula,such that (T, n) is convertible to the formula representing the nth positive integer q for which T(q) conv 0.

IX.—Generalised Derivatives and Integrals

1929 ◽  
Vol 48 ◽  
pp. 92-105
Author(s):  
W. L. Ferrar
Keyword(s):  
Positive Integer ◽  
Lower Limit ◽  
Infinite Series ◽  
Convergent Expansion ◽  
Definition Of ◽  
Image Position ◽  
Generalised Derivatives

The various definitions which have been adopted by one or more writers for Dnf(x), where n is real but not an integer, fall roughly into three classes:—(1) Liouville —a method which assumes a convergent expansionand defines Dnf(x) as ΣAαneax.(2) Riemann, Grünwald, Laurent and others—methods which, however they begin, ultimately come to the “integral definition,”the lower limit of integration being arbitrary:where k is a positive integer, 0<ρ<1, and n=k – ρ.(3) Pincherle5—a method which, seeking an operator with certain properties, gives an infinite series as a definition of Dnf(x)

Paradox of the class of all grounded classes

1953 ◽  
Vol 18 (2) ◽  
pp. 114-114 ◽  
Author(s):  
Shen Yuting
Keyword(s):  
Positive Integer ◽  
Natural Number ◽  
Image Position ◽  
Special Case

A class A for which there is an infinite progression of classes A1, A2, … (not necessarily all distinct) such thatis said to be groundless. A class which is not groundless is said to be grounded. Let K be the class of all grounded classes.Let us assume that K is a groundless class. Then there is an infinite progression of classes A1, A2, … such thatSince A1 ϵ K, A1 is a grounded class; sinceA1 is also a groundless class. But this is impossible.Therefore K is a grounded class. Hence K ϵ K, and we haveTherefore K is also a groundless class.This paradox forms a sort of triplet with the paradox of the class of all non-circular classes and the paradox of the class of all classes which are not n-circular (n a given natural number). The last of the three includes as a special case the paradox of the class of all classes which are not members of themselves (n = 1).More exactly, a class A1 is circular if there exists some positive integer n and classes A2, A3, …, An such thatFor any given positive integer n, a class A1 is n-circular if there are classes A2, …, An, such thatQuite obviously, by arguments similar to the above, we get a paradox of the class of all non-circular classes and a paradox of the class of all classes which are not n-circular, for each positive integer n.

scholarly journals ADDITIVE BASES AND NIVEN NUMBERS

2021 ◽  
pp. 1-8
Author(s):  
CARLO SANNA
Keyword(s):  
Positive Integer ◽  
Natural Number ◽  
Riemann Hypothesis ◽  
Additive Basis

Abstract Let $g \geq 2$ be an integer. A natural number is said to be a base-g Niven number if it is divisible by the sum of its base-g digits. Assuming Hooley’s Riemann hypothesis, we prove that the set of base-g Niven numbers is an additive basis, that is, there exists a positive integer $C_g$ such that every natural number is the sum of at most $C_g$ base-g Niven numbers.

Pascal's Prism

2012 ◽  
Vol 96 (536) ◽  
pp. 213-220
Author(s):  
Harlan J. Brothers
Keyword(s):  
Probability Theory ◽  
Fractal Geometry ◽  
Euclidean Geometry ◽  
Fibonacci Series ◽  
Pascal’S Triangle ◽  
Number Sequences ◽  
Definition Of ◽  
Image Position ◽  
Pascal's Triangle

Pascal's triangle is well known for its numerous connections to probability theory [1], combinatorics, Euclidean geometry, fractal geometry, and many number sequences including the Fibonacci series [2,3,4]. It also has a deep connection to the base of natural logarithms, e [5]. This link to e can be used as a springboard for generating a family of related triangles that together create a rich combinatoric object.2. From Pascal to LeibnizIn Brothers [5], the author shows that the growth of Pascal's triangle is related to the limit definition of e.Specifically, we define the sequence sn; as follows [6]:

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.

The Range of a Gap Series

1975 ◽  
Vol 18 (5) ◽  
pp. 753-754
Author(s):  
J. S. Hwang
Keyword(s):  
Natural Number ◽  
Gap Series ◽  
Image Position

Theorem. Letbe a function holomorphic in the disk, wherep is a natural number andIfthen then f(z) assumes every complex value infinitely often in every sector.The purpose of this note is to prove the above result. To do this, we first observe that from the condition a<∞, we can easily show that the derivative f′(z) satisfying

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.

Decomposing Complete 3-Uniform Hypergraph into 5-Cycles

2014 ◽  
Vol 672-674 ◽  
pp. 1935-1939
Author(s):  
Guan Ru Li ◽  
Yi Ming Lei ◽  
Jirimutu
Keyword(s):  
Positive Integer ◽  
Hamiltonian Cycles ◽  
Uniform Hypergraph ◽  
Design Algorithm ◽  
Uniform Hypergraphs ◽  
Edge Partition ◽  
Definition Of

About the Katona-Kierstead definition of a Hamiltonian cycles in a uniform hypergraph, a decomposition of complete k-uniform hypergraph Kn(k) into Hamiltonian cycles studied by Bailey-Stevens and Meszka-Rosa. For n≡2,4,5 (mod 6), we design algorithm for decomposing the complete 3-uniform hypergraphs into Hamiltonian cycles by using the method of edge-partition. A decomposition of Kn(3) into 5-cycles has been presented for all admissible n≤17, and for all n=4m +1, m is a positive integer. In general, the existence of a decomposition into 5-cycles remains open. In this paper, we use the method of edge-partition and cycle sequence proposed by Jirimutu and Wang. We find a decomposition of K20(3) into 5-cycles.

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