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Marilena Jianu

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Marilena Jianu received her M.Sc. degree in mathematics (statistics and probability) from the University of Bucharest, and her Ph.D. degree in civil engineering from the Technical University of Civil Engineering Bucharest. She has been teaching advanced mathematics to engineering students for over 20 years and has published several papers in pure and applied mathematics. She is an editorial board member of the Romanian Journal of Mathematics and Computer Science and a reviewer for Mathematical Reviews. https://orcid.org/0000-0002-2718-8133

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Papers by Marilena Jianu

Answers and Solutions to Exercises

Springer eBooks, 2022

On a Surface Associated to the Catalan Triangle

Axioms

We define a surface that interpolates the ballot numbers in the Catalan triangle corresponding to... more We define a surface that interpolates the ballot numbers in the Catalan triangle corresponding to every pair of nonnegative integers (except for the origin). We study the geometric properties of this surface and prove that it contains exactly five half-lines. The mean curvature and the Gauss curvature of the surface are also calculated.

Using Delta-Wye Transformations for Estimating Networks’ Reliability

Advances in intelligent systems and computing, Sep 25, 2022

On the Roots of Certain Reliability Polynomials

Advances in intelligent systems and computing, Sep 25, 2022

Consecutive k-out-of-n:F Systems Have Unbounded Roots

arXiv (Cornell University), Aug 9, 2022

This article is studying the roots of the reliability polynomials of linear consecutive-k-outof-n... more This article is studying the roots of the reliability polynomials of linear consecutive-k-outof-n:F systems. We are able to prove that these roots are unbounded in the complex plane, for any fixed k ≥ 2. In the particular case k = 2, we show that the reliability polynomials have only real roots and highlight the closure of these roots by establishing their explicit formulas. We also point out that in this case, for any fixed n the nonzero roots of the reliability polynomial are distinct numbers.

Full Hermite Interpolation and Approximation in Topological Fields

Mathematics

By using generalized divided differences, we study the simultaneous interpolation of an m times c... more By using generalized divided differences, we study the simultaneous interpolation of an m times continuously differentiable function and its derivatives up to a fixed order in a topological field K. If K is a valued field, then simultaneous Hermite interpolation and approximation are considered. Newton interpolating series are used in the case of an infinite number of conditions of interpolation. Applications to the numerical approximation of variational problems, the solution of a functional equation and, in the case of p-adic fields, the representation of solutions of a boundary value problem for an equation of the Fuchsian type illustrate the efficiency of the theoretical results.

The Shape of the Reliability Polynomial of a Hammock Network

Advances in Intelligent Systems and Computing, 2020

Motivated by the study of hammock (aka brick-wall) networks, we introduce in this paper the notio... more

On a Surface Associated with Pascal’s Triangle

Symmetry, 2022

An open problem in reliability theory is that of finding all the coefficients of the reliability ... more An open problem in reliability theory is that of finding all the coefficients of the reliability polynomial associated with particular networks. Because reliability polynomials can be expressed in Bernstein form (hence linked to binomial coefficients), it is clear that an extension of the classical discrete Pascal’s triangle (comprising all the binomial coefficients) to a continuous version (exhibiting infinitely many values in between the binomial coefficients) might be geometrically helpful and revealing. That is why we have decided to investigate the geometric properties of a continuous extension of Pascal’s triangle including: Gauss curvatures, mean curvatures, geodesics, and level curves, as well as their symmetries.

Characterization of Rectifying Curves by Their Involutes and Evolutes

Mathematics, 2021

A rectifying curve is a twisted curve with the property that all of its rectifying planes pass th... more A rectifying curve is a twisted curve with the property that all of its rectifying planes pass through a fixed point. If this point is the origin of the Cartesian coordinate system, then the position vector of the rectifying curve always lies in the rectifying plane. A remarkable property of these curves is that the ratio between torsion and curvature is a nonconstant linear function of the arc-length parameter. In this paper, we give a new characterization of rectifying curves, namely, we prove that a curve is a rectifying curve if and only if it has a spherical involute. Consequently, rectifying curves can be constructed as evolutes of spherical twisted curves; we present an illustrative example of a rectifying curve obtained as the evolute of a spherical helix. We also express the curvature and the torsion of a rectifying spherical curve and give necessary and sufficient conditions for a curve and its involute to be both rectifying curves.

Infinitely differentiable functions represented into Newton interpolating series

Carpathian Journal of Mathematics, 2014

We study infinitely differentiable functions which are representable into a Newton interpolating ... more We study infinitely differentiable functions which are representable into a Newton interpolating series at a suitable interpolation sequence with terms in [0, 1]. Applications of this series to approximate the solution of boundary value problems for linear systems of differential equations are presented.

Markov Chain Method for Computing the Reliability of Hammock Networks

Probability in the Engineering and Informational Sciences, 2020

In this paper, we develop a new method for evaluating the reliability polynomial of a hammock net... more In this paper, we develop a new method for evaluating the reliability polynomial of a hammock network. The method is based on a homogeneous absorbing Markov chain and provides the exact reliability for networks of width less than 5 and arbitrary length. Moreover, it produces a lower bound for the reliability polynomial for networks of width greater than or equal to 5. To investigate how sharp this lower bound is, we compare our method with other approximation methods and it proves to be the most accurate in terms of absolute as well as relative error. Using the fundamental matrix, we also calculate the average time to absorption, which provides the mean length of a network that is expected to work.

Full Hermite interpolation of the reliability of a hammock network

Applicable Analysis and Discrete Mathematics, 2020

Although the hammock networks were introduced more than sixty years ago, there is no general form... more Although the hammock networks were introduced more than sixty years ago, there is no general formula of the associated reliability polynomial. Using the full Hermite interpolation polynomial, we propose an approximation for the reliability polynomial of a hammock network of arbitrary size. In the second part of the paper, we provide combinatorial formulas for the first two non-zero coefficients of the reliability polynomial.

Optimization of the Solution of a Dispersion Model

Mathematics, 2020

The study of the combination of chemical kinetics with transport phenomena is the main step for r... more The study of the combination of chemical kinetics with transport phenomena is the main step for reactor design. It is possible to deviate from the model behaviour, the cause of which may be fluid channelling, fluid recirculation, or creation of stagnant regions in the vessel, by using a dispersion model. In this paper, the known general solution of the dispersion model for closed vessels is given in a new, straightforward form. In order to improve the classical theoretical solution, a hybrid of analytical and numerical methods is used. It is based on the general analytic solution and the least-squares method by fitting the results of a tracer test carried out on the vessel with the values of the analytic solution. Thus, the accuracy of the estimation for the vessel dispersion number is increased. The presented method can be used to similar problems modelled by a partial differential equation and some boundary conditions which are not sufficient to ensure the uniqueness of the solution.

$Research paper thumbnail of Functions represented into fractional Taylor series$

Functions represented into fractional Taylor series

ITM Web of Conferences, 2019

Fractional Taylor series are studied. Then solutions of fractional linear ordinary differential e... more Fractional Taylor series are studied. Then solutions of fractional linear ordinary differential equations (FODE), with respect to Caputo derivative, are approximated by fractional Taylor series. The Cauchy-Kowalevski theorem is proved to show the existence and uniqueness of local solutions for FODE with Cauchy initial data. Sufficient conditions for the global existence of the solution and the estimate of error are given for the method using fractional Taylor series. Two illustrative numerical examples are given to demonstrate the validity and applicability of this method.

A Combinatorial Solution for Bridge and Torch Problem

SN Operations Research Forum, 2020

In this note we give a combinatorial solution to the general case of the puzzle known as "bridge ... more In this note we give a combinatorial solution to the general case of the puzzle known as "bridge and torch problem". It is about finding the shortest time needed for n people to cross a bridge knowing that at most two persons can walk together on the bridge (at the speed of the slowest) and, because it is dark, a torch is needed for each crossing (and they have only one torch). The puzzle can be seen (and solved) as a discrete optimization problem.

Answers and Solutions to Exercises

Springer eBooks, 2022

On a Surface Associated to the Catalan Triangle

Axioms

We define a surface that interpolates the ballot numbers in the Catalan triangle corresponding to... more We define a surface that interpolates the ballot numbers in the Catalan triangle corresponding to every pair of nonnegative integers (except for the origin). We study the geometric properties of this surface and prove that it contains exactly five half-lines. The mean curvature and the Gauss curvature of the surface are also calculated.

Using Delta-Wye Transformations for Estimating Networks’ Reliability

Advances in intelligent systems and computing, Sep 25, 2022

On the Roots of Certain Reliability Polynomials

Advances in intelligent systems and computing, Sep 25, 2022

Consecutive k-out-of-n:F Systems Have Unbounded Roots

arXiv (Cornell University), Aug 9, 2022

This article is studying the roots of the reliability polynomials of linear consecutive-k-outof-n... more This article is studying the roots of the reliability polynomials of linear consecutive-k-outof-n:F systems. We are able to prove that these roots are unbounded in the complex plane, for any fixed k ≥ 2. In the particular case k = 2, we show that the reliability polynomials have only real roots and highlight the closure of these roots by establishing their explicit formulas. We also point out that in this case, for any fixed n the nonzero roots of the reliability polynomial are distinct numbers.

Full Hermite Interpolation and Approximation in Topological Fields

Mathematics

By using generalized divided differences, we study the simultaneous interpolation of an m times c... more By using generalized divided differences, we study the simultaneous interpolation of an m times continuously differentiable function and its derivatives up to a fixed order in a topological field K. If K is a valued field, then simultaneous Hermite interpolation and approximation are considered. Newton interpolating series are used in the case of an infinite number of conditions of interpolation. Applications to the numerical approximation of variational problems, the solution of a functional equation and, in the case of p-adic fields, the representation of solutions of a boundary value problem for an equation of the Fuchsian type illustrate the efficiency of the theoretical results.

The Shape of the Reliability Polynomial of a Hammock Network

Advances in Intelligent Systems and Computing, 2020

Motivated by the study of hammock (aka brick-wall) networks, we introduce in this paper the notio... more

On a Surface Associated with Pascal’s Triangle

Symmetry, 2022

An open problem in reliability theory is that of finding all the coefficients of the reliability ... more An open problem in reliability theory is that of finding all the coefficients of the reliability polynomial associated with particular networks. Because reliability polynomials can be expressed in Bernstein form (hence linked to binomial coefficients), it is clear that an extension of the classical discrete Pascal’s triangle (comprising all the binomial coefficients) to a continuous version (exhibiting infinitely many values in between the binomial coefficients) might be geometrically helpful and revealing. That is why we have decided to investigate the geometric properties of a continuous extension of Pascal’s triangle including: Gauss curvatures, mean curvatures, geodesics, and level curves, as well as their symmetries.

Characterization of Rectifying Curves by Their Involutes and Evolutes

Mathematics, 2021

A rectifying curve is a twisted curve with the property that all of its rectifying planes pass th... more A rectifying curve is a twisted curve with the property that all of its rectifying planes pass through a fixed point. If this point is the origin of the Cartesian coordinate system, then the position vector of the rectifying curve always lies in the rectifying plane. A remarkable property of these curves is that the ratio between torsion and curvature is a nonconstant linear function of the arc-length parameter. In this paper, we give a new characterization of rectifying curves, namely, we prove that a curve is a rectifying curve if and only if it has a spherical involute. Consequently, rectifying curves can be constructed as evolutes of spherical twisted curves; we present an illustrative example of a rectifying curve obtained as the evolute of a spherical helix. We also express the curvature and the torsion of a rectifying spherical curve and give necessary and sufficient conditions for a curve and its involute to be both rectifying curves.

Infinitely differentiable functions represented into Newton interpolating series

Carpathian Journal of Mathematics, 2014

We study infinitely differentiable functions which are representable into a Newton interpolating ... more We study infinitely differentiable functions which are representable into a Newton interpolating series at a suitable interpolation sequence with terms in [0, 1]. Applications of this series to approximate the solution of boundary value problems for linear systems of differential equations are presented.

Markov Chain Method for Computing the Reliability of Hammock Networks

Probability in the Engineering and Informational Sciences, 2020

In this paper, we develop a new method for evaluating the reliability polynomial of a hammock net... more In this paper, we develop a new method for evaluating the reliability polynomial of a hammock network. The method is based on a homogeneous absorbing Markov chain and provides the exact reliability for networks of width less than 5 and arbitrary length. Moreover, it produces a lower bound for the reliability polynomial for networks of width greater than or equal to 5. To investigate how sharp this lower bound is, we compare our method with other approximation methods and it proves to be the most accurate in terms of absolute as well as relative error. Using the fundamental matrix, we also calculate the average time to absorption, which provides the mean length of a network that is expected to work.

Full Hermite interpolation of the reliability of a hammock network

Applicable Analysis and Discrete Mathematics, 2020

Although the hammock networks were introduced more than sixty years ago, there is no general form... more Although the hammock networks were introduced more than sixty years ago, there is no general formula of the associated reliability polynomial. Using the full Hermite interpolation polynomial, we propose an approximation for the reliability polynomial of a hammock network of arbitrary size. In the second part of the paper, we provide combinatorial formulas for the first two non-zero coefficients of the reliability polynomial.

Optimization of the Solution of a Dispersion Model

Mathematics, 2020

The study of the combination of chemical kinetics with transport phenomena is the main step for r... more The study of the combination of chemical kinetics with transport phenomena is the main step for reactor design. It is possible to deviate from the model behaviour, the cause of which may be fluid channelling, fluid recirculation, or creation of stagnant regions in the vessel, by using a dispersion model. In this paper, the known general solution of the dispersion model for closed vessels is given in a new, straightforward form. In order to improve the classical theoretical solution, a hybrid of analytical and numerical methods is used. It is based on the general analytic solution and the least-squares method by fitting the results of a tracer test carried out on the vessel with the values of the analytic solution. Thus, the accuracy of the estimation for the vessel dispersion number is increased. The presented method can be used to similar problems modelled by a partial differential equation and some boundary conditions which are not sufficient to ensure the uniqueness of the solution.

$Research paper thumbnail of Functions represented into fractional Taylor series$

Functions represented into fractional Taylor series

ITM Web of Conferences, 2019

Fractional Taylor series are studied. Then solutions of fractional linear ordinary differential e... more Fractional Taylor series are studied. Then solutions of fractional linear ordinary differential equations (FODE), with respect to Caputo derivative, are approximated by fractional Taylor series. The Cauchy-Kowalevski theorem is proved to show the existence and uniqueness of local solutions for FODE with Cauchy initial data. Sufficient conditions for the global existence of the solution and the estimate of error are given for the method using fractional Taylor series. Two illustrative numerical examples are given to demonstrate the validity and applicability of this method.

A Combinatorial Solution for Bridge and Torch Problem

SN Operations Research Forum, 2020

In this note we give a combinatorial solution to the general case of the puzzle known as "bridge ... more In this note we give a combinatorial solution to the general case of the puzzle known as "bridge and torch problem". It is about finding the shortest time needed for n people to cross a bridge knowing that at most two persons can walk together on the bridge (at the speed of the slowest) and, because it is dark, a torch is needed for each crossing (and they have only one torch). The puzzle can be seen (and solved) as a discrete optimization problem.