The Fractal Dimension of Sets Derived from Complex Bases

1986 ◽  
Vol 29 (4) ◽  
pp. 495-500 ◽  
Author(s):  
William J. Gilbert
Keyword(s):  
Fractal Dimension ◽  
Positive Integer ◽  
Positive Root ◽  
Complex Numbers ◽  
Integer Part ◽  
Fixed Integer

AbstractFor each positive integer n, the radix representation of the complex numbers in the base —n + i gives rise to a tiling of the plane. Each tile consists of all the complex numbers representable in the base -n + i with a fixed integer part. We show that the fractal dimension of the boundary of each tile is 2 log λn/log(n2 + 1), where λn is the positive root of λ3 - (2n - 1) λ2 - (n - 1) 2λ - (n2 + 1).

scholarly journals Partitioning the positive integers with higher order recurrences

1991 ◽  
Vol 14 (3) ◽  
pp. 457-462 ◽  
Author(s):  
Clark Kimberling
Keyword(s):  
Positive Integer ◽  
Positive Root ◽  
Irrational Number ◽  
Higher Order ◽  
Typical Result ◽  
Algebraic Integers ◽  
Positive Integers

Associated with any irrational numberα>1and the functiong(n)=[αn+12]is an array{s(i,j)}of positive integers defined inductively as follows:s(1,1)=1,s(1,j)=g(s(1,j−1))for allj≥2,s(i,1)=the least positive integer not amongs(h,j)forh≤i−1fori≥2, ands(i,j)=g(s(i,j−1))forj≥2. This work considers algebraic integersαof degree≥3for which the rows of the arrays(i,j)partition the set of positive integers. Such an array is called a Stolarsky array. A typical result is the following (Corollary 2): ifαis the positive root ofxk−xk−1−…−x−1fork≥3, thens(i,j)is a Stolarsky array.

scholarly journals NORMALITY AND SHARED SETS

2009 ◽  
Vol 86 (3) ◽  
pp. 339-354 ◽  
Author(s):  
MINGLIANG FANG ◽  
LAWRENCE ZALCMAN
Keyword(s):  
Positive Integer ◽  
Meromorphic Functions ◽  
Complex Numbers ◽  
Finite Complex ◽  
Shared Sets ◽  
Zeros Of Functions

AbstractLet ℱ be a family of meromorphic functions defined in D, all of whose zeros have multiplicity at least k+1. Let a and b be distinct finite complex numbers, and let k be a positive integer. If, for each pair of functions f and g in ℱ, f(k) and g(k) share the set S={a,b}, then ℱ is normal in D. The condition that the zeros of functions in ℱ have multiplicity at least k+1 cannot be weakened.

scholarly journals A lower bound for a remainder term associated with the sum of digits function

1991 ◽  
Vol 34 (1) ◽  
pp. 121-142 ◽  
Author(s):  
D. M. E. Foster
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Upper Bound ◽  
Remainder Term ◽  
Fixed Integer ◽  
Sum Of Digits ◽  
Sum Of Digits Function ◽  
The Difference

For a fixed integer q≧2, every positive integer k = Σr≧0ar(q, k)qr where each ar(q, k)∈{0,1,2,…, q−1}. The sum of digits function α(q, k) Σr≧0ar(q, k) behaves rather erratically but on averaging has a uniform behaviour. In particular if , where n>1, then it is well known that A(q, n)∼½((q − 1)/log q)n logn as n → ∞. For odd values of q, a lower bound is now obtained for the difference 2S(q, n) = A(q, n)−½(q − 1))[log n/log q, where [log n/log q] denotes the greatest integer ≦log n /log q. This complements an upper bound already found.

On the decimal representation of integers

1952 ◽  
Vol 48 (4) ◽  
pp. 555-565 ◽  
Author(s):  
M. P. Drazin ◽  
J. Stanley Griffith
Keyword(s):  
Positive Integer ◽  
Fixed Integer ◽  
Usual Convention ◽  
Representation Of Integers ◽  
Image Position

Let r be any fixed integer with, r≥ 2; then, given any positive integer n, we can find* integers αk(r, n) (k = 0, 1, 2, …) such thatwhere, subject to the conditionsthe integers αk(r, n) are uniquely determined, and, in fact, clearlyαk(r, n) = [n/rk] − r[n/rk+1](square brackets denoting integral parts, according to the usual convention).

scholarly journals The Division Algorithm in Complex Bases

1996 ◽  
Vol 39 (1) ◽  
pp. 47-54 ◽  
Author(s):  
William J. Gilbert
Keyword(s):  
Iterated Function System ◽  
Complex Numbers ◽  
Integer Part ◽  
Division Algorithm ◽  
Fractal Boundary ◽  
Gaussian Integer ◽  
Gaussian Integers ◽  
Linear Maps ◽  
Iterated Function ◽  
Long Division

AbstractComplex numbers can be represented in positional notation using certain Gaussian integers as bases and digit sets. We describe a long division algorithm to divide one Gaussian integer by another, so that the quotient is a periodic expansion in such a complex base. To divide by the Gaussian integer w in the complex base b, using a digit set D, the remainder must be in the set wT(b,D) ∩ ℤ[i], where T(b,D) is the set of complex numbers with zero integer part in the base. The set T(b,D) tiles the plane, and can be described geometrically as the attractor of an iterated function system of linear maps. It usually has a fractal boundary. The remainder set can be determined algebraically from the cycles in a certain directed graph.

scholarly journals Periodic rings with commuting nilpotents

1984 ◽  
Vol 7 (2) ◽  
pp. 403-406
Author(s):  
Hazar Abu-Khuzam ◽  
Adil Yaqub
Keyword(s):  
Positive Integer ◽  
Commutative Ring ◽  
Finite Fields ◽  
Boolean Ring ◽  
Fixed Integer ◽  
Integer Coefficients ◽  
Nilpotent Elements ◽  
Periodic Rings

LetRbe a ring (not necessarily with identity) and letNdenote the set of nilpotent elements ofR. Suppose that (i)Nis commutative, (ii) for everyxinR, there exists a positive integerk=k(x)and a polynomialf(λ)=fx(λ)with integer coefficients such thatxk=xk+1f(x), (iii) the setIn={x|xn=x}wherenis a fixed integer,n>1, is an ideal inR. ThenRis a subdirect sum of finite fields of at mostnelements and a nil commutative ring. This theorem, generalizes the “xn=x” theorem of Jacobson, and (takingn=2) also yields the well known structure of a Boolean ring. An Example is given which shows that this theorem need not be true if we merely assume thatInis a subring ofR.

An Inclusion Theorem for Generalized Cesàro and Riesz Means

1968 ◽  
Vol 20 ◽  
pp. 735-738 ◽  
Author(s):  
A. Meir
Keyword(s):  
Positive Integer ◽  
Complex Numbers ◽  
Arbitrary Sequence ◽  
Riesz Means ◽  
Inclusion Theorem ◽  
Unbounded Sequence ◽  
Image Position

For a positive integer, p, a strictly increasing unbounded sequence of positive numbers {λn: n ⩾ 1} and an arbitrary sequence of complex numbers {an} let12

scholarly journals The Number of Unitarily k-Free Divisors of An Integer

1976 ◽  
Vol 21 (1) ◽  
pp. 19-35
Author(s):  
D. Suryanarayana ◽  
R. Sita Rama Chandra Rao
Keyword(s):  
Positive Integer ◽  
Prime Factor ◽  
Canonical Representation ◽  
Fixed Integer ◽  
Unitary Divisor ◽  
Free Divisors

Let k be a fixed integer ≧ 2. A positive integer n is called unitarily k-free, if the multiplicity of each prime factor of n is not a multiple of k; or equivalently, if n is not divisible unitarily by the k-th power of any integer > 1. By a unitary divisor, we mean as usual, a divisor d> 0 of n such that (d, n/d) = 1. The interger 1 is also considered to be unitarily k-free. The concept of a unitarily k-free integer was first introduced by Cohen (1961; §1). Let denote the set of unitarily k-free integers. When k = 2, the set coincides with the set Q* of exponentially odd integers (that is, integers in whose canonical representation each exponent is odd) discussed by Cohen himself in an earlier paper (1960; §1 and §6). A divisor d > 0 of the positive integer n is called a unitarily k-free divisor of n if d ∈ . Let (n) denote the number of unitarily k-free divisors of n.

scholarly journals Clausen’s Series 3F2(1) with Integral Parameter Differences

Symmetry ◽  
2021 ◽  
Vol 13 (10) ◽  
pp. 1783
Author(s):  
Kwang-Wu Chen
Keyword(s):  
Positive Integer ◽  
Closed Form ◽  
Open Problem ◽  
Group Symmetry ◽  
Complex Numbers ◽  
Integral Parameter ◽  
Arbitrary Complex

Ebisu and Iwassaki proved that there are three-term relations for 3F2(1) with a group symmetry of order 72. In this paper, we apply some specific three-term relations for 3F2(1) to partially answer the open problem raised by Miller and Paris in 2012. Given a known value 3F2((a,b,x),(c,x+1),1), if f−x is an integer, then we construct an algorithm to obtain 3F2((a,b,f),(c,f+n),1) in an explicit closed form, where n is a positive integer and a,b,c and f are arbitrary complex numbers. We also extend our results to evaluate some specific forms of p+1Fp(1), for any positive integer p≥2.

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