Magazine et Edition

Volume 80 (2021): Edition 3 (December 2021)

Volume 79 (2021): Edition 2 (December 2021)

Volume 78 (2021): Edition 1 (October 2021)

Volume 77 (2020): Edition 1 (December 2020)

Volume 76 (2020): Edition 1 (December 2020)
Real Functions, Dynamical Systems and their Applications

Volume 75 (2020): Edition 1 (April 2020)
Applied Mathematics'19

Volume 74 (2019): Edition 1 (December 2019)
Real Functons, Ideals, Measurable Functions, Functional Equations

Volume 73 (2019): Edition 1 (August 2019)
Number Theory, Algebra and Cryptology '18

Volume 72 (2018): Edition 1 (December 2018)

Volume 71 (2018): Edition 1 (December 2018)

Volume 70 (2017): Edition 1 (September 2017)

Volume 69 (2017): Edition 1 (June 2017)

Volume 68 (2017): Edition 1 (March 2017)
Special Edition: Real Functions ’16, Real Functions, Density Topologies, Porosity

Volume 67 (2016): Edition 1 (September 2016)

Volume 66 (2016): Edition 1 (June 2016)
Edition title: Applied Mathematics ’16

Volume 65 (2016): Edition 1 (March 2016)
Real Functions '15 — Measure Theory, Real Functions, General Topology. Editors: J. Borsík, 2016.

Volume 64 (2015): Edition 1 (September 2015)
Number Theory and Cryptology ’15

Volume 62 (2015): Edition 1 (March 2015)
Special Edition Title: Real Functions ’14

Volume 61 (2014): Edition 1 (December 2014)
Special Edition Title: Applied Mathematics ‘14

Volume 60 (2014): Edition 1 (September 2014)
Special Edition Title: Cryptology ’14

Volume 59 (2014): Edition 1 (June 2014)
Special Edition Title: Number Theory ‘14

Volume 58 (2014): Edition 1 (March 2014)
Real Functions ‘13 Real Functions, Topology, Real and Functional Analysis, Locally Convex Spaces

Volume 57 (2013): Edition 1 (December 2013)
Cryptology

Volume 56 (2013): Edition 1 (November 2013)
Number Theory

Volume 55 (2013): Edition 1 (August 2013)

Volume 54 (2013): Edition 1 (April 2013)
Differential and Difference Equations and Applications ‘2012

Volume 53 (2012): Edition 1 (December 2012)
TATRACRYPT ‘12

Volume 52 (2012): Edition 1 (August 2012)

Volume 51 (2012): Edition 1 (April 2012)
PROBASTAT ‘11

Volume 50 (2011): Edition 1 (December 2011)
Applied Mathematics and Informatics

Volume 49 (2011): Edition 1 (August 2011)
Real Functions ‘10

Volume 48 (2011): Edition 1 (April 2011)
Differential and Difference Equations and Applications 2010

Volume 47 (2010): Edition 1 (December 2010)
CCEC ‘09

Volume 46 (2010): Edition 1 (August 2010)
Real Functions ‘09

Volume 45 (2010): Edition 1 (April 2010)
NILCRYPT ‘10

Volume 44 (2009): Edition 1 (December 2009)
Real Function ’08 Functional Equation, Measures, Integration and Harmonic Analysis

Volume 43 (2009): Edition 1 (August 2009)
Differential and Difference Equations and Applications 2008

Volume 42 (2009): Edition 1 (April 2009)
Real Function ‘07

Détails du magazine
Format
Magazine
eISSN
1338-9750
Première publication
12 Nov 2012
Période de publication
3 fois par an
Langues
Anglais

Chercher

Volume 75 (2020): Edition 1 (April 2020)
Applied Mathematics'19

Détails du magazine
Format
Magazine
eISSN
1338-9750
Première publication
12 Nov 2012
Période de publication
3 fois par an
Langues
Anglais

Chercher

11 Articles
Accès libre

Parameter Estimation for the Forest Fire Propagation Model

Publié en ligne: 24 Apr 2020
Pages: 1 - 22

Résumé

Abstract

This paper first gives a brief overview of the Lagrangian forest fire propagation model [Ambroz, M.—Balažovjech, M.—Medl’a, M.—Mikula, K.: Numerical modeling of wildland surface fire propagation by evolving surface curves, Adv. Comput. Math. 45 (2019), no. 2, 1067–1103], which we apply to grass-field areas. Then, we aim to estimate the optimal model parameters. To achieve this goal, we use data assimilation of the measured data. From the data, we are able to estimate the normal velocity of the fire front (rate of spread), dominant wind direction and selected model parameters. In the data assimilation process, we use the Hausdorff distance as well as the Mean Hausdorff distance as a criterion. Moreover, we predict the fire propagation in small time intervals.

Mots clés

  • data assimilation
  • parameter estimation
  • curve evolution
  • forest fire

MSC 2010

  • Primary: 35R01, 65M08
  • Secondary: 35R37, 92F05
Accès libre

Bayesian Estimate of Parameters for ARMA Model Forecasting

Publié en ligne: 24 Apr 2020
Pages: 23 - 32

Résumé

Abstract

This paper presents a Bayesian approach to finding the Bayes estimator of parameters for ARMA model forecasting under normal-gamma prior assumption with a quadratic loss function in mathematical expression. Obtaining the conditional posterior predictive density is based on the normal-gamma prior and the conditional predictive density, whereas its marginal conditional posterior predictive density is obtained using the conditional posterior predictive density. Furthermore, the Bayes estimator of parameters is derived from the marginal conditional posterior predictive density.

Mots clés

  • ARMA model
  • Bayes estimator
  • normal-gamma prior
  • quadratic loss function

MSC 2010

  • 62C10
Accès libre

Estimating the Domestic Short Rate in a Convergence Model of Interest Rates

Publié en ligne: 24 Apr 2020
Pages: 33 - 48

Résumé

Abstract

In this paper we study the convergence model of interest rates by Corzo and Schwartz. It models the situation when a country is going to enter a monetary union, for example the eurozone. We are interested in estimating the underlying short rate, which is a theoretical variable, not observed on the market. We use the procedure already employed for the Vasicek model to the eurozone data and for the case of a zero correlation we show that a similar procedure can be used also for the estimation of the domestic parameters and the short rate values. The assumption of the zero correlation allows us to simplify the optimization problem, but using simulations we show that our algorithm is robust to the specification of the correlation. It estimates the short rate with a high precision also in the original case of a nonzero correlation, as well as in the case of a dynamic correlation, when the correlation is modelled as a function of time. Finally, we use the algorithm to real market data and estimate the short rate before adoption of the euro currency in Slovakia, Estonia, Latvia and Lithuania.

Mots clés

  • interest rates
  • short rate
  • convergence model
  • calibration

MSC 2010

  • 91G30
  • 35K15
  • 62M20
Accès libre

Finite Volume Scheme for AMSS Model

Publié en ligne: 24 Apr 2020
Pages: 49 - 62

Résumé

Abstract

We propose a new finite volume numerical scheme for the approximation of the Affine Morphological Scale Space (AMSS) model. We derive the basic scheme and its iterative improvement. For both schemes, several numerical experiments using examples where the exact solution is known are presented. Then the numerical errors and experimental order of convergence of the proposed schemes is studied.

Mots clés

  • affine morphological scale space
  • image processing
  • finite volume scheme
  • experimental order of convergence

MSC 2010

  • Primary: 65M08
  • Secondary: 35K20
Accès libre

The Finite Element Method as a Tool to Solve the Oblique Derivative Boundary Value Problem in Geodesy

Publié en ligne: 24 Apr 2020
Pages: 63 - 80

Résumé

Abstract

In this paper, we propose a novel approach to approximate the solution of the Laplace equation with an oblique derivative boundary condition by the finite element method. We present and analyse diverse testing experiments to study its behaviour and convergence. Finally, the usefulness of this approach is demonstrated by using it to gravity field modelling, namely, to approximate the solution of a geodetic boundary value problem in Himalayas.

Mots clés

  • oblique derivative boundary value problem
  • finite element method

MSC 2010

  • 65N30
Accès libre

Efficient 3D Shape Registration by Using Distance Maps and Stochastic Gradient Descent Method

Publié en ligne: 24 Apr 2020
Pages: 81 - 102

Résumé

Abstract

This paper presents an efficient 3D shape registration by using distance maps and stochastic gradient descent method. The proposed algorithm aims to find the optimal affine transformation parameters (translation, scaling and rotation) that maps two distance maps to each other. These distance maps represent the shapes as an interface and we apply level sets methods to calculate the signed distance to these interfaces. To maximize the similarity between the two distance maps, we apply sum of squared difference (SSD) optimization and gradient descent methods to minimize it. To address the shortcomings of the standard gradient descent method, i.e., many iterations to compute the minimum, we implemented the stochastic gradient descent method. The outcome of these two methods are compared to show the advantages of using stochastic gradient descent method. In addition, we implement computational optimization’s such as parallelization to speed up the registration process.

Mots clés

  • distance map
  • stochastic gradient method
  • registration
  • parallelization
  • affine transformation
  • optimization

MSC 2010

  • 65K10
  • 35Q68
  • 65Y05
  • 65Y20
Accès libre

Macrophage Image Segmentation by Thresholding and Subjective Surface Method

Publié en ligne: 24 Apr 2020
Pages: 103 - 120

Résumé

Abstract

We introduce two level-set method approaches to segmentation of 2D macrophage images. The first one is based on the Otsu thresholding and the second one on the information entropy thresholding, both followed by the classical subjective surface (SUBSURF) method. Results of both methods are compared with the semi-automatic Lagrangian method in which the segmentation curve evolves along the edge of the macrophage and it is controlled by an expert user. We present the comparison of all three methods with respect to the Hausdorff distance of resulting segmentation curves and we compare also their perimeter and enclosed area. We show that accuracy of the automatic SUBSURF method is comparable to the results of the semi-automatic Lagrangian segmentation.

Mots clés

  • image processing
  • segmentation
  • level-set methods
  • finite volume methods
  • thresholding
  • macrophages

MSC 2010

  • 68U10
  • 35K61
  • 65M08
Accès libre

Necessary and Sufficient Conditions for Oscillatory and Asymptotic Behaviour of Solutions to Second-Order Nonlinear Neutral Differential Equations with Several Delays

Publié en ligne: 24 Apr 2020
Pages: 121 - 134

Résumé

Abstract

In this paper, necessary and sufficient conditions are obtained for oscillatory and asymptotic behavior of solutions to second-order nonlinear neutral delay differential equations of the form ddt[r(t)[ddt(x(t)+p(t)x(t-τ))]α]+i=1mqi(t)H(x(t-σi))=0fortt0>0,{d \over {dt}}\left[ {r\left( t \right){{\left[ {{d \over {dt}}\left( {x\left( t \right) + p\left( t \right)x\left( {t - \tau } \right)} \right)} \right]}^\alpha }} \right] + \sum\limits_{i = 1}^m {{q_i}\left( t \right)H\left( {x\left( {t - {\sigma _i}} \right)} \right) = 0\,\,\,{\rm{for}}\,t \ge {t_0} > 0,}

under the assumption (r(n))−1/αdη=∞. Our main tool is Lebesque’s dominated convergence theorem. Further, some illustrative examples showing the applicability of the new results are included.

Mots clés

  • oscillation
  • non-oscillation
  • neutral
  • delay
  • nonlinear
  • Lebesgue’s dominated convergence theorem
  • Banach’s contraction principle

MSC 2010

  • 34C10
  • 34C15
  • 35K40
Accès libre

Necessary and Sufficient Conditions for Oscillation of Second-Order Delay Differential Equations

Publié en ligne: 24 Apr 2020
Pages: 135 - 146

Résumé

Abstract

In this work, we obtain necessary and sufficient conditions for the oscillation of all solutions of second-order half-linear delay differential equation of the form (r(x)γ)(t)+q(t)xα(τ(t))=0{\left( {r{{\left( {x'} \right)}^\gamma }} \right)^\prime }\left( t \right) + q\left( t \right){x^\alpha }\left( {\tau \left( t \right)} \right) = 0

Under the assumption (r(n))−1/γdη=∞, we consider the two cases when γ > α and γ < α. Further, some illustrative examples showing applicability of the new results are included, and state an open problem.

Mots clés

  • oscillation
  • non-oscillation
  • delay
  • linear
  • Lebesgue’s dominated convergence theorem

MSC 2010

  • 34C10
  • 35K40
  • 34K11
Accès libre

3D Cell Image Segmentation by Modified Subjective Surface Method

Publié en ligne: 24 Apr 2020
Pages: 147 - 162

Résumé

Abstract

In this work, we focused on 3D image segmentation where the segmented surface is reconstructed by the use of 3D digital image information and information from thresholded 3D image in a local neighbourhood. To this end, we applied a mathematical model based on the level set formulation and numerical method which is based on the so-called reduced diamond cell approach. The segmentation approach is based on surface evolution governed by a nonlinear PDE, the modified subjective surface equation. This is done by defining the input to the edge detector function as the weighted sum of norm of presmoothed 3D image and norm of presmoothed thresholded 3D image in a local neighbourhood. For the numerical discretization, we used a semi-implicit finite volume scheme. The method was applied to real data representing 3D microscopy images of cell nuclei within the zebrafish pectoral fin.

Mots clés

  • image segmentation
  • subjective surface method
  • level set method
  • finite volume method
  • semi-implicit scheme
  • cell microscopy images
  • zebrafish

MSC 2010

  • 65M08
  • 35K61
  • 68U10
Accès libre

Analysis of Cell Death by Image Processing

Publié en ligne: 24 Apr 2020
Pages: 163 - 190

Résumé

Abstract

In this paper, we present a graph theoretical approach to image processing with focus on the analysis of the biological data. We use the graph cut algorithms and extend them to obtain a segmentation of the biological cells. We introduce an utterly new algorithm for analysis of the resulting data and for sorting them into three main categories, which correspond to the biological cell death, based on the mathematical properties of the segmented elements.

Mots clés

  • graph cuts
  • segmentation
  • cell analysis
  • apoptosis
  • necrosis
  • computer morphometry

MSC 2010

  • 92C73
  • 92C55
  • 94A08
11 Articles
Accès libre

Parameter Estimation for the Forest Fire Propagation Model

Publié en ligne: 24 Apr 2020
Pages: 1 - 22

Résumé

Abstract

This paper first gives a brief overview of the Lagrangian forest fire propagation model [Ambroz, M.—Balažovjech, M.—Medl’a, M.—Mikula, K.: Numerical modeling of wildland surface fire propagation by evolving surface curves, Adv. Comput. Math. 45 (2019), no. 2, 1067–1103], which we apply to grass-field areas. Then, we aim to estimate the optimal model parameters. To achieve this goal, we use data assimilation of the measured data. From the data, we are able to estimate the normal velocity of the fire front (rate of spread), dominant wind direction and selected model parameters. In the data assimilation process, we use the Hausdorff distance as well as the Mean Hausdorff distance as a criterion. Moreover, we predict the fire propagation in small time intervals.

Mots clés

  • data assimilation
  • parameter estimation
  • curve evolution
  • forest fire

MSC 2010

  • Primary: 35R01, 65M08
  • Secondary: 35R37, 92F05
Accès libre

Bayesian Estimate of Parameters for ARMA Model Forecasting

Publié en ligne: 24 Apr 2020
Pages: 23 - 32

Résumé

Abstract

This paper presents a Bayesian approach to finding the Bayes estimator of parameters for ARMA model forecasting under normal-gamma prior assumption with a quadratic loss function in mathematical expression. Obtaining the conditional posterior predictive density is based on the normal-gamma prior and the conditional predictive density, whereas its marginal conditional posterior predictive density is obtained using the conditional posterior predictive density. Furthermore, the Bayes estimator of parameters is derived from the marginal conditional posterior predictive density.

Mots clés

  • ARMA model
  • Bayes estimator
  • normal-gamma prior
  • quadratic loss function

MSC 2010

  • 62C10
Accès libre

Estimating the Domestic Short Rate in a Convergence Model of Interest Rates

Publié en ligne: 24 Apr 2020
Pages: 33 - 48

Résumé

Abstract

In this paper we study the convergence model of interest rates by Corzo and Schwartz. It models the situation when a country is going to enter a monetary union, for example the eurozone. We are interested in estimating the underlying short rate, which is a theoretical variable, not observed on the market. We use the procedure already employed for the Vasicek model to the eurozone data and for the case of a zero correlation we show that a similar procedure can be used also for the estimation of the domestic parameters and the short rate values. The assumption of the zero correlation allows us to simplify the optimization problem, but using simulations we show that our algorithm is robust to the specification of the correlation. It estimates the short rate with a high precision also in the original case of a nonzero correlation, as well as in the case of a dynamic correlation, when the correlation is modelled as a function of time. Finally, we use the algorithm to real market data and estimate the short rate before adoption of the euro currency in Slovakia, Estonia, Latvia and Lithuania.

Mots clés

  • interest rates
  • short rate
  • convergence model
  • calibration

MSC 2010

  • 91G30
  • 35K15
  • 62M20
Accès libre

Finite Volume Scheme for AMSS Model

Publié en ligne: 24 Apr 2020
Pages: 49 - 62

Résumé

Abstract

We propose a new finite volume numerical scheme for the approximation of the Affine Morphological Scale Space (AMSS) model. We derive the basic scheme and its iterative improvement. For both schemes, several numerical experiments using examples where the exact solution is known are presented. Then the numerical errors and experimental order of convergence of the proposed schemes is studied.

Mots clés

  • affine morphological scale space
  • image processing
  • finite volume scheme
  • experimental order of convergence

MSC 2010

  • Primary: 65M08
  • Secondary: 35K20
Accès libre

The Finite Element Method as a Tool to Solve the Oblique Derivative Boundary Value Problem in Geodesy

Publié en ligne: 24 Apr 2020
Pages: 63 - 80

Résumé

Abstract

In this paper, we propose a novel approach to approximate the solution of the Laplace equation with an oblique derivative boundary condition by the finite element method. We present and analyse diverse testing experiments to study its behaviour and convergence. Finally, the usefulness of this approach is demonstrated by using it to gravity field modelling, namely, to approximate the solution of a geodetic boundary value problem in Himalayas.

Mots clés

  • oblique derivative boundary value problem
  • finite element method

MSC 2010

  • 65N30
Accès libre

Efficient 3D Shape Registration by Using Distance Maps and Stochastic Gradient Descent Method

Publié en ligne: 24 Apr 2020
Pages: 81 - 102

Résumé

Abstract

This paper presents an efficient 3D shape registration by using distance maps and stochastic gradient descent method. The proposed algorithm aims to find the optimal affine transformation parameters (translation, scaling and rotation) that maps two distance maps to each other. These distance maps represent the shapes as an interface and we apply level sets methods to calculate the signed distance to these interfaces. To maximize the similarity between the two distance maps, we apply sum of squared difference (SSD) optimization and gradient descent methods to minimize it. To address the shortcomings of the standard gradient descent method, i.e., many iterations to compute the minimum, we implemented the stochastic gradient descent method. The outcome of these two methods are compared to show the advantages of using stochastic gradient descent method. In addition, we implement computational optimization’s such as parallelization to speed up the registration process.

Mots clés

  • distance map
  • stochastic gradient method
  • registration
  • parallelization
  • affine transformation
  • optimization

MSC 2010

  • 65K10
  • 35Q68
  • 65Y05
  • 65Y20
Accès libre

Macrophage Image Segmentation by Thresholding and Subjective Surface Method

Publié en ligne: 24 Apr 2020
Pages: 103 - 120

Résumé

Abstract

We introduce two level-set method approaches to segmentation of 2D macrophage images. The first one is based on the Otsu thresholding and the second one on the information entropy thresholding, both followed by the classical subjective surface (SUBSURF) method. Results of both methods are compared with the semi-automatic Lagrangian method in which the segmentation curve evolves along the edge of the macrophage and it is controlled by an expert user. We present the comparison of all three methods with respect to the Hausdorff distance of resulting segmentation curves and we compare also their perimeter and enclosed area. We show that accuracy of the automatic SUBSURF method is comparable to the results of the semi-automatic Lagrangian segmentation.

Mots clés

  • image processing
  • segmentation
  • level-set methods
  • finite volume methods
  • thresholding
  • macrophages

MSC 2010

  • 68U10
  • 35K61
  • 65M08
Accès libre

Necessary and Sufficient Conditions for Oscillatory and Asymptotic Behaviour of Solutions to Second-Order Nonlinear Neutral Differential Equations with Several Delays

Publié en ligne: 24 Apr 2020
Pages: 121 - 134

Résumé

Abstract

In this paper, necessary and sufficient conditions are obtained for oscillatory and asymptotic behavior of solutions to second-order nonlinear neutral delay differential equations of the form ddt[r(t)[ddt(x(t)+p(t)x(t-τ))]α]+i=1mqi(t)H(x(t-σi))=0fortt0>0,{d \over {dt}}\left[ {r\left( t \right){{\left[ {{d \over {dt}}\left( {x\left( t \right) + p\left( t \right)x\left( {t - \tau } \right)} \right)} \right]}^\alpha }} \right] + \sum\limits_{i = 1}^m {{q_i}\left( t \right)H\left( {x\left( {t - {\sigma _i}} \right)} \right) = 0\,\,\,{\rm{for}}\,t \ge {t_0} > 0,}

under the assumption (r(n))−1/αdη=∞. Our main tool is Lebesque’s dominated convergence theorem. Further, some illustrative examples showing the applicability of the new results are included.

Mots clés

  • oscillation
  • non-oscillation
  • neutral
  • delay
  • nonlinear
  • Lebesgue’s dominated convergence theorem
  • Banach’s contraction principle

MSC 2010

  • 34C10
  • 34C15
  • 35K40
Accès libre

Necessary and Sufficient Conditions for Oscillation of Second-Order Delay Differential Equations

Publié en ligne: 24 Apr 2020
Pages: 135 - 146

Résumé

Abstract

In this work, we obtain necessary and sufficient conditions for the oscillation of all solutions of second-order half-linear delay differential equation of the form (r(x)γ)(t)+q(t)xα(τ(t))=0{\left( {r{{\left( {x'} \right)}^\gamma }} \right)^\prime }\left( t \right) + q\left( t \right){x^\alpha }\left( {\tau \left( t \right)} \right) = 0

Under the assumption (r(n))−1/γdη=∞, we consider the two cases when γ > α and γ < α. Further, some illustrative examples showing applicability of the new results are included, and state an open problem.

Mots clés

  • oscillation
  • non-oscillation
  • delay
  • linear
  • Lebesgue’s dominated convergence theorem

MSC 2010

  • 34C10
  • 35K40
  • 34K11
Accès libre

3D Cell Image Segmentation by Modified Subjective Surface Method

Publié en ligne: 24 Apr 2020
Pages: 147 - 162

Résumé

Abstract

In this work, we focused on 3D image segmentation where the segmented surface is reconstructed by the use of 3D digital image information and information from thresholded 3D image in a local neighbourhood. To this end, we applied a mathematical model based on the level set formulation and numerical method which is based on the so-called reduced diamond cell approach. The segmentation approach is based on surface evolution governed by a nonlinear PDE, the modified subjective surface equation. This is done by defining the input to the edge detector function as the weighted sum of norm of presmoothed 3D image and norm of presmoothed thresholded 3D image in a local neighbourhood. For the numerical discretization, we used a semi-implicit finite volume scheme. The method was applied to real data representing 3D microscopy images of cell nuclei within the zebrafish pectoral fin.

Mots clés

  • image segmentation
  • subjective surface method
  • level set method
  • finite volume method
  • semi-implicit scheme
  • cell microscopy images
  • zebrafish

MSC 2010

  • 65M08
  • 35K61
  • 68U10
Accès libre

Analysis of Cell Death by Image Processing

Publié en ligne: 24 Apr 2020
Pages: 163 - 190

Résumé

Abstract

In this paper, we present a graph theoretical approach to image processing with focus on the analysis of the biological data. We use the graph cut algorithms and extend them to obtain a segmentation of the biological cells. We introduce an utterly new algorithm for analysis of the resulting data and for sorting them into three main categories, which correspond to the biological cell death, based on the mathematical properties of the segmented elements.

Mots clés

  • graph cuts
  • segmentation
  • cell analysis
  • apoptosis
  • necrosis
  • computer morphometry

MSC 2010

  • 92C73
  • 92C55
  • 94A08

Planifiez votre conférence à distance avec Sciendo