Improved Iterative Solution of Linear Fredholm Integral Equations of Second Kind via Inverse-Free Iterative Schemes

This work is devoted to Fredholm integral equations of second kind with non-separable kernels. Our strategy is to approximate the non-separable kernel by using an adequate Taylor’s development. Then, we adapt an already known technique used for separable kernels to our case. First, we study the local convergence of the proposed iterative scheme, so we obtain a ball of starting points around the solution. Then, we complete the theoretical study with the semilocal convergence analysis, that allow us to obtain the domain of existence for the solution in terms of the starting point. In this case, the existence of a solution is deduced. Finally, we illustrate this study with some numerical experiments.

An improved smooth-windowed Wigner-Ville distribution analysis for voltage variation signal

This paper outlines research conducted using bilinear time-frequency distribution (TFD), a smooth-windowed wigner-ville distribution (SWWVD) used to represent time-varying signals in time-frequency representation (TFR). Good time and frequency resolutions offer superiority in SWWVD to analyze voltage variation signals that consist of variations in magnitude. The separable kernel parameters are estimated from the signal in order to get an accurate TFR. The TFR for various kernel parameters is compared by a set of performance measures. The evaluation shows that different kernel settings are required for different signal parameters. Verification of the TFD that operated at optimal kernel parameters is then conducted. SWWVD exhibits a good performance of TFR which gives high peak-to-side lobe ratio (PSLR) and signal-to-cross-terms ratio (SCR) accompanied by low main-lobe width (MLW) and absolute percentage error (APE). This proved that the technique is appropriate for voltage variation signal analysis and it essential for development in an advanced embedded system.

Exact Mechanical Hierarchy of Non-Linear Fractional-Order Hereditariness

Non-local time evolution of material stress/strain is often referred to as material hereditariness. In this paper, the widely used non-linear approach to single integral time non-local mechanics named quasi-linear approach is proposed in the context of fractional differential calculus. The non-linear model of the springpot is defined in terms of a single integral with separable kernel endowed with a non-linear transform of the state variable that allows for the use of Boltzmann superposition. The model represents a self-similar hierarchy that allows for a time-invariance as the result of the application of the conservation laws at any resolution scale. It is shown that the non-linear springpot possess an equivalent mechanical hierarchy in terms of a functionally-graded elastic column resting on viscous dashpots with power-law decay of the material properties. Some numerical applications are reported to show the capabilities of the proposed model.

ON A CONNECTION BETWEEN A CLASS OF SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS AND INTEGRAL OPERATORS WITH SEMI-SEPARABLE KERNEL

In the present paper the connection between a class of systems of differential equations and integral operators with semi-separable kernel is established. Using a matrix of the system the inverse to a given integral operator is constructed. Moreover by putting some additional conditions on the kernel of integral operator and by the help of inverse of integral operator a fundamental matrix of the system is constructed.

Big Data Clustering with Kernel k-Means: Resources, Time and Performance

Data clustering is an unsupervised learning task that has found many applications in various scientific fields. The goal is to find subgroups of closely related data samples (clusters) in a set of unlabeled data. A classic clustering algorithm is the so-called k-Means. It is very popular, however, it is also unable to handle cases in which the clusters are not linearly separable. Kernel k-Means is a state of the art clustering algorithm, which employs the kernel trick, in order to perform clustering on a higher dimensionality space, thus overcoming the limitations of classic k-Means regarding the non-linear separability of the input data. With respect to the challenges of Big Data research, a field that has established itself in the last few years and involves performing tasks on extremely large amounts of data, several adaptations of the Kernel k-Means have been proposed, each of which has different requirements in processing power and running time, while also incurring different trade-offs in performance. In this paper, we present several issues and techniques involving the usage of Kernel k-Means for Big Data clustering and how the combination of each component in a clustering framework fares in terms of resources, time and performance. We use experimental results, in order to evaluate several combinations and provide a recommendation on how to approach a Big Data clustering problem.

On the Relativistic Separable Functions for the Breakup Reactions

In the paper the so-called modified Yamaguchi function for the Bethe-Salpeter equation with a separable kernel is discussed. The type of the functions is defined by the analytic stucture of the hadron current with breakup - the reactions with interacting nucleon-nucleon pair in the final state (electro-, photo-, and nucleon-disintegration of the deuteron).