ABSTRACT. We investigate a numerical method for the construction of an optimal set of quadrature rules in the sense of Borges for r⩾2 definite integrals with the same integrand and interval of integration, but with different weight functions, related to an arbitrary multi-index. The presented method is illustrated by numerical examples.

ABSTRACT. Generalized averaged Gaussian quadrature formulas, introduced by Spalević [3], may yield a smaller error than Gauss quadrature rules. When moments or modified moments are difficult to compute, these formulas can serve as good substitutes. However, generalized averaged Gaussian quadrature formulas may have external nodes, i.e. nodes outside the convex hull of the measure corresponding to the Gauss rules. This would make them unusable when the domain of the integrand is limited to this convex hull. In this paper we investigate whether removing some of the last rows and columns of the matrices determining generalized averaged Gaussian quadrature rules (cf. [2]) will produce quadrature rules with no external nodes. The results that will be presented have been recently published in [1].

[1] D.Lj. Djukić, L. Reichel, M.M. Spalević: Truncated generalized averaged Gauss quadrature rules. J. Comput. Appl. Math. 308 (2016), 408--418.

[2] L. Reichel, M.M. Spalević, T. Tang: Generalized averaged Gauss quadrature rules for the approximation of matrix functionals. BIT Numer. Math. 56 (2016), 1045--1057.

[3] M.M. Spalević: On generalized averaged Gaussian formulas. Math. Comp. 76 (2007), 1483--1492.

ABSTRACT. We estimate the error of selected cubature formulae constructed by the product of Gaussian quadrature rules. The cases of multiple and (hyper-)surface integrals over n-dimensional cube, simplex, sphere and ball are considered. The error estimates are obtained as the absolute value of the difference between cubature formula constructed by the product of Gaussian quadrature rules and cubature formula constructed by the product of corresponding generalized averaged Gaussian quadrature rules. Generaziled averaged Gaussian quadrature rule \widehat{G}_{2l+1} is (2l+1)-point quadrature rule. It has 2l+1 nodes and the nodes of the corresponding Gauss rule G_l with l nodes form a subset, similar to the situation for the (2l+1)-point Gauss-Kronrod rule H_{2l+1} associated with G_l. The advantages of \widehat{G}_{2l+1} are that it exists also when H_{2l+1} does not, and that the numerical construction, based on recently proposed effective numerical procedure, of \widehat{G}_{2l+1} is simpler than the construction of H_{2l+1}.

ABSTRACT. We consider Kronrod extension of generalizations of the well known Micchelli-Rivlin quadrature formula, of highest algebraic degree of precision, for the Fourier-Chebyshev coefficients. For analytic functions the remainder term of these quadrature formulas can be represented as a contour integral with a complex kernel. We study the kernel, on elliptic contours with foci at the points 1 and -1 and a sum of semi-axes r>1, for the quoted quadrature formulas. Starting from the explicit expression of the kernel, we determine the locations on the ellipses where maximum modulus of the kernel is attained. So we derive effective L-infinity error bounds for these quadrature formulas. Complex-variable methods are used to obtain expansions of the error in these quadrature formulas over the interval [-1,1]. Finally, effective L-1 error bounds are also derived for these quadrature formulas.

ABSTRACT. One of the unlimited field of study of number theory is the binary quadratic Diophantine equations. Continued fractions are one of the computational approach to obtain the integer solutions of Diophantine equations. The aim of this study is to show how it works to obtain the general solution of some binary quadratic Diophantine equations in terms of Fibonacci and Lucas sequences.

ABSTRACT. In this paper we present a new family of numbers, called generalized Whitney numbers. This family is a generalization of different types of Whitney and Stirling numbers. Basic properties of these numbers such as recurrence relations, explicit formulas and generating functions are given. Also some interesting combinatorial identities are obtained.

ABSTRACT. In this paper, we deal with a class of analytic functions given by means of Rodrigues type formula and their some properties. For this family, we first obtain a generating function by use of Cauchy's integral formula and then give several recurrence formulas by means of this generating function. We also derive several families of bilateral and bilinear generating functions for these analytic functions and present some special cases. Furthermore, we give some examples of analytic functions belonging to this family and find differential equations verified by these functions.

ABSTRACT. The generalized lightning traveling current source return stroke model (also called GTCS model) represents generalization of all engineering lightning return stroke
models that is generalization of the transmission line (TL models, also called current propagation models) 
and traveling current source (TCS models, also knownas current generation models) models.
Within the GTCS model the channel discharge function was introduced and calculated
for special cases. We applied new numerical method for the calculation of the channel discharge function.
The proposed method is highly accurate, extremely efficient and relatively simple.

ABSTRACT. We study the growth of gradient of mappings which satisfy certain PDE equa- tions (or inequalities) using Green-Laplacian formula for functions and its derivatives. If, in addition, the considered mappings are quasiconformal between C2 domains, we show that they are Lipschitz. Some of the obtained results can be considered as versions of Kellogg-Wa- rshawski type theorem for quasiconformal mappings.

ABSTRACT. This paper presents a discussion on application of second-order- like search directions in the Stochastic Approximation methods together with convergence conditions and some results on numerical implementation. We consider strictly convex problems in noisy environment and assume that only noisy values for the objective function and the gradient are available, as well as some approximate Hessian value. Under the zero mean assumption on noise a convergence analysis is presented for methods that use some approximate second-order direction. We prove that there exists a level of inexactness, governed by the usual gain sequence in SA methods, that does not interfere with the convergence and hence derive the set of convergence conditions that are applicable to a number of search directions. These directions include the so called mini-batch subsampled Hessian in statistical learning and similar directions. A set of numerical tests is presented in order to demonstrate efficiency and implementation issues of the proposed methods.

ABSTRACT. S. Li in [Studia Sci. Math. Hungar. 29 (1994) 71-83] proposed a Kronrod type extension to the well-known Turan formula. He showed that such an extension exists for any weight function. For the classical Chebyshev weight function of the first kind, Li found the Kronrod extension of Turan formula that has all its nodes real and belonging to the interval of integration, [-1,1]. In this paper we show the existence and the uniqueness of the additional two cases - the Kronrod exstensions of corresponding Gauss-Turan quadrature formulas for special case of Gori-Micchelli weight function and for generalized Chebyshev weight function of the second kind, that have all their nodes real and belonging to the integration interval [-1,1]. Numerical results for the weight coefficients in these cases are presented, while the analytic formulas of the nodes are known.

ABSTRACT. We consider the Bayesian estimation of the future order statistics and the mean of a future sample based on lower record values from the Generalized exponential distribution family. Bayesian credible interval sets are presented for the derived estimators. A real data set is provided to show the implementation of the procedures presented.

ABSTRACT. In reservoir engineering, accurate well modeling is crucial for reliable fluid flow simulations. Flow in the entire reservoir is induced mainly by wells, therefore poor near-well modeling results in accuracy loss throughout the model. Groundwater flow equation (Richards equation) is obtained from the conservation law, Boussinesq approximation and Darcy's \cite{Vid14} \[ \frac{\partial \theta}{\partial t}=\nabla\cdot \left( k_{\text{r}}(s)\mathbb{K}({\bf x})\nabla h \right), \] where $\theta$ is the water content, $k_{\text{r}}$ is relative conductivity, $\mathbb{K}({\bf x})$ is symmetric and positive definite hydraulic conductivity tensor, and $h$ is the hydraulic head.

Hydraulic head varies logarithmically and its gradient changes sharply in the well viscinity. Thus, linear approximation of hydraulic head is inappropriate and numerical methods based on it are inaccurate in the near-well region. Local grid refinement can alleviate the problem, but this comes at a computational cost.

Non-linear multi-point flux approximation \cite{Dro} is obtained as a combination of two one-sided linear fluxes. Non-linear multi-point scheme is second order accurate and preserves the elliptic local maximum principle. Nevertheless, linear approximation is employed and therefore the accuracy is lost if a well is present. Not only that the hydraulic head is inconsistent, but also the well extraction rate is wrong.

Two correction methods presented in \cite{Dot16,Dot14} for non-linear two-point scheme are also applicable for multi-point scheme. The WFC scheme modifies flux on the well faces, while the fluxes through other faces are approximated using the unmodified non-linear multi-point scheme. The NWC scheme modifies fluxes in the specified near well region. The NWC scheme is further generalized in \cite{Kra17} for polyhedral grids and arbitrary wells. Both of these corrections change only the approximation of one-sided linear fluxes, but use the same logic for their combining as the non-linear multi-point scheme.

Obtained results indicate that WFC scheme greatly improves the well extraction rate compared to the uncorrected scheme, but the hydraulic head is still inconsistent even though it is improved. On the other hand, NWC scheme gives not only improved well extraction rate, but also obtained hydraulic head is second order accurate.

Both correction methods are implemented in WODA \cite{Woda} (Well Outline and Design Aid), an open-source tool for simulation of unsaturated groundwater flows in discontinous and anisotropic enviroment.

\textbf{Keywords:} Multi-point nonlinear finite volume method, Near-well modeling, Groundwater flow simulations

ABSTRACT. This paper focuses on the development of automated techniques for proving mixed trigonometric polynomial inequalities of the form: $$ f(x)=\sum_{i=1}^{n}{\alpha_i x^{p_i} \sin ^{q_i} x \cos ^{r_i} x}>0, $$ where $p_i,q_i,r_i \in N_0$, $\alpha_i \in R \;\backslash\!\left\{ 0\right\}$ and $x\in(0,\frac{\pi}{2})$. An algorithm that reduces proving of such inequalities to proving of the corresponding polynomial inequalities is developed. It is shown that many open problems as well as various inequalities recently published in renowned journals can be proved using the proposed algorithm.

ABSTRACT. Disruptive powers of applied mathematics and computer science are changing many industries, including financial industry. One example is application of machine learning algorithms in the field of high frequency trading which aims in introducing predictive power into stochastic environment thus securing even more investments and leveraging portfolio risks. Current traditional approaches in price prediction which rely on domain experts knowledge and human traders, not that rarely, tend to lack in speed and reliability. Therefore, in order to address this issue, we introduced approach presented in this paper, that is based on automated process that incorporates data gathering, data transformation, training predictive model and generating predictions that later facilitate making financial investments. In the process of mitigating high dimensionality problem within our approach we compared Support Vector Machine and Boosting Classifiers since they tend to be more robust in such case. As a result, Gradient Boosting proved to be faster in learning and superior in performance leading to prediction results comparable to other research efforts. Presented approach was tested but it is not limited to data obtained from Google Finance service.

ABSTRACT. We will solve the Suskevic problem on right zero divisors in the ring of infinite upper triangular matrices for a special type of infinite upper bidiagonal matrices.