On Local Smoothness Classes of Periodic Functions

We introduce a special class of functions for mathematical modeling of periodic signals of special shape with irregularly spaced arguments. This method was developed for determination of phenomenological characteristics of the light curves, which are necessary for registration in the "General Catalogue of Variable Stars" and other databases. For eclipsing binary stars with smooth light curves - of types EB and EW - it is recommended a trigonometric polynomial of optimal degree in a complete or symmetric form. For eclipsing binary systems with narrow minima (EA-type), statistically optimal is an approximation in a class of non-polynomial spline functions. It is used a combination of the second-order trigonometric polynomial (TP2, what describes effects of "reflection", "ellipsoidality" and "spotness") and localized contributions of minima (parametrized in depth and profile separately for primary and secondary minima). Such an approach is characterized by a statistical accuracy of the smoothing curve, which is up to ~1.5-2 times better than the trigonometric polynomial of statistically optimal degree, and the absence of false "waves" in the light curve associated with the effect of Gibbs. In addition to the minimum width, which can not be determined by a trigonometric polynomial approximation, the method allows to determine with better accuracy its depth, as well as to separate the effects of the eclipse and out-of-eclipse parts. For multi-color observations, improving the accuracy of the smoothing of the curve in each filter will allow to obtain with better accuracy the curves of the color index variations. Effectivity of the proposed method increases with decreasing eclipse depth. The method called NAV ("New Algol Variable"), was applied to eclipsing binary systems VSX J022427.8-104034=USNO-B1.0 0793-0023471 and BM UMa. For VSX0224, an alternative model of "double period" is discussed.

Let L2 be the space of 2π-periodic square-summable functions and E(f, X)2 be the best approximation of f by the space X in L2. For n ∈ ℕ and B ∈ L2, let $${{\Bbb S}_{B,n}}$$ be the space of functions s of the form $$s\left( x \right) = \sum\limits_{j = 0}^{2n - 1} {{\beta _j}B\left( {x - \frac{{j\pi }}{n}} \right)} $$ . This paper describes all spaces $${{\Bbb S}_{B,n}}$$ that satisfy the exact inequality $$E{\left( {f,{S_{B,n}}} \right)_2} \leqslant \frac{1}{{^{{n^r}}}}\parallel {f^{\left( r \right)}}{\parallel _2}$$ . (2n–1)-dimensional subspaces fulfilling the same estimate are specified. Well-known inequalities are for approximation by trigonometric polynomials and splines obtained as special cases.

Flicker-Noise Spectroscopy (FNS), a general approach to the extraction and parameterization of resonant and stochastic components contained in medical time series, is presented. The basic idea of FNS is to treat the correlation links present in sequences of different irregularities, such as spikes, jumps, and discontinuities in derivatives of different orders, on all levels of the spatiotemporal hierarchy of the system under study as main information carriers. The tools to extract and analyze the information are power spectra and difference moments (structural functions), which complement the information of each other. The structural function stochastic component is formed exclusively by jumps of the dynamic variable while the power spectrum stochastic component is formed by both spikes and jumps on every level of the hierarchy. The information passport characteristics that are determined by fitting the derived expressions to the experimental variations for the stochastic components of power spectra and structural functions are interpreted as the correlation times and parameters that describe the rate of memory loss on these correlation time intervals for different irregularities. The number of the extracted parameters is determined by the requirements of the problem under study. Application of this approach to the analysis of tremor velocity signals for a Parkinsonian patient is discussed.

We consider real trigonometric polynomial Bernoulli equations of the form A(θ)y′=B1(θ)+Bn(θ)yn where n≥2, with A,B1,Bn being trigonometric polynomials of degree at most μ≥1 in variables θ and Bn(θ)≢0. We also consider trigonometric polynomials of the form A(θ)yn−1y′=B0(θ)+Bn(θ)yn where n≥2, being A,B0,Bn trigonometric polynomials of degree at most μ≥1 in the variable θ and Bn(θ)≢0. For the first equation, we show that when n≥4, it has at most 3 real trigonometric polynomial solutions when n is even and 5 real trigonometric polynomial solutions when n is odd. For the second equation, we show that when n≥3, it has at most 3 real trigonometric polynomial solutions when n is odd and 5 real trigonometric polynomial solutions when n is even. We also provide trigonometric polynomial equations of the two types mentioned above where the maximum number of trigonometric polynomial solutions is achieved. The proof method will be to apply extended Fermat problems to polynomial equations.

We prove the right Lax-type inequality on subarcs of the unit circle of the complex plane for complex algebraic polynomials of degree n having no zeros in the open unit disk. This is done by establishing the right Bernstein–Szegő–Videnskii type inequality for real trigonometric polynomials of degree at most n on intervals shorter than the period. The paper is closely related to recent work by B. Nagy and V. Totik. In fact, their asymptotically sharp Bernstein-type inequality for complex algebraic polynomials of degree at most n on subarcs of the unit circle is recaptured by using more elementary methods. Our discussion offers a somewhat new way to see V.S. Videnskii’s Bernstein and Markov type inequalities for trigonometric polynomials of degree at most n on intervals shorter than a period, two classical polynomial inequalities first published in 1960. A new Riesz–Schur type inequality for trigonometric polynomials is also established. Combining this with Videnskii’s Bernstein-type inequality gives Videnskii’s Markov-type inequality immediately.

Analysis of fractional order Bonhoeffer–van der Pol oscillator

In this contribution, we analyze the memory effect associated with fractional derivatives in a Cournot-type oligopoly model. Such an analysis will be made from the comparison of the solutions of a duopoly with fractional derivatives of different orders. We show that the firm that has more memory on the competition, given by the fractional derivative, will have advantages over its competitor in the long run. Moreover, we show that the choices of fractional derivatives αj ∈ (0,3/5) for both firms j = 1,2 will result in advantages in profit compared to the profit of firms without memory.

<p>Two inverse source problems for a space-time fractional differential equation involving bi-fractional Laplacian operators in the spatial variable and Caputo time-fractional derivatives of different orders between 1 and 2 are studied. In the first inverse source problem, the space-dependent term along with the diffusion concentration is recovered, while in the second inverse source problem, the time-dependent term along with the diffusion concentration is identified. Both inverse source problems are ill-posed in the sense of Hadamard. The existence and uniqueness of solutions for both inverse source problems are investigated. Finally, several examples are presented to illustrate the obtained results for the inverse source problems.</p>

This paper presents a novel space-time feature-based human activity analysis system. We detect Space Time Interest Points (STIP) and generate their description based on the facet model. The proposed approach detects interest points in video data using the three-dimensional facet model efficiently. Then we describe each interest point by three-dimensional Haar wavelet transform and time derivatives of different order obtained from said facet model. Here we represent each video clip following the bag-of-words approach by learning feature specific dictionary. Finally, classification is done using non-linear SVM with χ 2 -kernel. We evaluate the performance of our system on standard datasets like Weizmann, KTH, UCF sports, ICD, UCF YouTube, and UCF50 and get better, or at least comparable results compared to other state-of-the-art systems.

In this article the existence as well as the uniqueness (EU) of the solutions for nonlinear multiorder fractional-differential equations (FDE) with local boundary conditions and fractional derivatives of different orders (Caputo and Riemann–Liouville) are covered. The existence result is derived from Krasnoselskii’s fixed point theorem and its uniqueness is shown using the Banach contraction mapping principle. To illustrate the reliability of the results, two examples are given.

Algorithm ALESCAF (alternating least squares identification based on the characteristic function) uses the derivatives of the second characteristic function (c.f.) of observations, without any need of sparsity assumption on sources, but assuming their statistical independence. ALESCAF was already proposed by the authors in P. Comon and M. Rajih (2005), where only one derivative order was considered. In this paper, new versions of ALESCAF are proposed, that jointly use derivatives of different orders. We also propose ALESCAS, a new algorithm that uses the knowledge of source c.f.'s. Computer simulations demonstrate that both algorithms accelerate the convergence

In this paper, we investigate the existence of solutions for a nonlinear fractional-order coupled system involving both Caputo and Riemann–Liouville generalized fractional derivatives of different orders equipped with coupled integral boundary conditions. We transform the given system into an equivalent fixed point problem and solve it by applying the standard fixed point theorems. Examples are constructed for the illustration of the obtained results.

In addition to the well-known and widely-used adaptive strategies for region subdivision and the choice of local quadrature rules, there still exists a third possibility for doing numerical integration adaptively, when interval computation is considered. Based on the Peano's kernels theorem and interval arithmetic, we are able to estimate the truncation error of a quadrature rule by means of different derivatives of the integrand, where the available orders of the derivatives depend on the degree of smoothness of the integrand and the exactness degree of the underlying quadrature rule. We classify the methods as the adaptive orders strategies, if they make use of the derivatives of different orders to improve each single local error estimation. In this paper, alternatives for adaptive error estimation are discussed. Moreover, a practical way for realization of the optimal error estimation is suggested. Numerical results for integrands of different classes as well as numerical comparisons of different methods are given.

Decades ago, fractional calculus arose to generalize ordinary derivation and integration, and then became a means of modeling and interpreting many phenomena in various fields such as engineering, physics, chemistry, biology and signal processing. The definition of the fractional derivative began with a derivative with a singular kernel, such as the Riemann-Liouville and Caputo derivative. The study of existence and uniqueness of solution of fractional partial differential equations are then proved by mani methods as: the well-known Lax–Milgram theorem, Energy estimate and fixed point theorem... Among them, In this work we investigate the existence and uniqueness of solutions for a new class of tripled system, which containing of mixed fractional derivatives between the Riemann-Liouville and the Caputo fractional derivatives of different orders with Laplacian operator, by using Leray-Schauder alternative fixed point theorem type in generalized Banach space and Banach contraction principle. The stability of the Ulam type of the proposed coupled system is also studied. Due to the importance of variable-order fractional calculus, both from a theoretical and an applied perspective, we think that the results obtained can be of interest to the research pursuits of many readers. At the end, to conclude this paper we provide an example illustrating our findings and approach.

The generalized generating function (GGF), which can exploit the statistical information carried on complex-valued signals efficiently by treating its real and imaginary parts as a whole and offer an elegant algebraic structure, has been proposed by authors for blind identification of mixtures. In this paper, we extend the GGF-based method to be able to deal with the challenging underdetermined mixtures with complex-valued sources. A new algorithm named ALSGGF, in which the mixing matrix is estimated by decomposing the tensor constructed from the higher conjugate derivative of the second GGF of the observations with alternating least squares algorithm, is proposed. Furthermore, we show that the conjugate derivatives of different orders of the second GGF can be used jointly in a simple way to improve the performance. Simulation experiments validate the superiority of the proposed ALSGGF algorithms.