Pub Date : 2021-01-01DOI: 10.33039/ami.2021.10.002
F. Laudano
In this paper, we study the possibility of building a learning path that allows students to develop trigonometric knowledge and skills by the end of Grade 10 of secondary science-based schools. In particular, we describe an action research experiment, in part done through distance learning, aimed at incorporating all trigonometry topics within the framework of the study of Euclidean geometry. The inquiry-based learning methodology and the support of dynamic geometry software with a laboratory teaching approach were used. The learning path is based on several “visual/dynamic proof” and is explained by an example lesson on the Cosines Law. The experiment could be extended by teachers into physical/virtual classrooms and could offer practical strategies and tools for teaching trigonometry.
{"title":"Visual argumentations in teaching trigonometry","authors":"F. Laudano","doi":"10.33039/ami.2021.10.002","DOIUrl":"https://doi.org/10.33039/ami.2021.10.002","url":null,"abstract":"In this paper, we study the possibility of building a learning path that allows students to develop trigonometric knowledge and skills by the end of Grade 10 of secondary science-based schools. In particular, we describe an action research experiment, in part done through distance learning, aimed at incorporating all trigonometry topics within the framework of the study of Euclidean geometry. The inquiry-based learning methodology and the support of dynamic geometry software with a laboratory teaching approach were used. The learning path is based on several “visual/dynamic proof” and is explained by an example lesson on the Cosines Law. The experiment could be extended by teachers into physical/virtual classrooms and could offer practical strategies and tools for teaching trigonometry.","PeriodicalId":43454,"journal":{"name":"Annales Mathematicae et Informaticae","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"86567513","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.33039/ami.2021.07.001
Anikó Vágner
GTFS (General Transit Feed Specification) is a standard of Google for public transportation schedules. The specification describes stops, routes, dates, trips, etc. of one or more public transportation company for a city or a country. Examining a GTFS feed it can be considered as a graph. In addition in the last decades new database management systems was born in order to support the big data era and/or help to write program codes. Their collective name is the NoSQL databases, which covers many types of database systems. One type of them is the graph databases, from which the Neo4j is the most widespread. In this paper I try to find the answer for the question how the Neo4j can support the usage of the GTFS. The most obvious usage of the GTFS is the route planning for which the Neo4j offers some algorithms. I built some storage structures on which the tools provided by Neo4j can be effectively used to plan routes on GTFS data.
{"title":"Route planning on GTFS using Neo4j","authors":"Anikó Vágner","doi":"10.33039/ami.2021.07.001","DOIUrl":"https://doi.org/10.33039/ami.2021.07.001","url":null,"abstract":"GTFS (General Transit Feed Specification) is a standard of Google for public transportation schedules. The specification describes stops, routes, dates, trips, etc. of one or more public transportation company for a city or a country. Examining a GTFS feed it can be considered as a graph. In addition in the last decades new database management systems was born in order to support the big data era and/or help to write program codes. Their collective name is the NoSQL databases, which covers many types of database systems. One type of them is the graph databases, from which the Neo4j is the most widespread. In this paper I try to find the answer for the question how the Neo4j can support the usage of the GTFS. The most obvious usage of the GTFS is the route planning for which the Neo4j offers some algorithms. I built some storage structures on which the tools provided by Neo4j can be effectively used to plan routes on GTFS data.","PeriodicalId":43454,"journal":{"name":"Annales Mathematicae et Informaticae","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84746153","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.33039/AMI.2021.03.014
S. Rihane, A. Togbé
In this paper, we find all the Padovan and Perrin numbers which are Pell or Pell-Lucas numbers.
在本文中，我们找到了所有属于Pell或Pell- lucas数的Padovan和Perrin数。
{"title":"On the intersection of Padovan, Perrin sequences and Pell, Pell-Lucas sequences","authors":"S. Rihane, A. Togbé","doi":"10.33039/AMI.2021.03.014","DOIUrl":"https://doi.org/10.33039/AMI.2021.03.014","url":null,"abstract":"In this paper, we find all the Padovan and Perrin numbers which are Pell or Pell-Lucas numbers.","PeriodicalId":43454,"journal":{"name":"Annales Mathematicae et Informaticae","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80593117","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.33039/AMI.2021.04.005
N. X. Tho
This paper shows that the equation in the title does not have positive integer solutions when n is divisible by 4. This gives a partial answer to a question by Melvyn Knight. The proof is a mixture of elementary p-adic analysis and elliptic curve theory.
{"title":"What positive integers n can be presented in the form n = (x + y + z)(1/x + 1/y + 1/z)?","authors":"N. X. Tho","doi":"10.33039/AMI.2021.04.005","DOIUrl":"https://doi.org/10.33039/AMI.2021.04.005","url":null,"abstract":"This paper shows that the equation in the title does not have positive integer solutions when n is divisible by 4. This gives a partial answer to a question by Melvyn Knight. The proof is a mixture of elementary p-adic analysis and elliptic curve theory.","PeriodicalId":43454,"journal":{"name":"Annales Mathematicae et Informaticae","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"76332545","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.33039/AMI.2021.02.001
S. Almousa, M. Telek
This paper presents numerical methods for finding high order concentrated matrix-exponential (ME) distributions, whose squared coefficient of variation (SCV) is very low. Due to the absence of symbolic construction to obtain the most concentrated ME distributions, non-linear optimization problems are defined to obtain high order concentrated matrix-exponential (CME) distributions . The number of parameters to optimize increases with the order in the “full” version of the optimization problem. For orders, where “full” optimization is infeasible (n > 184), a “heuristic” optimization procedure, optimizing only 3 parameters independent of the order, was proposed in [6]. In this work we present an enhanced version of this heuristic optimization procedure, optimizing only 6 parameters independent of the order, which results in CME distributions with lower SCV than the existing 3-parameter method. The SCV gain of the new procedure compared to the old one is ∗This work is partially supported by the OTKA K-123914 and the NKFIH BME NC TKP2020 projects. Annales Mathematicae et Informaticae 53 (2021) pp. 5–19 doi: https://doi.org/10.33039/ami.2021.02.001 url: https://ami.uni-eszterhazy.hu
{"title":"Enhanced optimization of high order concentrated matrix-exponential distributions","authors":"S. Almousa, M. Telek","doi":"10.33039/AMI.2021.02.001","DOIUrl":"https://doi.org/10.33039/AMI.2021.02.001","url":null,"abstract":"This paper presents numerical methods for finding high order concentrated matrix-exponential (ME) distributions, whose squared coefficient of variation (SCV) is very low. Due to the absence of symbolic construction to obtain the most concentrated ME distributions, non-linear optimization problems are defined to obtain high order concentrated matrix-exponential (CME) distributions . The number of parameters to optimize increases with the order in the “full” version of the optimization problem. For orders, where “full” optimization is infeasible (n > 184), a “heuristic” optimization procedure, optimizing only 3 parameters independent of the order, was proposed in [6]. In this work we present an enhanced version of this heuristic optimization procedure, optimizing only 6 parameters independent of the order, which results in CME distributions with lower SCV than the existing 3-parameter method. The SCV gain of the new procedure compared to the old one is ∗This work is partially supported by the OTKA K-123914 and the NKFIH BME NC TKP2020 projects. Annales Mathematicae et Informaticae 53 (2021) pp. 5–19 doi: https://doi.org/10.33039/ami.2021.02.001 url: https://ami.uni-eszterhazy.hu","PeriodicalId":43454,"journal":{"name":"Annales Mathematicae et Informaticae","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75172971","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.33039/AMI.2021.03.013
Oktavian Lantang, G. Terdik, A. Hajdu, Attila Tiba
In this work, we investigated the ability of several Convolutional Neural Network (CNN) models for predicting the spread of cancer using medical images. We used a dataset released by the Kaggle, namely PatchCamelyon. The dataset consists of 220,025 pathology images digitized by a tissue scanner. A clinical expert labeled each image as cancerous or non-cancerous. We used 70% of the images as a training set and 30% of them as a validation set. We design three models based on three commonly used modules: VGG, Inception, and Residual Network (ResNet), to develop an ensemble model and implement a voting system to determine the final decision. Then, we compared the performance of this ensemble model to the performance of each single model. Additionally, we used a weighted majority voting system, where the final prediction is equal to the weighted average of the prediction produced by each network. Our results show that the classification of the two ensemble models reaches 96%. Thus these results prove that the ensemble model outperforms single network architectures.
{"title":"Comparison of single and ensemble-based convolutional neural networks for cancerous image classification","authors":"Oktavian Lantang, G. Terdik, A. Hajdu, Attila Tiba","doi":"10.33039/AMI.2021.03.013","DOIUrl":"https://doi.org/10.33039/AMI.2021.03.013","url":null,"abstract":"In this work, we investigated the ability of several Convolutional Neural Network (CNN) models for predicting the spread of cancer using medical images. We used a dataset released by the Kaggle, namely PatchCamelyon. The dataset consists of 220,025 pathology images digitized by a tissue scanner. A clinical expert labeled each image as cancerous or non-cancerous. We used 70% of the images as a training set and 30% of them as a validation set. We design three models based on three commonly used modules: VGG, Inception, and Residual Network (ResNet), to develop an ensemble model and implement a voting system to determine the final decision. Then, we compared the performance of this ensemble model to the performance of each single model. Additionally, we used a weighted majority voting system, where the final prediction is equal to the weighted average of the prediction produced by each network. Our results show that the classification of the two ensemble models reaches 96%. Thus these results prove that the ensemble model outperforms single network architectures.","PeriodicalId":43454,"journal":{"name":"Annales Mathematicae et Informaticae","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84455968","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.33039/AMI.2021.03.004
V. Skala
The Taylor expansion [19] is used in many applications for a value estimation of scalar functions of one or two variables in the neighbour point. Usually, only the first two elements of the Taylor expansion are used, i.e. a value in the given point and derivatives estimation. The Taylor expansion can be also used for vector functions, too. The usual formulae are well known, but if the second element of the expansion, i.e. with the second derivatives are to be used, mathematical formulations are getting too complex for efficient programming, as it leads to the use of multi-dimensional matrices. This contribution describes a new form of the Taylor expansion for multidimensional vector functions. The proposed approach uses "standard" formalism of linear algebra, i.e. using vectors and matrices, which is simple, easy to implement. It leads to efficient computation on the GPU in the three dimensional case, as the GPU offers fast vector-vector computation and many parts can be done in parallel.
{"title":"Efficient Taylor expansion computation of multidimensional vector functions on GPU","authors":"V. Skala","doi":"10.33039/AMI.2021.03.004","DOIUrl":"https://doi.org/10.33039/AMI.2021.03.004","url":null,"abstract":"The Taylor expansion [19] is used in many applications for a value estimation of scalar functions of one or two variables in the neighbour point. Usually, only the first two elements of the Taylor expansion are used, i.e. a value in the given point and derivatives estimation. The Taylor expansion can be also used for vector functions, too. The usual formulae are well known, but if the second element of the expansion, i.e. with the second derivatives are to be used, mathematical formulations are getting too complex for efficient programming, as it leads to the use of multi-dimensional matrices. This contribution describes a new form of the Taylor expansion for multidimensional vector functions. The proposed approach uses \"standard\" formalism of linear algebra, i.e. using vectors and matrices, which is simple, easy to implement. It leads to efficient computation on the GPU in the three dimensional case, as the GPU offers fast vector-vector computation and many parts can be done in parallel.","PeriodicalId":43454,"journal":{"name":"Annales Mathematicae et Informaticae","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80514604","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.33039/AMI.2021.03.009
Gábor Kusper, C. Biró, A. Adamkó, Imre Baják
In this paper we generalize the well-known notions of Horn and 𝑞 -Horn formulae. A Horn clause, by definition, contains at most one positive literal. A Horn formula contains only Horn clauses. We generalize these notions as follows. A clause is a 𝑤 -Horn clause if and only if it contains at least one negative literal or it is a unit or it is the empty clause. A formula is a 𝑤 -Horn formula if it contains only 𝑤 -Horn clauses after exhaustive unit propagation, i.e., after a Boolean Constraint Propagation (BCP) step. We show that the set of 𝑤 -Horn formulae properly includes the set of Horn formulae. A function 𝛽 ( 𝑥 ) is a valuation function if 𝛽 ( 𝑥 ) + 𝛽 ( ¬ 𝑥 ) = 1 and 𝛽 ( 𝑥 ) ∈ { 0 , 0 . 5 , 1 } , where 𝑥 is a Boolean variable. A formula ℱ is a 𝑞 -Horn formula if and only if there is a valuation function 𝛽 ( 𝑥 ) such that for each clause 𝐶 in ℱ we have that ∑︀ 𝑥 ∈ 𝐶 𝛽 ( 𝑥 ) ≤ 1 . In this case we call 𝛽 ( 𝑥 ) a 𝑞 -feasible valuation for ℱ . In other words, a formula is 𝑞 -Horn if and only if each clause in it contains at most one “positive” literal (where 𝛽 ( 𝑥 ) = 1 ) or at most two half ones (where 𝛽 ( 𝑥 ) = 0 . 5 ). We generalize these notions as follows. A formula ℱ is a 𝑧 -Horn formula if and only if ℱ ′ = BCP ( ℱ ) and either ℱ ′ is trivially satisfiable or trivially unsatisfiable or there is a valuation function 𝛾 ( 𝑥 ) such that for each clause 𝒞 in ℱ ′ we have that ( 1 or ∑︀ 𝑥 ∈ 𝐶 ∧ 𝛾 ( 𝑥 )=0 . 5 𝛾 ( 𝑥 ) = 1 . In this case we call 𝛾 ( 𝑥 ) to be a 𝑧 -feasible valuation for 𝐹 ′ . In other words, a formula is 𝑧 -Horn if and only if each clause in it after a BCP step contains at least one “negative” literal (where 𝛾 ( 𝑥 ) = 0 ) or exactly two half ones (where 𝛾 ( 𝑥 ) = 0 . 5 ). We show that the set of 𝑧 -Horn formulae properly includes the set of 𝑞 -Horn formulae. We also show that the 𝑤 -Horn SAT problem can be decided in polynomial time. We also show that each satisfiable formula is 𝑧
{"title":"Introducing w-Horn and z-Horn: A generalization of Horn and q-Horn formulae","authors":"Gábor Kusper, C. Biró, A. Adamkó, Imre Baják","doi":"10.33039/AMI.2021.03.009","DOIUrl":"https://doi.org/10.33039/AMI.2021.03.009","url":null,"abstract":"In this paper we generalize the well-known notions of Horn and 𝑞 -Horn formulae. A Horn clause, by definition, contains at most one positive literal. A Horn formula contains only Horn clauses. We generalize these notions as follows. A clause is a 𝑤 -Horn clause if and only if it contains at least one negative literal or it is a unit or it is the empty clause. A formula is a 𝑤 -Horn formula if it contains only 𝑤 -Horn clauses after exhaustive unit propagation, i.e., after a Boolean Constraint Propagation (BCP) step. We show that the set of 𝑤 -Horn formulae properly includes the set of Horn formulae. A function 𝛽 ( 𝑥 ) is a valuation function if 𝛽 ( 𝑥 ) + 𝛽 ( ¬ 𝑥 ) = 1 and 𝛽 ( 𝑥 ) ∈ { 0 , 0 . 5 , 1 } , where 𝑥 is a Boolean variable. A formula ℱ is a 𝑞 -Horn formula if and only if there is a valuation function 𝛽 ( 𝑥 ) such that for each clause 𝐶 in ℱ we have that ∑︀ 𝑥 ∈ 𝐶 𝛽 ( 𝑥 ) ≤ 1 . In this case we call 𝛽 ( 𝑥 ) a 𝑞 -feasible valuation for ℱ . In other words, a formula is 𝑞 -Horn if and only if each clause in it contains at most one “positive” literal (where 𝛽 ( 𝑥 ) = 1 ) or at most two half ones (where 𝛽 ( 𝑥 ) = 0 . 5 ). We generalize these notions as follows. A formula ℱ is a 𝑧 -Horn formula if and only if ℱ ′ = BCP ( ℱ ) and either ℱ ′ is trivially satisfiable or trivially unsatisfiable or there is a valuation function 𝛾 ( 𝑥 ) such that for each clause 𝒞 in ℱ ′ we have that ( 1 or ∑︀ 𝑥 ∈ 𝐶 ∧ 𝛾 ( 𝑥 )=0 . 5 𝛾 ( 𝑥 ) = 1 . In this case we call 𝛾 ( 𝑥 ) to be a 𝑧 -feasible valuation for 𝐹 ′ . In other words, a formula is 𝑧 -Horn if and only if each clause in it after a BCP step contains at least one “negative” literal (where 𝛾 ( 𝑥 ) = 0 ) or exactly two half ones (where 𝛾 ( 𝑥 ) = 0 . 5 ). We show that the set of 𝑧 -Horn formulae properly includes the set of 𝑞 -Horn formulae. We also show that the 𝑤 -Horn SAT problem can be decided in polynomial time. We also show that each satisfiable formula is 𝑧","PeriodicalId":43454,"journal":{"name":"Annales Mathematicae et Informaticae","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77115665","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-01-01DOI: 10.33039/AMI.2021.04.004
C. Szabó, Csilla Bereczky-Zámbó, Júlia Szenderák, J. Szeibert
Recently more and more ethical issues arise in several sciences. We think that didactics of mathematics is not an exception. In this paper we investigate the question whether we can allow from mathematical precision in talent care. We suggest that these questions origin even from the formulation of a problem. The formulation of three well-known math problem is analized.
{"title":"On a metamatemEthical question in talent care","authors":"C. Szabó, Csilla Bereczky-Zámbó, Júlia Szenderák, J. Szeibert","doi":"10.33039/AMI.2021.04.004","DOIUrl":"https://doi.org/10.33039/AMI.2021.04.004","url":null,"abstract":"Recently more and more ethical issues arise in several sciences. We think that didactics of mathematics is not an exception. In this paper we investigate the question whether we can allow from mathematical precision in talent care. We suggest that these questions origin even from the formulation of a problem. The formulation of three well-known math problem is analized.","PeriodicalId":43454,"journal":{"name":"Annales Mathematicae et Informaticae","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2021-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88810860","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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