We study properties of β-numeration systems, where β > 1 is the real root of the polynomial x −mx − x−1, m ∈ N, m ≥ 1. We consider arithmetic operations on the set of β-integers, i.e., on the set of numbers whose greedy expansion in base β has no fractional part. We show that the number of fractional digits arising under addition of β-integers is at most 5 for m ≥ 3 and 6 for m = 2, whereas und...

and Applied Analysis 3 Let Γ(⋅) denote the gamma function. For any positive integer n and real number θ (n − 1 < θ < n), there are different definitions of fractional derivatives with order θ in [8]. During this paper, we consider the left, (right) Caputo derivative and left (right) Riemann-Liouville derivative defined as follows: (i) the left Caputo derivative: C 0 D θ t V (t) = 1 Γ (n − θ) ∫ ...

We consider an aggregation equation in Rn, n ≥ 2, with fractional dissipation, namely, ut + ∇ · (u∇K ∗ u) = −ν(−∆)γ/2u , where 0 ≤ γ ≤ 2 and K is a nonnegative decreasing radial kernel with a Lipschitz point at the origin, e.g. K(x) = e−|x|. We prove that for 0 ≤ γ < 1 the solutions develop blow-up in finite for a general class of initial data. In contrast we prove that for 1 < γ ≤ 2 the equati...

In the recent paper Communications in Nonlinear Science and Numerical Simulation. Vol.18. No.11. (2013) 2945-2948, it was demonstrated that a violation of the Leibniz rule is a characteristic property of derivatives of non-integer orders. It was proved that all fractional derivatives Dα, which satisfy the Leibniz rule Dα(fg) = (Dαf) g+ f (Dαg), should have the integer order α = 1, i.e. fraction...

Fractional Fourier transform (FRFT) is a generalization of Fourier transform, and it indicates a rotation of signal in the time-frequency plane[l). The FRFT has been widely successfully used in the many areas[l][2]. Because of the importance of FRFT, discrete fractional Fourier transform (DFRFT) becomes an important issue in recent years[3][4] [5][6]. In the development of DFRFT, the DFRFT has ...

In view of the images polluted by mixed noise in actual image processing, in this paper, the hybrid filter algorithm combined by median filtering and wavelet transform is generalized to two-dimensional fractional wavelet domain, verified the effectiveness and advantages of hybrid filtering algorithm combined by median filtering and two-dimensional fractional wavelet transform. And put forward a...

Images are widely used to convey information in an easy to understand format; however, the conventional representation of an image as 3 channels per pixel per image creates difficulty in implementing certain widely used classes of algorithms. In this paper, we summarize two different applications of the fractional Fourier transform, which transforms the representation of the data in the image a...