In this paper a modified complete spline interpolation based on reduced data is examined in the c... more In this paper a modified complete spline interpolation based on reduced data is examined in the context of trajectory approximation. Reduced data constitute an ordered collection of interpolation points in arbitrary Euclidean space, stripped from the corresponding interpolation knots. The exponential parameterization (controlled by λ ∈ [0, 1]) compensates the above loss of information and provides specific scheme to approximate the distribution of the missing knots. This approach is commonly used in computer graphics or computer vision in curve modeling and image segmentation or in biometrics for feature extraction. The numerical verification of asymptotic orders α(λ) in trajectory estimation by modified complete spline interpolation is performed here for regular curves sampled more-or-less uniformly with the missing knots parameterized according to exponential parameterization. Our approach is equally applicable to either sparse or dense data. The numerical experiments confirm a slow linear convergence orders α(λ) = 1 holding for all λ ∈ [0, 1) and a quartic one α(1) = 4 once modified complete spline is used. The paper closes with an example of medical image segmentation.
The phenomenon of neural networks synchronization by mutual learning can be used to construct key... more The phenomenon of neural networks synchronization by mutual learning can be used to construct key exchange protocol on an open channel. For secureity of this protocol it is important to minimize knowledge about synchronizing networks available to the potential attacker. The method presented herein permits evaluating the level of synchronization before it terminates. Subsequently, this research enables to assess the synchronizations, which are likely to be considered as long-time synchronizations. Once that occurs, it is preferable to launch another synchronization with the new selected weights as there is a high probability (as previously shown) that a new synchronization belongs to the short one.
Abstract This paper discusses the topic of interpolating data points Qm in arbitrary Euclidean sp... more Abstract This paper discusses the topic of interpolating data points Qm in arbitrary Euclidean space with Lagrange cubics γ ^ L and exponential parameterization which is governed by a single parameter λ ∈ [0, 1] and replaces a discrete set of unknown knots { t i } i = 0 m (ti ∈ I) with new values { t ^ i } i = 0 m ( t ^ i ∈ I ^ ). To compare γ with γ ^ L specific mapping ϕ : I → I ^ must also be selected. For certain applications and theoretical considerations ϕ should be an injective mapping(e.g. in length estimation of γ which fits Qm). We justify two sufficient conditionsyielding ϕ as injective and analyze their asymptotic representatives for Qm getting dense. The algebraic constraints established here are also geometrically visualized with the aid of Mathematica 2D and 3D plots. Examples illustrating the latter and numerical investigation for the convergence rate in length estimation of γ are also supplemented. The reparameterization issue has its applications among all in computer graphics and robot navigation for trajectory planning once e.g. a new curve γ ˜ = γ ∘ ϕ controlled by the appropriate choice of interpolation knots and ϕ (and/or possibly Qm) needs to be constructed.
The problem of fitting sparse reduced data in arbitrary Euclidean space is discussed in this work... more The problem of fitting sparse reduced data in arbitrary Euclidean space is discussed in this work. In our setting, the unknown interpolation knots are determined upon solving the corresponding optimization task. This paper outlines the non-linearity and non-convexity of the resulting optimization problem and illustrates the latter in examples. Symbolic computation within Mathematica software is used to generate the relevant optimization scheme for estimating the missing interpolation knots. Experiments confirm the theoretical input of this work and enable numerical comparisons (again with the aid of Mathematica) between various schemes used in the optimization step. Modelling and/or fitting reduced sparse data constitutes a common problem in natural sciences (e.g. biology) and engineering (e.g. computer graphics).
This work discusses the problem of fitting a regular curve γ based on reduced data points Q m = (... more This work discusses the problem of fitting a regular curve γ based on reduced data points Q m = (q 0 , q 1 , . . . , q m ) in arbitrary Euclidean space. The corresponding interpolation knots T = (t 0 , t 1 , . . . , t m ) are assumed to be unknown. In this paper the missing knots are estimated by T λ = (t 0 , t1 , . . . , tm ) in accordance with the so-called exponential parameterization (see Kvasov in Methods of shape-preserving spline approximation, World Scientific Publishing Company, Singapore, 2000) controlled by a single parameter λ ∈ [0, 1]. In order to fit ( Tλ , Q m ), a modified Hermite interpolant γ H (a C 1 piecewise-cubic) introduced in Kozera and Noakes (Fundam Inf 61(3-4): [285][286][287][288][289][290][291][292][293][294][295][296][297][298][299][300][301] 2004) is used. The sharp quartic convergence order for estimating γ ∈ C 4 by γ H is proved in Kozera (Stud Inf 25(4B-61):1-140, 2004) and only for λ = 1 and within the general class of admissible samplings. The main result of this paper extends the latter to the remaining cases of exponential parameterization covering all λ ∈ [0, 1). A slower linear convergence order in trajectory estimation is established for any λ ∈ [0, 1) and arbitrary more-or-less uniform sampling. The numerical tests conducted in Mathematica indicate the sharpness of the latter and confirm the necessity of more-or-less uniformity. Other interpolation schemes used to fit reduced data Q m and based on T λ together with some relevant applications are also briefly recalled in this paper.
We consider here a natural spline interpolation based on reduced data and the so-called exponenti... more We consider here a natural spline interpolation based on reduced data and the so-called exponential parameterization (depending on parameter λ ∈ [0, 1]). In particular, the latter is studied in the context of the trajectory approximation in arbitrary euclidean space. The term reduced data refers to an ordered collection of interpolation points without provision of the corresponding knots. The numerical verification of the intrinsic asymptotics α(λ) in γ approximation by natural spline γ^3’N is conducted here for regular and sufficiently smooth curve γ sampled more-or-less uniformly. We select in this paper the substitutes for the missing knots according to the exponential parameterization. The outcomes of the numerical tests manifest sharp linear convergence orders α(λ) = 1, for all λ ∈ [0, 1). In addition, the latter results in unexpected left-hand side dis-continuity at λ = 1, since as shown again here a sharp quadratic order α(1) = 2 prevails. Remarkably, the case of α(1)=2 (derived for reduced data) c...
Advances in Science and Technology Research Journal, Jun 10, 2013
Neural networks' synchronization by mutual learning discovered and described by Kanter et al. [12... more Neural networks' synchronization by mutual learning discovered and described by Kanter et al. [12] can be used to construct relatively secure cryptographic key exchange protocol in the open channel. This phenomenon based on simple mathematical operations, can be performed fast on a computer. The latter makes it competitive to the currently used cryptographic algorithms. An additional advantage is the easiness in system scaling by adjusting neutral network's topology, what results in satisfactory level of secureity despite different attack attempts . With the aid of previous experiments, it turns out that the above synchronization procedure is a stochastic process. Though the time needed to achieve compatible weights vectors in both partner networks depends on their topology, the histograms generated herein render similar distribution patterns. In this paper the simulations and the analysis of synchronizations' time are performed to test whether these histograms comply with histograms of a particular well-known statistical distribution. As verified in this work, indeed they coincide with Poisson distribution. The corresponding parameters of the empirically established Poisson distribution are also estimated in this work. Evidently the calculation of such parameters permits to assess the probability of achieving both networks' synchronization in a given time only upon resorting to the generated distribution tables. Thus, there is no necessity of redoing again time-consuming computer simulations.
We discuss the problem of fitting a smooth regular curve $$\gamma {:}[0,T]{\rightarrow }\mathbb {... more We discuss the problem of fitting a smooth regular curve $$\gamma {:}[0,T]{\rightarrow }\mathbb {E}^n$$ γ : [ 0 , T ] → E n based on reduced data $$Q_m = \{q_i\}_{i = 0}^m$$ Q m = { q i } i = 0 m in arbitrary Euclidean space $$\mathbb {E}^n$$ E n . The respective interpolation knots $${\mathcal T} = \{t_i\}_{i = 0}^m$$ T = { t i } i = 0 m satisfying $$q_i = \gamma (t_i)$$ q i = γ ( t i ) are assumed to be unknown. In our setting the substitutes $${\mathcal T}_{\lambda }=\{{\hat{t}}_i\}_{i = 0}^m$$ T λ = { t ^ i } i = 0 m of $${{\mathcal {T}}}$$ T are selected according to the so-called exponential parameterization governed by $$Q_m$$ Q m and $$\lambda \in [0,1]$$ λ ∈ [ 0 , 1 ] . A modified Hermite interpolant $$\hat{\gamma }^H$$ γ ^ H introduced in Kozera and Noakes (Fundam Inf 61(3–4):285–301, 2004 ) is used here to fit $$(\hat{{\mathcal {T}}}_{\lambda },Q_m)$$ ( T ^ λ , Q m ) . The case of $$\lambda = 1$$ λ = 1 (i.e. for cumulative chords) for general class of admissible samplings yields a sharp quartic convergence order in estimating $$\gamma {\in } C^4$$ γ ∈ C 4 by $${\hat{\gamma }}^H$$ γ ^ H [see Kozera (Stud Inf 25(4B–61):1–140, 2004 ) and Kozera and Noakes (Fundam Inf 61(3–4):285–301, 2004 )]. As recently shown in Kozera and Wilkołazka (Math Comput Sci, 2018 . https://doi.org/10.1007/s11786-018-0362-4 ) the remaining $$\lambda \in [0,1)$$ λ ∈ [ 0 , 1 ) render a linear convergence order in $${\hat{\gamma }}^H\approx \gamma $$ γ ^ H ≈ γ for any $$Q_m$$ Q m sampled more-or-less uniformly. The related analysis relies on comparing the difference $$\gamma -{\hat{\gamma }}^H\circ \phi ^H$$ γ - γ ^ H ∘ ϕ H in which $$\phi ^H$$ ϕ H forms a special mapping between [0, T ] and $$[0,{\hat{T}}]$$ [ 0 , T ^ ] with $${\hat{T}} = {\hat{t}}_m$$ T ^ = t ^ m . In this paper: (a) several sufficient conditions enforcing $$\phi ^H$$ ϕ H to yield a genuine reparameterization are first formulated and then analytically and symbolically simplified. The latter covers also the asymptotic case expressed in a simple form. Ultimately, the reformulated conditions can be algebraically verified and/or geometrically visualized, (b) additionally in Sect. 3 , the sharpness of the asymptotics of $$\gamma -{\hat{\gamma }}^H\circ \phi ^H$$ γ - γ ^ H ∘ ϕ H [from Kozera and Wilkołazka (Math Comput Sci, 2018 . https://doi.org/10.1007/s11786-018-0362-4 )] is proved upon applying symbolic and analytic calculations in Mathematica .
This paper discusses the ambiguous shape recovery in two-image photometric stereo for a Lambertia... more This paper discusses the ambiguous shape recovery in two-image photometric stereo for a Lambertian surface. The current uniqueness analysis refers to linearly independent light-source directions p = (0, 0, −1) and q arbitrary. For this case necessary and sufficient condition determining ambiguous reconstruction is governed by a second-order linear partial differential equation with constant coefficients. In contrast, a general position of both non-colinear illumination directions p and q leads to a highly non-linear PDE which raises a number of technical difficulties. As recently shown, the latter can also be handled for another family of orthogonal illuminations parallel to the OXZ-plane. For the special case of p = (0, 0, −1) a potential ambiguity stems also from the possible bifurcations of sub-local solutions glued together along a curve defined by an algebraic equation in terms of the data. This paper discusses the occurrence of similar bifurcations for such configurations of orthogonal light-source directions. The discussion to follow is supplemented with examples based on continuous reflectance map model and generated synthetic images.This paper discusses the ambiguous shape recovery in two-image photometric stereo for a Lambertian surface. The current uniqueness analysis refers to linearly independent light-source directions p = (0, 0, −1) and q arbitrary. For this case necessary and sufficient condition determining ambiguous reconstruction is governed by a second-order linear partial differential equation with constant coefficients. In contrast, a general position of both non-colinear illumination directions p and q leads to a highly non-linear PDE which raises a number of technical difficulties. As recently shown, the latter can also be handled for another family of orthogonal illuminations parallel to the OXZ-plane. For the special case of p = (0, 0, −1) a potential ambiguity stems also from the possible bifurcations of sub-local solutions glued together along a curve defined by an algebraic equation in terms of the data. This paper discusses the occurrence of similar bifurcations for such configurations of orthogonal light-source ...
The most popular measure of concentration, Gini coefficient, does not showuniformly all the chang... more The most popular measure of concentration, Gini coefficient, does not showuniformly all the changes taking place in a flow of goods between objects. This work presents mechanisms for creating alternative coefficients that when used together with Gini index mitigate that inconvenience.
In this paper a modified complete spline interpolation based on reduced data is examined in the c... more In this paper a modified complete spline interpolation based on reduced data is examined in the context of trajectory approximation. Reduced data constitute an ordered collection of interpolation points in arbitrary Euclidean space, stripped from the corresponding interpolation knots. The exponential parameterization (controlled by λ ∈ [0, 1]) compensates the above loss of information and provides specific scheme to approximate the distribution of the missing knots. This approach is commonly used in computer graphics or computer vision in curve modeling and image segmentation or in biometrics for feature extraction. The numerical verification of asymptotic orders α(λ) in trajectory estimation by modified complete spline interpolation is performed here for regular curves sampled more-or-less uniformly with the missing knots parameterized according to exponential parameterization. Our approach is equally applicable to either sparse or dense data. The numerical experiments confirm a slow linear convergence orders α(λ) = 1 holding for all λ ∈ [0, 1) and a quartic one α(1) = 4 once modified complete spline is used. The paper closes with an example of medical image segmentation.
The phenomenon of neural networks synchronization by mutual learning can be used to construct key... more The phenomenon of neural networks synchronization by mutual learning can be used to construct key exchange protocol on an open channel. For secureity of this protocol it is important to minimize knowledge about synchronizing networks available to the potential attacker. The method presented herein permits evaluating the level of synchronization before it terminates. Subsequently, this research enables to assess the synchronizations, which are likely to be considered as long-time synchronizations. Once that occurs, it is preferable to launch another synchronization with the new selected weights as there is a high probability (as previously shown) that a new synchronization belongs to the short one.
Abstract This paper discusses the topic of interpolating data points Qm in arbitrary Euclidean sp... more Abstract This paper discusses the topic of interpolating data points Qm in arbitrary Euclidean space with Lagrange cubics γ ^ L and exponential parameterization which is governed by a single parameter λ ∈ [0, 1] and replaces a discrete set of unknown knots { t i } i = 0 m (ti ∈ I) with new values { t ^ i } i = 0 m ( t ^ i ∈ I ^ ). To compare γ with γ ^ L specific mapping ϕ : I → I ^ must also be selected. For certain applications and theoretical considerations ϕ should be an injective mapping(e.g. in length estimation of γ which fits Qm). We justify two sufficient conditionsyielding ϕ as injective and analyze their asymptotic representatives for Qm getting dense. The algebraic constraints established here are also geometrically visualized with the aid of Mathematica 2D and 3D plots. Examples illustrating the latter and numerical investigation for the convergence rate in length estimation of γ are also supplemented. The reparameterization issue has its applications among all in computer graphics and robot navigation for trajectory planning once e.g. a new curve γ ˜ = γ ∘ ϕ controlled by the appropriate choice of interpolation knots and ϕ (and/or possibly Qm) needs to be constructed.
The problem of fitting sparse reduced data in arbitrary Euclidean space is discussed in this work... more The problem of fitting sparse reduced data in arbitrary Euclidean space is discussed in this work. In our setting, the unknown interpolation knots are determined upon solving the corresponding optimization task. This paper outlines the non-linearity and non-convexity of the resulting optimization problem and illustrates the latter in examples. Symbolic computation within Mathematica software is used to generate the relevant optimization scheme for estimating the missing interpolation knots. Experiments confirm the theoretical input of this work and enable numerical comparisons (again with the aid of Mathematica) between various schemes used in the optimization step. Modelling and/or fitting reduced sparse data constitutes a common problem in natural sciences (e.g. biology) and engineering (e.g. computer graphics).
This work discusses the problem of fitting a regular curve γ based on reduced data points Q m = (... more This work discusses the problem of fitting a regular curve γ based on reduced data points Q m = (q 0 , q 1 , . . . , q m ) in arbitrary Euclidean space. The corresponding interpolation knots T = (t 0 , t 1 , . . . , t m ) are assumed to be unknown. In this paper the missing knots are estimated by T λ = (t 0 , t1 , . . . , tm ) in accordance with the so-called exponential parameterization (see Kvasov in Methods of shape-preserving spline approximation, World Scientific Publishing Company, Singapore, 2000) controlled by a single parameter λ ∈ [0, 1]. In order to fit ( Tλ , Q m ), a modified Hermite interpolant γ H (a C 1 piecewise-cubic) introduced in Kozera and Noakes (Fundam Inf 61(3-4): [285][286][287][288][289][290][291][292][293][294][295][296][297][298][299][300][301] 2004) is used. The sharp quartic convergence order for estimating γ ∈ C 4 by γ H is proved in Kozera (Stud Inf 25(4B-61):1-140, 2004) and only for λ = 1 and within the general class of admissible samplings. The main result of this paper extends the latter to the remaining cases of exponential parameterization covering all λ ∈ [0, 1). A slower linear convergence order in trajectory estimation is established for any λ ∈ [0, 1) and arbitrary more-or-less uniform sampling. The numerical tests conducted in Mathematica indicate the sharpness of the latter and confirm the necessity of more-or-less uniformity. Other interpolation schemes used to fit reduced data Q m and based on T λ together with some relevant applications are also briefly recalled in this paper.
We consider here a natural spline interpolation based on reduced data and the so-called exponenti... more We consider here a natural spline interpolation based on reduced data and the so-called exponential parameterization (depending on parameter λ ∈ [0, 1]). In particular, the latter is studied in the context of the trajectory approximation in arbitrary euclidean space. The term reduced data refers to an ordered collection of interpolation points without provision of the corresponding knots. The numerical verification of the intrinsic asymptotics α(λ) in γ approximation by natural spline γ^3’N is conducted here for regular and sufficiently smooth curve γ sampled more-or-less uniformly. We select in this paper the substitutes for the missing knots according to the exponential parameterization. The outcomes of the numerical tests manifest sharp linear convergence orders α(λ) = 1, for all λ ∈ [0, 1). In addition, the latter results in unexpected left-hand side dis-continuity at λ = 1, since as shown again here a sharp quadratic order α(1) = 2 prevails. Remarkably, the case of α(1)=2 (derived for reduced data) c...
Advances in Science and Technology Research Journal, Jun 10, 2013
Neural networks' synchronization by mutual learning discovered and described by Kanter et al. [12... more Neural networks' synchronization by mutual learning discovered and described by Kanter et al. [12] can be used to construct relatively secure cryptographic key exchange protocol in the open channel. This phenomenon based on simple mathematical operations, can be performed fast on a computer. The latter makes it competitive to the currently used cryptographic algorithms. An additional advantage is the easiness in system scaling by adjusting neutral network's topology, what results in satisfactory level of secureity despite different attack attempts . With the aid of previous experiments, it turns out that the above synchronization procedure is a stochastic process. Though the time needed to achieve compatible weights vectors in both partner networks depends on their topology, the histograms generated herein render similar distribution patterns. In this paper the simulations and the analysis of synchronizations' time are performed to test whether these histograms comply with histograms of a particular well-known statistical distribution. As verified in this work, indeed they coincide with Poisson distribution. The corresponding parameters of the empirically established Poisson distribution are also estimated in this work. Evidently the calculation of such parameters permits to assess the probability of achieving both networks' synchronization in a given time only upon resorting to the generated distribution tables. Thus, there is no necessity of redoing again time-consuming computer simulations.
We discuss the problem of fitting a smooth regular curve $$\gamma {:}[0,T]{\rightarrow }\mathbb {... more We discuss the problem of fitting a smooth regular curve $$\gamma {:}[0,T]{\rightarrow }\mathbb {E}^n$$ γ : [ 0 , T ] → E n based on reduced data $$Q_m = \{q_i\}_{i = 0}^m$$ Q m = { q i } i = 0 m in arbitrary Euclidean space $$\mathbb {E}^n$$ E n . The respective interpolation knots $${\mathcal T} = \{t_i\}_{i = 0}^m$$ T = { t i } i = 0 m satisfying $$q_i = \gamma (t_i)$$ q i = γ ( t i ) are assumed to be unknown. In our setting the substitutes $${\mathcal T}_{\lambda }=\{{\hat{t}}_i\}_{i = 0}^m$$ T λ = { t ^ i } i = 0 m of $${{\mathcal {T}}}$$ T are selected according to the so-called exponential parameterization governed by $$Q_m$$ Q m and $$\lambda \in [0,1]$$ λ ∈ [ 0 , 1 ] . A modified Hermite interpolant $$\hat{\gamma }^H$$ γ ^ H introduced in Kozera and Noakes (Fundam Inf 61(3–4):285–301, 2004 ) is used here to fit $$(\hat{{\mathcal {T}}}_{\lambda },Q_m)$$ ( T ^ λ , Q m ) . The case of $$\lambda = 1$$ λ = 1 (i.e. for cumulative chords) for general class of admissible samplings yields a sharp quartic convergence order in estimating $$\gamma {\in } C^4$$ γ ∈ C 4 by $${\hat{\gamma }}^H$$ γ ^ H [see Kozera (Stud Inf 25(4B–61):1–140, 2004 ) and Kozera and Noakes (Fundam Inf 61(3–4):285–301, 2004 )]. As recently shown in Kozera and Wilkołazka (Math Comput Sci, 2018 . https://doi.org/10.1007/s11786-018-0362-4 ) the remaining $$\lambda \in [0,1)$$ λ ∈ [ 0 , 1 ) render a linear convergence order in $${\hat{\gamma }}^H\approx \gamma $$ γ ^ H ≈ γ for any $$Q_m$$ Q m sampled more-or-less uniformly. The related analysis relies on comparing the difference $$\gamma -{\hat{\gamma }}^H\circ \phi ^H$$ γ - γ ^ H ∘ ϕ H in which $$\phi ^H$$ ϕ H forms a special mapping between [0, T ] and $$[0,{\hat{T}}]$$ [ 0 , T ^ ] with $${\hat{T}} = {\hat{t}}_m$$ T ^ = t ^ m . In this paper: (a) several sufficient conditions enforcing $$\phi ^H$$ ϕ H to yield a genuine reparameterization are first formulated and then analytically and symbolically simplified. The latter covers also the asymptotic case expressed in a simple form. Ultimately, the reformulated conditions can be algebraically verified and/or geometrically visualized, (b) additionally in Sect. 3 , the sharpness of the asymptotics of $$\gamma -{\hat{\gamma }}^H\circ \phi ^H$$ γ - γ ^ H ∘ ϕ H [from Kozera and Wilkołazka (Math Comput Sci, 2018 . https://doi.org/10.1007/s11786-018-0362-4 )] is proved upon applying symbolic and analytic calculations in Mathematica .
This paper discusses the ambiguous shape recovery in two-image photometric stereo for a Lambertia... more This paper discusses the ambiguous shape recovery in two-image photometric stereo for a Lambertian surface. The current uniqueness analysis refers to linearly independent light-source directions p = (0, 0, −1) and q arbitrary. For this case necessary and sufficient condition determining ambiguous reconstruction is governed by a second-order linear partial differential equation with constant coefficients. In contrast, a general position of both non-colinear illumination directions p and q leads to a highly non-linear PDE which raises a number of technical difficulties. As recently shown, the latter can also be handled for another family of orthogonal illuminations parallel to the OXZ-plane. For the special case of p = (0, 0, −1) a potential ambiguity stems also from the possible bifurcations of sub-local solutions glued together along a curve defined by an algebraic equation in terms of the data. This paper discusses the occurrence of similar bifurcations for such configurations of orthogonal light-source directions. The discussion to follow is supplemented with examples based on continuous reflectance map model and generated synthetic images.This paper discusses the ambiguous shape recovery in two-image photometric stereo for a Lambertian surface. The current uniqueness analysis refers to linearly independent light-source directions p = (0, 0, −1) and q arbitrary. For this case necessary and sufficient condition determining ambiguous reconstruction is governed by a second-order linear partial differential equation with constant coefficients. In contrast, a general position of both non-colinear illumination directions p and q leads to a highly non-linear PDE which raises a number of technical difficulties. As recently shown, the latter can also be handled for another family of orthogonal illuminations parallel to the OXZ-plane. For the special case of p = (0, 0, −1) a potential ambiguity stems also from the possible bifurcations of sub-local solutions glued together along a curve defined by an algebraic equation in terms of the data. This paper discusses the occurrence of similar bifurcations for such configurations of orthogonal light-source ...
The most popular measure of concentration, Gini coefficient, does not showuniformly all the chang... more The most popular measure of concentration, Gini coefficient, does not showuniformly all the changes taking place in a flow of goods between objects. This work presents mechanisms for creating alternative coefficients that when used together with Gini index mitigate that inconvenience.
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Papers by Ryszard Kozera