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Plenary Speakers

Ioan Rasa, Technical University of Cluj-Napoca, Romania

Francesco Altomare, University of Bari, Italy

Gianluca Vinti, University of Perugia, Italy

Stefano De Marchi, University of Padua, Italy



Name Institution Title Abstract
Francesco Altomare
Department of Mathematics, University of Bari
Approximation processes and representation formulae for operator semigroups in terms of integrated means
The representation/approximation formulae for strongly continuous operator semigroups (in short, C0 - semigroups) are of interest both from a theoretical point of view and from an applied one, especially when they are involved, for instance, in the numerical analysis of the partial differential equations governed by such C0 -semigroups. Various methods and results are known in this field, often accompanied by a through study of the rate of convergence of the given representations/approximations. The talk will be devoted to report some recent results on such a topic which are documented in the paper [1] and which has been jointly written with M. Cappelletti Montano and V. Leonessa. Of concern are some representation formulae for C0 -semigroups on Banach spaces, in terms of integrated means with respect to a given family of probability Borel measures and other parameters. Such representation formulae have been suggested by some recent studies devoted to new sequences of positive linear operators which, among other things, generalize both Bernstein operators as well as Kantorovich operators ([2], [3]). They extend the representation formulae which have been obtained in [4] with purely probabilistic methods. Some estimates of the rate of convergence in terms of the rectified modulus of continuity and the second modulus of continuity will be discussed as well.
Stefano De Marchi
Dipartimento di Matematica "Tullio Levi-Civita", University of Padova (UNIPD), Italy
(\beta,\gamma)-Chebyshev functions and points of the interval and some extensions
In this talk, we introduce the class of (\beta,\gamma)-Chebyshev functions and corresponding points, which can be seen as a family of generalized Chebyshev polynomials and points. For the (\beta,\gamma)-Chebyshev functions, we prove that they are orthogonal in certain subintervals of [-1,1] with respect to a weighted arc-cosine measure. In particular we investigate the cases where they become polynomials, deriving new results concerning classical Chebyshev polynomials of first kind. Besides, we show that subsets of Chebyshev and Chebyshev-Lobatto points are instances of (\beta,\gamma)-Chebyshev points. We also study the behavior of the Lebesgue constants of the polynomial interpolant at these points on varying the parameters (\beta) and (\gamma). If time allows, we discuss their extension to the square [-1,1]^2. This is a joint work with Giacomo Elefante and Francesco Marchetti (both from UNIPD).
Carmen-Violeta Muraru (Popescu) (joint work with Acu Ana-Maria, Voichita Adriana Radu)
"Vasile Alecsandri" University of Bacau
On semi-exponential Baskakov operators
Recently, the idea of exponential type operators is extended in capturing semi-exponential operators as generalization of the first ones. The present study is focused on the approximation properties of the semi-exponential Baskakov operators type, connected with . A Voronovskaja type results are obtained and some approximation properties in the weighted spaces of continuous functions are also studied.
George Popescu
University of Craiova, Department of Applied Mathematics
Rhaly Operators, Boundedness, Compactness, Zero Statistical Density
We present a study upon Rhaly operators on Hilbert spaces, that is operators defined by "terraced matrices". We prove a necessary condition for a terraced matrix to define a bounded Rhaly operator, or a compact operator, involving subsequences with zero statistical density. We show that boundedness and compactness of Rhaly operators are reduced to study the Rhaly operators defined by subsequences of the reciprocal of natural numbers. This leads to study the reduced operators of the Cesaro operator and investigate harmonic series defined by subsequences of natural numbers with zero statistical density.
Ioan Rasa (joint work with Ana Maria Acu)
Technical University of Cluj-Napoca
Analytic inequalities and stochastic orders
The talk is devoted to a family of analytic inequalities connected with positive linear operators and stochastic orders. The results are formulated in analytic terms and/or probabilistic terms.
Nicusor Minculete
Transilvania University of Brasov
Several inequalities related the numerical radius
The aim of this presentation is to give new upper bounds of \omega(T), which denotes the numerical radius of an operator T on a Hilbert space (H,\langle\cdot,\cdot\rangle). Next, we give certain inequalities about radius \omega(S^*T).
Ariana Pitea
University Politehnica of Bucharest
Numerical reckoning of fixed points in a geometric framework
Numerical schemes for the reckoning of fixed points with adequate properties are presented, regarding three step independent iterative procedures. Convergence results are established. A qualitative study is made, from the point of view of T-stability and data dependence.
Diana Curila (Popescu) (joint work with Alexandru Mihai Bica)
University of Oradea, Department of Mathematics and Informatics
The Akima.s Fitting Method for Quartic Splines
For the Hermite type quartic spline interpolating on the partition knots and at the midpoint of each subinterval, we consider the estimation of the derivatives on the knots, and the values of these derivatives are obtained by constructing an algorithm of Akima.s type. For computing the derivatives on end-points are also considered alternatives that request optimal properties near the end-points. The error estimate in the interpolation with this quartic spline is generally obtained in terms of the modulus of continuity. In the case of interpolating smooth functions, the corresponding error estimate reveal the maximal order of approximation O(h3). A numerical experiment is presented for making the comparison between the Akima.s cubic spline and the Akima.s variant quartic spline having de.ciency 2 and natural end-point conditions.
Zoltan Satmari (joint work with Alexandru Mihai Bica)
University of Oradea, Department of Mathematics and Informatics
Bernstein numerical method for solving nonlinear fractional and weakly singular Volterra integral equations of the second kind
We will show a method of approximation for a fractional nonlinear Volterra integral equation. This method is based on Banach's Fixed Point Theorem. At each iteration step of this method, we will apply a Bernstein polynomial approximation of the functions involved in the integral. At the end of the article, we will demonstrate our results, providing some numerical examples.
Ioan Cristian Buscu, Ancuta Emilia Steopoaie
Technical University, Cluj-Napoca
Voronovskaja type results for generalized Baskakov operators preserving monomials
We consider generalized Baskakov operators preserving the constant function 1 and a monomial x^j. For them we establish Voronovskaja type results.
Ioan Cristian Buscu, Andra Mihaela Seserman
Technical University, Cluj-Napoca
Convergence of special sequences of positive operators
Some modified sequences of positive linear operators converge to other positive linear operators. We present a general result in this direction and illustrate it by several examples.
Irina Savu
University Politehnica of Bucharest
Orbital Fuzzy Iterated Function Systems
In this paper we introduce the concept of orbital fuzzy iterated function system and prove that the fuzzy operator associated to such a system is weakly Picard. An example is provided.
Marta Enachioiu, Gabriela Denisa Motronea
Technical University, Cluj-Napoca
Positive solutions of algebraic systems with positive coefficients
We consider a special family of nonlinear algebraic systems of equations with positive coefficients and investigate the existence of positive solutions. The uniqueness of a positive solution is also studied.
Benaissa Zerroudi
Faculty of science, Ibn Zohr University - Agadir
Enhancing the approximation order of Shepherd operators through barycentric coordinates
This paper presents a new approach to constructing approximation operators, with cubic and quartic approximation order, which interpolates functional values on a set of scattered data. We based on a combination of multinodes Shepard basis functions with rational interpolants based on set of nodes. Numerical results show the efficiency and accuracy of the proposed operators and are implemented by fast algorithms that are useful in multiple applications.
Ionut Tudor Iancu
University of Oradea, Department of Mathematics and Computer Science
Statistical convergence in a Korovkin-type theorem for monotone and sublinear operators
In this paper we generalize the result on statistical uniform convergence in the Korovkin theorem for positive and linear operators in C([a, b]), to the more general case of monotone and sublinear operators. Our result is illustrated by concrete examples.
Daniela Inoan, Daniela Marian
Department of Mathematics, Technical University of Cluj-Napoca
Semi-Hyers-Ulam-Rassias Stability of a Volterra Integro-Differential Equation with a Convolution Type Kernel via Laplace Transform
We present some results regarding semi-Hyers-Ulam-Rassias stability of a Volterra integro-differential equation of order I, II and n, with a convolution type kernel. To this purpose the Laplace transform is used. The results obtained show that the stability holds for problems formulated with various functions: exponential and polynomial functions.
Radu Paltanea, Alexandra Diana Melesteu
Transilvania University of Brasov
On a method for uniform summation of the Fourier-Jacobi series
We point out that there is a matrix method of summation of Fourier-Jacobi series attached to all continuous functions, which is given by the representation of Durrmeyer operators with Jacobi weight. It is shown that this method of summation is stronger than the Cesaro methods of summation of all orders.
Miruna-Stefana Sorea
Lucian Blaga University of Sibiu
Poincare-Reeb graphs of real algebraic domains
An algebraic domain is a closed topological subsurface of a real affine plane whose boundary consists of disjoint smooth connected components of real algebraic plane curves. We study the non-convexity of an algebraic domain by collapsing all vertical segments contained in it: this yields a Poincare-Reeb graph, which is naturally transversal to the foliation by vertical lines. Using an adapted version of a Weierstrass-type approximation theorem, we show that any transversal graph whose vertices have only valencies 1 and 3 and are situated on distinct vertical lines can be realized as a Poincaré-Reeb graph. This talk is based on joint work with Arnaud Bodin and Patrick Popescu-Pampu.
Adrian Girjoaba
Universitatea Lucian Blaga din Sibiu
On the log-convexity of a Bernstein-like polynomials sequence
We prove that the sequence of the sum of the squares of the Bernstein polynomials is log-convex. There are given two proofs of this result: one by relating our sequence to the Legendre polynomials sequence and one by induction. I know of this problem from Professor Ioan Rasa, Cluj-Napoca.
Gabriel Prajitura, Andrei Vernescu
State University of New York, United States; Valahia University of Târgoviște (retired) Romania
About uniform continuous functions which are not Lipschitz
To approximate a function, some simpler functions are used: simpler as definitory structure (formula) and also as regularity, continuity and/or smoothness. Any Lipschitz function is a uniformly continuous function, but conversely it not always true. We add to the usual examples some other, especially of the class of differentiable functions.
Andra Malina
Babes-Bolyai University of Cluj-Napoca, Romania
Univariate Shepard operators combined with least squares fitting polynomials
We construct three new univariate Shepard operators using linear, quadratic and cubic polynomials in order to improve the approximation results of the Shepard method. The polynomials are constructed such that they fit the interpolation data in a weighted least squares approximation way.
Alexandru-Mihai Bica
University of Oradea, Romania
Catmul-Rom quartic splines
The quartic degree variant of Catmul-Rom splines that interpolates on mesh points and at midpoints is defined. The error estimates are obtained providing the optimal order of convergence O(h^5). The usefulness of these splines for the numerical solution of two-point boundary value problems, associated to fifth order functional differential equations, is pointed out.
Doru Dumitrescu
University Politehnica Bucharest, Romania
Fixed point results on Jleli-Samet metric spaces
In the setting of Jleli-Samet metric spaces, existence and uniqueness fixed point theorems are stated and proved. We used generalized contractive operators obtained as a sum of mappings with adequate properties. As consequences, some classic results in literature are obtained.
Gulen Bascanbaz-Tunca
Ankara University, Turkey
Generalized Kantorovich Operators on Multidimensional Hypercube
In this talk, based on the construction of Stancu's operators [1], we introduce a new generalization of Kantorovich operators acting on $L^p( Q)$, $1\leq p<\infty $, where $Q$ is a multidimensional unit hypercube. We study approximation in $L^p$-norm by the sequence of such operators. And, for the rate of the approximation, we present some estimates via multivariate averaged modulus of smoothness of the first order for functions from $L^p( Q)$ as well as from a smooth subspace of $L^p( Q)$.
Jose A. Adell and Daniel Cardenas-Morales (speaker)
University of Zaragoza and University of Ja\'{e}n, Spain
On the constant in a direct inequality for the Szasz-Mirakyan operator
We are concerned with the approximation of continuous functions by the classical Sz\'{a}sz-Mirakyan operators $S_t$, $t>1$, in terms of the Ditzian-Totik modulus of smoothness $\omega_2^{\varphi}$ with step-weight function $\varphi (x)=\sqrt{x}$. Information about the rate of uniform convergence is given by the direct inequality $$\|S_tf-f\|_{[0,\infty )}\leq K_S\omega_2^{\varphi}\left(f;\frac{1}{\sqrt{t}}\right).$$ Here, we focus on the absolute constant $K_S$, and prove that it can be taken smaller than $2.5$.
Gianluca Vinti
University of Perugia, Italy
A mathematical model for the study of vascular patologies
I will present a mathematical model based on the study of some sampling type operators whose approximation results lead to applications to the reconstruction and the enhancement of digital images. These, in turn, will allow to solve a diagnostic problem, concerning vascular pathologies, studied within the CARE project.
Mirella Cappelletti Montano (joint work with Benedetta Lisena)
Università degli Studi di Bari Aldo Moro, Italy
A diffusive two predators-one prey model on periodically evolving domains
The talk deals with a diffusive two predators-one prey model with Holling-type II functional response. We assume that the density of prey and predators are spatially inhomogeneous on a (periodically) evolving domain and are subject to homogeneous Neumann boundary conditions. We study the asymptotic properties of the solutions of this reaction-diffusion model. More specifically, we introduce suitable conditions in order that one predator species faces extinction and the surviving predator and its prey coexist stably, showing that their density, as time increases, tends to the periodic solution of the corresponding kinetic two species predator-prey model. We also consider an autonomous model on a fixed domain.
Marius-Mihai Birou
Technical University of Cluj Napoca, România
New bounds for the complete elliptic integral of the first kind
The complete elliptic integral of the first kind has applications in mathematics, physics and engineering. A lot of researchers has attracted to obtain inequalities for this integral. In this article we give new bounds for the complete elliptic integral of the first kind.
Alina Ramona Baias
Technical University of Cluj Napoca, România
On the Best Ulam constant of some linear difference equations
Abstract: An equation is called Ulam stable if for every approximate solution of it there exists an exact solution near it. In this talk, we present some stability results for the $p$ order linear difference equation with constant coefficients $$x_{n+p}=a_1x_{n+p-1}+\ldots+a_px_n.$$ For some particular cases the Best Ulam constant is also obtained.
Maryam Mohammadi, Mohammad Heidari,
Stefano De Marchi
Faculty of Mathematical Sciences and Computer, Kharazmi University, Tehran, Iran
Department of Mathematics "Tullio Levi-Civita", University of Padova, Italy
On the new positive definite RBFs by using Fourier cosine transform
In this article, we introduce a new family of infinitely smooth positive definite (PD) radial basis functions (RBFs) by using the Fourier cosine transform of the squared of a completely monotone function. These bases are represented in terms of positive Borel measures and their Fourier transforms are also given. The proposed theory is used for reconstructing the well-known Matérn RBF and presenting a new positive definite RBF. Numerical results show an accurate reconstruction of functions in which both the Runge and Gibbs phenomena are substantially mitigated.
Nesibe Manav Mutlu,
Nursel Cetin
Nisantasi University, Turkey
Ankara Haci Bayram Veli University, Turkey
On Schurer Type Complex Stancu Operators
In this work, we examine some approximation properties of complex Stancu-Schurer operators. We obtain a quantitative upper estimate for the simultaneous approximation and a qualitative Voronovskaja type result. Finally, we obtain the exact order of approximation by these operators attached to analytic functions in closed disks.
Heinz-Joachim Rack,
Robert Vajda
Dr. Rack AG, Hagen, Germany
University of Szeged, Hungary
Algebraic construction of Zolotarev polynomials with interchanged parameters
The inverse polynomial image (IPI) of [-1,1] under the monic proper aka hard-core Zolotarev polynomial Z(x; s) of degree n > 1, with prescribed parameter s, consists of [-1,1] and [\alpha, \beta], and there is a point \gamma (1 < \gamma < \alpha) where Z´(x = \gamma; s) = 0. The four occurring parameters are interrelated by (\alpha + \beta) = 2*(\gamma + s) and Z(x; s) was originally represented in terms of elliptic functions, see [1]-[5] for details. In [5] the problem is solved to construct by algebraic means the related Zolotarev polynomial Z(x; \beta), with prescribed \beta. This amounts to determine the compatible \alpha to retain the said IPI with a corresponding new s = s_0, say. We obtain here solutions for the two analogous problems: To construct by algebraic means Z(x; \alpha) and Z(x; \gamma), which amounts to determine the compatible \beta respectively \alpha and \beta. Advantage is taken of reduced relation curves, cylindrical algebraic decomposition, and the CAS Mathematica. This is joint work with Robert Vajda.
Bianca Ioana Vasian
Transilvania University of Brasov, Romania
Approximation properties of some non-positive Kantorovich type operators
In this paper we will establish a convergence result for some operators which are obtained as a Kantorovich modification of Szasz-Mirakjan operators. The operators obtained this way are not positive on the entire domain of definition.
Stefan Garoiu
Transilvania University of Brasov, Romania
Voronovskaya type results for geometric series of Durrmeyer operators
In this paper we will introduce some Voronovskaya type results concerning the operators introduce by U. Abel. Namely, we will obtain some convergence results concerning the geometric series of Durrmeyer operators.
Diana Otrocol
Technical University of Cluj-Napoca, Romania
Applications of fibre contraction principle to some classes of functional integral equations
By a new variant of fibre contraction principle we give existence, uniqueness and convergence of successive approximations results for some functional integral equations. In the case of ordered Banach space, Gronwall-type and comparison-type results are also given.
Alexandra Diana Melesteu,
Maria Dimitriu
Transilvania University of Brasov, Romania
Transilvania University of Brasov, Romania
A Stancu type generalization of the Balazs operator
In this paper we investigate certain properties of Stancu type generalization of the Balazs operator.
Michele Campiti
Department of Mathematics and Physics, University of Salento, Italy
One-sided Korovkin approximation
We consider a Korovkin-type approximation problem which allows to distinguish between upper and lower approximation. The results are based on a classical characterization of Korovkin closures in terms of upper and lower envelopes and are exsposed both in the context of continuous than integrable functions. In this last case an analogous of the classical characterization is obtained which allows to consider Korovkin systems including integrable non continuous functions.
Cristina Maria Pacurar
Transilvania University of Brasov, Romania
On some operators appearing in fractal interpolation theory
In this paper we study some operators that appear in fractal interpolation theory. Firstly, we present operators that are more general compared to the classical case, as the scaling in the y-coordinate is replaced by a non-linear contraction. Moreover, we study smooth operators that provide differentiable interpolation functions.
Dorian Popa
Technical University of Cluj-Napoca, Romania
On some results in Ulam stability.
In this talk I will present some results in Ulam stability for some functional equations, differential operators and positive linear operators from the papers written in collaboration with Ioan Rasa.
Pop Emilia
Faculty of Mathematics and Computer Science, Babes-Bolyai University, Cluj-Napoca, Romania
The intermediate point from a mean value theorem
In this paper we consider two continuous functions f, g:[a,b]->R and we study for these ones, under which circumstances the intermediate point function is four order differentiable at the point x=a and we calculate its derivate.
Antonio-Jesus Lopez-Moreno
University of Jaen, Spain
On a conjecture of G. T. Thachev
In [6], for the classical Bernstein operators, $B_n$, Thachev formulated the following conjecture: $$\lim_{n\to\infty}\frac{B_n\big((e_1-x)^{q+1},x\big)}{B_n\big(\mid e_1-x\mid^q,x\big)}=0,$$ for all $q\in\mathbb{N}$. In this note we give some different proofs for this conjecture that can be applied for a general class of linear positive operators. We can also offer constants for the limit involved in the conjecture. We use techniques presented by Adell, Bustamante and Quesada [1], López and Gupta [3,4], Xiang [7] and others.
Margareta Heilmann
(joint work A.M. Acu, I. Rasa, A.E. Stepoaie)
University of Wuppertal, Germany
Voronovskaja type results for the Aldaz, Kounchev, Render modification of Baskakov type operators
For $j>1$, $j\in {\mathbb N}$ fixed and $n\geq j$, Aldaz, Kounchev and Render [1] introduced a modification of the Bernstein operator that preserves the monomials $e_0$ and $e_j$. This operator $B_{n,j}: C[0,1]\to C[0,1]$ is constructed using the classical original Bernstein basis functions $p_{n,k}(x) = {n \choose k} x^k (1-x)^{n-k}$, $ x \in [0,1]$ and is explicitly given by $$ B_{n,j}(f;x)=\displaystyle\sum_{k=0}^n f\left(t_{n,k}^{(j)}\right)p_{n,k}(x),$$ where $$t_{n,k}^{(j)}=\left(\dfrac{k(k-1)\dots(k-j+1)}{n(n-1)\dots(n-j+1)}\right)^{1/j}.$$ We generalize the definition to Baskakov type operators and prove a corresponding Voronovskaja type result.
Vlad Pasca
(joint work with A.M. Acu, M. Heilmann, I. Rasa and M. Dancs)
"Lucian Blaga" University of Sibiu, Romania
Voronovskaya type results for special sequences of operators
The aim of this paper is twofold. First, we present Voronovskaya type formulas that can be ”differentiated” and other ones which are associated with operators preserving two prescribed functions. Then the Bernstein-Schnabl type operators Ln are considered. This sequence of positive linear operators was studied in the literature; we provide new properties of it. In particular, its Voronovskaya formula can be ”differentiated” and each Ln is invariant under the Kantorovich type modification. The moments of each operator Ln form a sequence of Appell polynomials, while the central moments are constant functions.
Ana Maria Acu,
Nursel Cetin,
Gancho Tachev
"Lucian Blaga" University of Sibiu, Romania
Ankara Haci Bayram Veli University, Turkey
University of Architecture Civil Engineering and Geodesy, Bulgaria
Perturbed Baskakov-type operators
This talk deals with modifications of Baskakov-type operator. Firstly, we obtain the rate of convergence by using modulus of continuity and then Voronovskaja type asymptotic formula for these operators. Finally, we consider some numerical examples which show the relevance of the results.
Mirela Vinerean,
Niclas Bernhoff
Karlstad University, Sweden
Normal Discrete Velocity Models for Boltzmann Equation for Mixtures
Discretization methods for Boltzmann equation (BE) have been developed on the idea of replacing the original equation by a finite set of nonlinear hyperbolic PDEs corresponding to the densities linked to a suitable finite set of velocities. Making a transition from the BE to its discrete version, we often gain some unphysical (spurious) conservation laws. The well-known theorem of uniqueness of collision invariants for the continuous velocity space R^d very often does not work for a finite set of discrete velocities. One open problem related to the discrete BE is the construction of normal (fulfilling only physical conservation laws) discrete velocity models (DVMs). In many papers on DVMs, authors postulate from the beginning that a finite velocity space with such properties is given and after that they study the discrete BE. Our aim is not to study the equations for DVMs, but to discuss possible choices of finite sets of velocities satisfying this type of restrictions. In this presentation, we will address ways of constructing DVMs for mixtures.