In this paper, we consider an iterative approximation problem of split variational inequalities in Hilbert spaces. In order to solve this split problem, we construct an iterative algorithm which combines a forward-backward-forward method and a self-adaptive rule to update the step-sizes. We prove that the constructed algorithm converges strongly to a solution of the split variational inequalities under some mild assumptions.
{"title":"A self-adaptive forward-backward-forward algorithm for solving split variational inequalities","authors":"Tzu-Chien Yin, N. Hussain, A. Asiri","doi":"10.37193/cjm.2023.02.15","DOIUrl":"https://doi.org/10.37193/cjm.2023.02.15","url":null,"abstract":"In this paper, we consider an iterative approximation problem of split variational inequalities in Hilbert spaces. In order to solve this split problem, we construct an iterative algorithm which combines a forward-backward-forward method and a self-adaptive rule to update the step-sizes. We prove that the constructed algorithm converges strongly to a solution of the split variational inequalities under some mild assumptions.","PeriodicalId":50711,"journal":{"name":"Carpathian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42311689","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Given a set of vertex-norm, or distances from the origin, in two-dimensional space, there does not always exist a convex configuration, or convex polygon, whose vertices satisfy the vertex-norm. In this research, we provide the necessary and sufficient conditions, based on the angles spanned by the polygon around the origin, for the existence of such convex configuration. General strategies for constructing a convex configuration satisfying a given vertex-norm set as well as examples of vertex-norm sets for which no such convex configuration exists are also illustrated.
{"title":"The Non-Existence of Convex Configuration for a Given Set of Vertex-Norm in Two-Dimensional Space","authors":"Kirati Sriamorn, S. Sriswasdi, Supanut Chaidee","doi":"10.37193/cjm.2023.02.11","DOIUrl":"https://doi.org/10.37193/cjm.2023.02.11","url":null,"abstract":"Given a set of vertex-norm, or distances from the origin, in two-dimensional space, there does not always exist a convex configuration, or convex polygon, whose vertices satisfy the vertex-norm. In this research, we provide the necessary and sufficient conditions, based on the angles spanned by the polygon around the origin, for the existence of such convex configuration. General strategies for constructing a convex configuration satisfying a given vertex-norm set as well as examples of vertex-norm sets for which no such convex configuration exists are also illustrated.","PeriodicalId":50711,"journal":{"name":"Carpathian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42374939","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
"In this paper we present a mathematical model for studying the interactions between human immune system and a pathogenic virus, such as Covid-19. A mathematical analysis based on dynamical systems theory is performed. More exactly, we model the interactions between the immune system and the virus by a modified predator-prey method. Several conclusions emerge from this study, and the main two of them are the followings: 1) a deficiency in the concentration of a single type of white blood cells in the early stages of virus proliferation may lead to the virus victory, and 2) if the number of at least one type of white blood cells can be increased beyond the normal threshold by medical interventions in the early stages of virus infection, then the immune system has a better chance to win against the virus."
{"title":"Analysis of interactions between human immune system and a pathogenic virus","authors":"G. Moza, L. F. Vesa","doi":"10.37193/CJM.2023.02.06","DOIUrl":"https://doi.org/10.37193/CJM.2023.02.06","url":null,"abstract":"\"In this paper we present a mathematical model for studying the interactions between human immune system and a pathogenic virus, such as Covid-19. A mathematical analysis based on dynamical systems theory is performed. More exactly, we model the interactions between the immune system and the virus by a modified predator-prey method. Several conclusions emerge from this study, and the main two of them are the followings: 1) a deficiency in the concentration of a single type of white blood cells in the early stages of virus proliferation may lead to the virus victory, and 2) if the number of at least one type of white blood cells can be increased beyond the normal threshold by medical interventions in the early stages of virus infection, then the immune system has a better chance to win against the virus.\"","PeriodicalId":50711,"journal":{"name":"Carpathian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47872322","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this paper, we make a comprehensive comparison in terms of the quality of the achieved solutions, the corresponding execution time and impact of the genetic operators on the quality of the results between the Haploid Genetic Algorithms (HGAs) and Diploid Genetic Algorithms (DGAs). The standard genetic algorithms, referred to in our paper as HGAs are characterized by the fact that they are using a haploid representation relating an individual with a chromosome, while the DGAs are using diploid individuals which are made of two chromosomes corresponding to the dominant and recessive genes. Even though the general opinion is that DGAs do not provide much benefit as compared to classical GAs, based on extensive computational experiments, we do show that the DGAs are robust, have a high degree of consistency and perform better, sometimes almost twice as well, than the HGAs, but are slower due to the high number of operations to be performed, caused by the duplication of the genetic information. However, the quality of the solutions achieved by the DGAs compensate their relative high execution time. The better quality of the DGAs, proving the efficiency of using diploid genes, is given by the homogeneity of the population which covers the search space thoroughly and in this way being capable of avoiding the local optima.
{"title":"A Comparative Study between Haploid Genetic Algorithms and Diploid Genetic Algorithms","authors":"A. Petrovan, Oliviu Matei, P. Pop","doi":"10.37193/cjm.2023.02.08","DOIUrl":"https://doi.org/10.37193/cjm.2023.02.08","url":null,"abstract":"In this paper, we make a comprehensive comparison in terms of the quality of the achieved solutions, the corresponding execution time and impact of the genetic operators on the quality of the results between the Haploid Genetic Algorithms (HGAs) and Diploid Genetic Algorithms (DGAs). The standard genetic algorithms, referred to in our paper as HGAs are characterized by the fact that they are using a haploid representation relating an individual with a chromosome, while the DGAs are using diploid individuals which are made of two chromosomes corresponding to the dominant and recessive genes. Even though the general opinion is that DGAs do not provide much benefit as compared to classical GAs, based on extensive computational experiments, we do show that the DGAs are robust, have a high degree of consistency and perform better, sometimes almost twice as well, than the HGAs, but are slower due to the high number of operations to be performed, caused by the duplication of the genetic information. However, the quality of the solutions achieved by the DGAs compensate their relative high execution time. The better quality of the DGAs, proving the efficiency of using diploid genes, is given by the homogeneity of the population which covers the search space thoroughly and in this way being capable of avoiding the local optima.","PeriodicalId":50711,"journal":{"name":"Carpathian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42039426","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper addresses the issue of which strong duality holds between parametric robust semi-definite linear optimization problems and their dual programs. In the case of a spectral norm uncertainty set, it yields a corresponding strong duality result with a semi-definite programming as its dual. We also show that the dual can be reformulated as a second-order cone programming problem or a linear programming problem when the constraint uncertainty sets of parametric robust semi-definite linear programs are given in terms of affinely parameterized diagonal matrix.
{"title":"Strong duality in parametric robust semi-definite linear programming and exact relaxations","authors":"Nithirat Sisarat, R. Wangkeeree, R. Wangkeeree","doi":"10.37193/cjm.2023.02.10","DOIUrl":"https://doi.org/10.37193/cjm.2023.02.10","url":null,"abstract":"This paper addresses the issue of which strong duality holds between parametric robust semi-definite linear optimization problems and their dual programs. In the case of a spectral norm uncertainty set, it yields a corresponding strong duality result with a semi-definite programming as its dual. We also show that the dual can be reformulated as a second-order cone programming problem or a linear programming problem when the constraint uncertainty sets of parametric robust semi-definite linear programs are given in terms of affinely parameterized diagonal matrix.","PeriodicalId":50711,"journal":{"name":"Carpathian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45877554","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The crossing number $mathrm{cr}(G)$ of a graph $G$ is the minimum number of edge crossings over all drawings of $G$ in the plane. In the paper, the crossing number of the join product $G^ast + D_n$ for the connected graph $G^ast$ on six vertices consisting of one path on four vertices $P_4$ and two leaves adjacent with the same outer vertex of the path $P_4$ is given, where $D_n$ consists of $n$ isolated vertices. Finally, by adding some edges to the graph $G^ast$, we obtain the crossing numbers of the join products of other four graphs with $D_n$.
{"title":"On the crossing numbers of the join products of five graphs on six vertices with discrete graph","authors":"Stefan Berezný, M. Staš","doi":"10.37193/cjm.2023.02.03","DOIUrl":"https://doi.org/10.37193/cjm.2023.02.03","url":null,"abstract":"The crossing number $mathrm{cr}(G)$ of a graph $G$ is the minimum number of edge crossings over all drawings of $G$ in the plane. In the paper, the crossing number of the join product $G^ast + D_n$ for the connected graph $G^ast$ on six vertices consisting of one path on four vertices $P_4$ and two leaves adjacent with the same outer vertex of the path $P_4$ is given, where $D_n$ consists of $n$ isolated vertices. Finally, by adding some edges to the graph $G^ast$, we obtain the crossing numbers of the join products of other four graphs with $D_n$.","PeriodicalId":50711,"journal":{"name":"Carpathian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43539048","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In mathematical psychology, the model of decision practice represents the development of moral judgment that deals with the time to decide the meaning of the various choices and selecting one of them for use. Most animal behavior research classifies such situations as two distinct phenomena. On the other hand, reward plays a big part in this kind of study since, based on the selected side and food location, such circumstances may be classified into four categories. This paper intends to investigate such types of behavior and establish a general functional equation for it. The proposed functional equation can be used to describe several psychological and learning theory models in the existing literature. By using the fixed point theory tools, we obtain the results related to the existence, uniqueness, and stability of a solution to the proposed functional equation. Finally, we give two examples to support our main results.
{"title":"On the solution of the generalized functional equation arising in mathematical psychology and theory of learning approached by the Banach fixed point theorem","authors":"","doi":"10.37193/cjm.2023.02.14","DOIUrl":"https://doi.org/10.37193/cjm.2023.02.14","url":null,"abstract":"In mathematical psychology, the model of decision practice represents the development of moral judgment that deals with the time to decide the meaning of the various choices and selecting one of them for use. Most animal behavior research classifies such situations as two distinct phenomena. On the other hand, reward plays a big part in this kind of study since, based on the selected side and food location, such circumstances may be classified into four categories. This paper intends to investigate such types of behavior and establish a general functional equation for it. The proposed functional equation can be used to describe several psychological and learning theory models in the existing literature. By using the fixed point theory tools, we obtain the results related to the existence, uniqueness, and stability of a solution to the proposed functional equation. Finally, we give two examples to support our main results.","PeriodicalId":50711,"journal":{"name":"Carpathian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46341192","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
"Let $mathcal P_{n}:=H^{*}((mathbb{R}P^{infty})^{n}) cong mathbb Z_2[x_{1},x_{2},ldots,x_{n}]$ be the graded polynomial algebra over $mathcal K,$ where $mathcal K$ denotes the prime field of two elements. We investigate the Peterson hit problem for the polynomial algebra $mathcal P_{n},$ viewed as a graded left module over the mod-$2$ Steenrod algebra, $mathcal{A}.$ For $n>4,$ this problem is still unsolved, even in the case of $n=5$ with the help of computers. In this paper, we study the hit problem for the case $n=6$ in degree $d_{k}=6(2^{k} -1)+9.2^{k},$ with $k$ an arbitrary non-negative integer. By considering $mathcal K$ as a trivial $mathcal A$-module, then the hit problem is equivalent to the problem of finding a basis of $mathcal K$-graded vector space $mathcal K {otimes}_{mathcal{A}}mathcal P_{n}.$ The main goal of the current paper is to explicitly determine an admissible monomial basis of the $mathcal K$-graded vector space $mathcal K{otimes}_{mathcal{A}}mathcal P_6$ in some degrees. At the same time, the behavior of the sixth Singer algebraic transfer in degree $d_{k}=6(2^{k} -1)+9.2^{k}$ is also discussed at the end of this article. Here, the Singer algebraic transfer is a homomorphism from the homology of the mod-$2$ Steenrod algebra, $mbox{Tor}^{mathcal{A}}_{n, n+d}(mathcal K, mathcal K),$ to the subspace of $mathcal Kotimes_{mathcal{A}}mathcal P_{n}$ consisting of all the $GL_n$-invariant classes of degree $d.$"
{"title":"A note on the generators of the polynomial algebra of six variables and application","authors":"Nguyen Khac Tin","doi":"10.37193/cjm.2023.02.13","DOIUrl":"https://doi.org/10.37193/cjm.2023.02.13","url":null,"abstract":"\"Let $mathcal P_{n}:=H^{*}((mathbb{R}P^{infty})^{n}) cong mathbb Z_2[x_{1},x_{2},ldots,x_{n}]$ be the graded polynomial algebra over $mathcal K,$ where $mathcal K$ denotes the prime field of two elements. We investigate the Peterson hit problem for the polynomial algebra $mathcal P_{n},$ viewed as a graded left module over the mod-$2$ Steenrod algebra, $mathcal{A}.$ For $n>4,$ this problem is still unsolved, even in the case of $n=5$ with the help of computers. In this paper, we study the hit problem for the case $n=6$ in degree $d_{k}=6(2^{k} -1)+9.2^{k},$ with $k$ an arbitrary non-negative integer. By considering $mathcal K$ as a trivial $mathcal A$-module, then the hit problem is equivalent to the problem of finding a basis of $mathcal K$-graded vector space $mathcal K {otimes}_{mathcal{A}}mathcal P_{n}.$ The main goal of the current paper is to explicitly determine an admissible monomial basis of the $mathcal K$-graded vector space $mathcal K{otimes}_{mathcal{A}}mathcal P_6$ in some degrees. At the same time, the behavior of the sixth Singer algebraic transfer in degree $d_{k}=6(2^{k} -1)+9.2^{k}$ is also discussed at the end of this article. Here, the Singer algebraic transfer is a homomorphism from the homology of the mod-$2$ Steenrod algebra, $mbox{Tor}^{mathcal{A}}_{n, n+d}(mathcal K, mathcal K),$ to the subspace of $mathcal Kotimes_{mathcal{A}}mathcal P_{n}$ consisting of all the $GL_n$-invariant classes of degree $d.$\"","PeriodicalId":50711,"journal":{"name":"Carpathian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43544306","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
"In this work, we show that a slight change in the Sklar's formula tremendously affects the class of aggregation functions it represents. While the original formula can only be used to construct $d$-increasing aggregation functions, this new formula can be used to construct any continuous aggregation function excepted possibly those belong to the boundary of this set. In particular, all continuous aggregation functions can be approximated by aggregation functions in this form. This shows that it is sufficient to only consider aggregation functions in this form for most cases. Construction examples via this method are also given."
{"title":"Aggregation Function Constructed from Copula","authors":"Varayut Boonyasri, S. Tasena","doi":"10.37193/cjm.2023.02.04","DOIUrl":"https://doi.org/10.37193/cjm.2023.02.04","url":null,"abstract":"\"In this work, we show that a slight change in the Sklar's formula tremendously affects the class of aggregation functions it represents. While the original formula can only be used to construct $d$-increasing aggregation functions, this new formula can be used to construct any continuous aggregation function excepted possibly those belong to the boundary of this set. In particular, all continuous aggregation functions can be approximated by aggregation functions in this form. This shows that it is sufficient to only consider aggregation functions in this form for most cases. Construction examples via this method are also given.\"","PeriodicalId":50711,"journal":{"name":"Carpathian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49445749","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
"Let $(X,|cdot|)$ be a Banach space and $T:Ato X$ a contraction mapping, where $Asubset X$ is a closed set. Consider a sequence ${x_n}subset A$ and define the sequence ${y_n}subset X$, by $y_n=x_n+Tleft(x_{sigma(n)}right)$, where ${sigma(n)}$ is a sequence of natural numbers. We highlight some general conditions so that the two sequences ${x_n}$ and ${y_n}$ are simultaneously convergent. Both cases: 1) $sigma(n)
{"title":"On the transfer of convergence between two sequences in Banach spaces","authors":"D. Marinescu, Eugen Păltănea","doi":"10.37193/cjm.2023.02.05","DOIUrl":"https://doi.org/10.37193/cjm.2023.02.05","url":null,"abstract":"\"Let $(X,|cdot|)$ be a Banach space and $T:Ato X$ a contraction mapping, where $Asubset X$ is a closed set. Consider a sequence ${x_n}subset A$ and define the sequence ${y_n}subset X$, by $y_n=x_n+Tleft(x_{sigma(n)}right)$, where ${sigma(n)}$ is a sequence of natural numbers. We highlight some general conditions so that the two sequences ${x_n}$ and ${y_n}$ are simultaneously convergent. Both cases: 1) $sigma(n)<n$, for all $n$, and 2) $sigma(n)ge n$, for all $n$, are discussed. In the first case, a general Picard iteration procedure is inferred. The results are then extended to sequences of mappings and some appropriate applications are also proposed.\"","PeriodicalId":50711,"journal":{"name":"Carpathian Journal of Mathematics","volume":" ","pages":""},"PeriodicalIF":1.4,"publicationDate":"2022-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49466881","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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