Related papers: Neighboring Fractions in Farey Subsequences
We generalize classical results on the gap distribution (and other fine-scale statistics) for the one-dimensional Farey sequence to arbitrary dimension. This is achieved by exploiting the equidistribution of horospheres in the space of…
An astonishingly simple analytical frequency approximation formula for a class of strongly nonlinear oscillators is derived and applied to various example systems yielding useful quick estimates.
We give a formula for matrix exponentials and partial fraction decompositions.
We propose a fast and stable method for constructing matrix approximations to fractional integral operators applied to series in the Chebyshev fractional polynomials. This method utilizes a recurrence relation satisfied by the fractional…
Let $F_Q$ be the set of Farey fractions of order $Q$. Given the integers $\d\ge 2$ and $0\le \c \le \d-1$, let $F_Q(c,d)$ be the subset of $F_Q$ of those fractions whose denominators are $\equiv c \pmod d$, arranged in ascending order. The…
We show that highly accurate approximations can often be obtained from constructing Thiele interpolating continued fractions by a Greedy selection of the interpolation points together with an early termination condition. The obtained…
Let F be a holomorphic map whose components satisfy some polynomial relations. We present an algorithm for constructing Nash maps locally approximating F, whose components satisfy the same relations.
Recently, we have proposed a new diffusive representation for fractional derivatives and, based on this representation, suggested an algorithm for their numerical computation. From the construction of the algorithm, it is immediately…
We describe monotone maps between subsequences of the Farey sequences.
This paper introduces Farey Recursive Functions and investigates their basic properties. Farey Recursive Functions are a special type of recursive function from the rationals to a commutative ring. The recursion of these functions is…
We prove that the Farey sequences can be express into equivalence classes labeled by a fractal parameter which looks like a Hausdorff dimension $h$ defined within the interval 1 < h < 2. The classes $h$ satisfy the same properties of the…
We obtain a new decomposition of the Riemann-Liouville operators of fractional integration as a series involving derivatives (of integer order). The new formulas are valid for functions of class $C^n$, $n \in \mathbb{N}$, and allow us to…
In this paper, the authors design a trial to count rational ratios on the interval [0, 1], and plot a normalized frequency statistical graph. Patterns, symmetry and co-linear properties reflected in the graph are confirmed. The main…
In this paper, we introduce a new method for calculating fractional integrals and differentials. The method involves an equation that we have obtained from infinite applied integration by parts. The equation works for special class of…
Two approximations, derived from continuous expansions of Riemann-Liouville fractional derivatives into series involving integer order derivatives, are studied. Using those series, one can formally transform any problem that contains…
We give an algorithm to compute the series expansion for the inverse of a given function. The algorithm is extremely easy to implement and gives the first $N$ terms of the series. We show several examples of its application in calculating…
We study solutions to the Egyptian fractions equation with the prime factors of the denominators constrained to lie in a fixed set of primes. We evaluate the effectiveness of the greedy algorithm in establishing bounds on such solutions.…
Continued fractions are linked to Stern's diatomic sequence 0,1,1,2,1,3,2,3,1,4,... (given by the recursion relation a_2n=a_n and a_{2n+1} = a_n + a_{n+1}, where a_0=0 and a_1=1), which has long been known. Using a particular…
Several conjectural continued fractions found with the help of various algorithms are published in this paper.
Franel and Landau derived an arithmetic statement involving the Farey sequence that is equivalent to the Riemann hypothesis. Since there is a relationship between the Mertens function and the Riemann hypothesis, there should be a…