AN EXACT FORMULA FOR THE HARMONIC CONTINUED FRACTION

2020 ◽  
pp. 1-11
Author(s):  
MARTIN BUNDER ◽  
PETER NICKOLAS ◽  
JOSEPH TONIEN
Keyword(s):  
Real Number ◽  
Continued Fraction ◽  
Positive Real Number ◽  
Exact Formula ◽  
Positive Real ◽  
Image Position

For a positive real number $t$ , define the harmonic continued fraction $$\begin{eqnarray}\text{HCF}(t)=\biggl[\frac{t}{1},\frac{t}{2},\frac{t}{3},\ldots \biggr].\end{eqnarray}$$ We prove that $$\begin{eqnarray}\text{HCF}(t)=\frac{1}{1-2t(\frac{1}{t+2}-\frac{1}{t+4}+\frac{1}{t+6}-\cdots \,)}.\end{eqnarray}$$

scholarly journals GENERALIZED CONTINUED FRACTION EXPANSIONS WITH CONSTANT PARTIAL DENOMINATORS

2018 ◽  
Vol 107 (02) ◽  
pp. 272-288
Author(s):  
TOPI TÖRMÄ
Keyword(s):  
Positive Integer ◽  
Real Number ◽  
Continued Fraction ◽  
Positive Real Number ◽  
Rational Numbers ◽  
Positive Real ◽  
Image Position ◽  
Fixed Positive Integer ◽  
Continued Fraction Expansions ◽  
Positive Integers

We study generalized continued fraction expansions of the form $$\begin{eqnarray}\frac{a_{1}}{N}\frac{}{+}\frac{a_{2}}{N}\frac{}{+}\frac{a_{3}}{N}\frac{}{+}\frac{}{\cdots },\end{eqnarray}$$ where $N$ is a fixed positive integer and the partial numerators $a_{i}$ are positive integers for all $i$ . We call these expansions $\operatorname{dn}_{N}$ expansions and show that every positive real number has infinitely many $\operatorname{dn}_{N}$ expansions for each $N$ . In particular, we study the $\operatorname{dn}_{N}$ expansions of rational numbers and quadratic irrationals. Finally, we show that every positive real number has, for each $N$ , a $\operatorname{dn}_{N}$ expansion with bounded partial numerators.

The limit theorem for aperiodic discrete renewal processes

1964 ◽  
Vol 4 (1) ◽  
pp. 122-128
Author(s):  
P. D. Finch
Keyword(s):  
Limit Theorem ◽  
Real Number ◽  
Renewal Process ◽  
Positive Real Number ◽  
Random Variables ◽  
Renewal Processes ◽  
Positive Real ◽  
Image Position

A discrete renewal process is a sequence {X4} of independently and inentically distributed random variables which can take on only those values which are positive integral multiples of a positive real number δ. For notational convenience we take δ = 1 and write where .

Diophantine Approximation and Horocyclic Groups

1957 ◽  
Vol 9 ◽  
pp. 277-290 ◽  
Author(s):  
R. A. Rankin
Keyword(s):  
Real Number ◽  
Diophantine Approximation ◽  
Modular Group ◽  
Positive Real Number ◽  
Irrational Number ◽  
Infinitely Many Solutions ◽  
Positive Real ◽  
Closed Set ◽  
Coprime Integers ◽  
Image Position

1. Introduction. Let ω be an irrational number. It is well known that there exists a positive real number h such that the inequality(1)has infinitely many solutions in coprime integers a and c. A theorem of Hurwitz asserts that the set of all such numbers h is a closed set with supremum √5. Various proofs of these results are known, among them one by Ford (1), in which he makes use of properties of the modular group. This approach suggests the following generalization.

An asymptotic bound for the residual area of a packing of discs

1966 ◽  
Vol 62 (4) ◽  
pp. 699-704 ◽  
Author(s):  
D. G. Larman
Keyword(s):  
Real Number ◽  
Positive Real Number ◽  
Positive Real ◽  
Extremal Case ◽  
Empirical Results ◽  
Asymptotic Bound ◽  
Image Position

Suppose that a sequence of discsarranged in decreasing order of diameters, forms a packing within the unit plane square I2. It has been shown, by Florian(1), that the area ofis at least O(a), where a is the radius of θn. However, Gilbert (2) has produced some empirical results for the Apollonius packing 71 of discs which seem to suggest that for such a packing, the area of the setis at least O(as) for some positive real number s, less than one. As Gilbert remarks, it is difficult to imagine that the Apollonius packing is not the extremal case, and so, that it would seem likely that there exists a positive real number s, less than one, such that for a general packing, the area ofis at least O(as). The purpose of this paper is to establish this result by showing that 0·97 is an allowable value for s.

scholarly journals Mixed periodic Jacobi continued fractions

1986 ◽  
Vol 104 ◽  
pp. 129-148 ◽  
Author(s):  
Yoshifumi Kato
Keyword(s):  
Real Number ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Positive Real Number ◽  
Jacobi Matrix ◽  
Padé Approximant ◽  
Positive Real ◽  
Pade Approximant

Let b0 be a positive real number andbe a Jacobi matrix. We can associate with them a Jacobi continued fraction, which will be abbreviated to a J fraction from the next section, as followswhere An(z)/Bn(z) is the n-th Padé approximant of φ(z).

scholarly journals On p–valent starlike functions with reference to the Bernardi integral operator

1984 ◽  
Vol 30 (1) ◽  
pp. 37-43
Author(s):  
Vinod Kumar ◽  
S.L. Shukla
Keyword(s):  
Integral Operator ◽  
Real Number ◽  
Positive Real Number ◽  
Starlike Functions ◽  
Positive Real ◽  
Powerful Technique ◽  
Image Position

Let (A, B) denote the class of certain p-valent starlike functions. Recently G. Lakshma Reddy and K.S. Padmanabhan [Bull. Austral. Math. Soc. 25 (1982), 387–396] have shown that the function g defined bybelongs to the class (A, B) if f ∈ (A, B). The technique used by them fails when c is any positive real number. In this paper, by employing a more powerful technique, we improve their result to the case when c is any real number such that c ≥ −p(1+A)/(1+B).

scholarly journals The asymmetric product of three inhomogeneous linear forms

1981 ◽  
Vol 31 (4) ◽  
pp. 439-455 ◽  
Author(s):  
A. C. Woods
Keyword(s):  
Real Number ◽  
Positive Real Number ◽  
Positive Real ◽  
Linear Forms ◽  
Inhomogeneous Linear Forms ◽  
Image Position

AbstractIt is shown, given any positive real number λ and any point (x1, x2, x3) of R3 and any lattice λ R3; that there exists a point (z1, z2, z3) of λ for whichwhich generalizes a theorem due to Remak.

On convex sets and measures

1966 ◽  
Vol 62 (1) ◽  
pp. 33-42
Author(s):  
D. G. Larman ◽  
D. J. Ward
Keyword(s):  
Real Number ◽  
Positive Real Number ◽  
Convex Sets ◽  
Positive Real ◽  
Countable Collection ◽  
Dimensional Measure ◽  
Image Position

If α is a positive real number then, for each set E in R3, we definewhere U(ρ, E) is any countable collection of convex sets, each of diameter at most ρ, whose union covers E, and Aα is a positive real number. Then the convex α-dimensional measure Λα(E), of E, is . In this paper we shall only be considering the cases α = 1,2 where, as is usual, we take A1 = 1 and A2 = ¼π. The symbols ,Λs(E) will denote the spherical (circular) measure of E, i.e. when the coverings are restricted to being collections of spheres (circles), and E(z) is the intersection of a set E (in R3) with the plane (x, y) at z.

scholarly journals Finding of principle square root of a real number by using interpolation method

2018 ◽  
Vol 7 (1) ◽  
pp. 77-83
Author(s):  
Rajendra Prasad Regmi
Keyword(s):  
Real Number ◽  
Positive Real Number ◽  
Interpolation Method ◽  
Square Root ◽  
Real Numbers ◽  
Positive Real ◽  
Unknown Quantity ◽  
Square Roots

There are various methods of finding the square roots of positive real number. This paper deals with finding the principle square root of positive real numbers by using Lagrange’s and Newton’s interpolation method. The interpolation method is the process of finding the values of unknown quantity (y) between two known quantities.

Some remarks on the solvability of non-local elliptic problems with the Hardy potential

2014 ◽  
Vol 16 (04) ◽  
pp. 1350046 ◽  
Author(s):  
B. Barrios ◽  
M. Medina ◽  
I. Peral
Keyword(s):  
Real Number ◽  
Sobolev Inequality ◽  
Positive Real Number ◽  
Fractional Power ◽  
Existence Of Solution ◽  
Sharp Constant ◽  
Hardy Potential ◽  
Positive Real ◽  
Existence And Multiplicity ◽  
Non Local

The aim of this paper is to study the solvability of the following problem, [Formula: see text] where (-Δ)s, with s ∈ (0, 1), is a fractional power of the positive operator -Δ, Ω ⊂ ℝN, N > 2s, is a Lipschitz bounded domain such that 0 ∈ Ω, μ is a positive real number, λ < ΛN,s, the sharp constant of the Hardy–Sobolev inequality, 0 < q < 1 and [Formula: see text], with αλ a parameter depending on λ and satisfying [Formula: see text]. We will discuss the existence and multiplicity of solutions depending on the value of p, proving in particular that p(λ, s) is the threshold for the existence of solution to problem (Pμ).

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