Shalu Saini, Rajeev Kumar, Deeksha, Rishu Arora, Kamal Kumar
In the present article, the time fractional Fisher equation is considered in conformal form to study the application of the Lie classical method and quantitative analysis. The Lie symmetry method has been applied to find the infinitesimal generators and symmetry reductions of the fractional Fisher equation. The obtained reduced form of the equation is solved by the method of G ′ / G , which gives different forms of solutions. The theory of bifurcation has been utilized in the reduced form to check the stability and nature of critical points by transforming the equations into an autonomous system. Some phase portraits have been drawn at different critical points by the use of maple.
{"title":"Symmetry Analysis and Wave Solutions of the Fisher Equation Using Conformal Fractional Derivatives","authors":"Shalu Saini, Rajeev Kumar, Deeksha, Rishu Arora, Kamal Kumar","doi":"10.1155/2023/1633450","DOIUrl":"https://doi.org/10.1155/2023/1633450","url":null,"abstract":"In the present article, the time fractional Fisher equation is considered in conformal form to study the application of the Lie classical method and quantitative analysis. The Lie symmetry method has been applied to find the infinitesimal generators and symmetry reductions of the fractional Fisher equation. The obtained reduced form of the equation is solved by the method of \u0000 \u0000 \u0000 \u0000 G\u0000 \u0000 \u0000 ′\u0000 \u0000 \u0000 /\u0000 G\u0000 \u0000 , which gives different forms of solutions. The theory of bifurcation has been utilized in the reduced form to check the stability and nature of critical points by transforming the equations into an autonomous system. Some phase portraits have been drawn at different critical points by the use of maple.","PeriodicalId":49251,"journal":{"name":"Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43425243","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}
A picture in a group presentation is a geometric configuration with an arrangement of discs and arcs within a boundary disc. The drawing of this picture does not have to follow a particular rule, only using the generator as discs and the relation as arcs. It will form a picture label pattern if drawn with a particular rule. This paper discusses the label pattern of a picture in the presentation of direct product groups. Direct product presentation is used with two cyclic groups, and where and . The method for forming a picture label pattern is to arrange the first generator in the initial arrangement, compile a second generator, and add a number of commutators. Furthermore, the pattern is used to calculate the length of the label on the picture. It is obtained that the picture’s label is
分组演示中的图片是一种几何配置，在边界圆盘内排列圆盘和圆弧。这张图的绘制不必遵循特定的规则，只需将生成器用作圆盘，将关系用作圆弧。如果使用特定规则绘制，它将形成图片标签图案。本文讨论了直接产品组表示中图片的标签模式。直接乘积表示与两个循环基团一起使用，ℤ p和ℤ 其中p，q∈ℤ + 且p、q≥2。形成图片标签图案的方法是将第一生成器排列在初始排列中，编译第二生成器，并添加多个换向器。此外，该图案用于计算图片上标签的长度。可以得出图片的标签是q−1b n a b q−n，并且标签的长度为p+2n-q，其中n是换向器片的数量。
{"title":"The Picture on the Presentation of Direct Product Group of Two Cyclic Groups","authors":"Y. Yanita, Budi Rudianto","doi":"10.1155/2023/8018645","DOIUrl":"https://doi.org/10.1155/2023/8018645","url":null,"abstract":"<jats:p>A picture in a group presentation is a geometric configuration with an arrangement of discs and arcs within a boundary disc. The drawing of this picture does not have to follow a particular rule, only using the generator as discs and the relation as arcs. It will form a picture label pattern if drawn with a particular rule. This paper discusses the label pattern of a picture in the presentation of direct product groups. Direct product presentation is used with two cyclic groups, <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M1\">\u0000 <msub>\u0000 <mrow>\u0000 <mi>ℤ</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>p</mi>\u0000 </mrow>\u0000 </msub>\u0000 </math>\u0000 </jats:inline-formula> and <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M2\">\u0000 <msub>\u0000 <mrow>\u0000 <mi>ℤ</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>q</mi>\u0000 </mrow>\u0000 </msub>\u0000 </math>\u0000 </jats:inline-formula> where <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M3\">\u0000 <mi>p</mi>\u0000 <mo>,</mo>\u0000 <mi>q</mi>\u0000 <mo>∈</mo>\u0000 <msup>\u0000 <mrow>\u0000 <mi>ℤ</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mo>+</mo>\u0000 </mrow>\u0000 </msup>\u0000 </math>\u0000 </jats:inline-formula> and <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M4\">\u0000 <mi>p</mi>\u0000 <mo>,</mo>\u0000 <mi>q</mi>\u0000 <mo>≥</mo>\u0000 <mn>2</mn>\u0000 </math>\u0000 </jats:inline-formula>. The method for forming a picture label pattern is to arrange the first generator in the initial arrangement, compile a second generator, and add a number of commutators. Furthermore, the pattern is used to calculate the length of the label on the picture. It is obtained that the picture’s label is <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M5\">\u0000 <msup>\u0000 <mrow>\u0000 <mi>a</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mi>q</mi>\u0000 ","PeriodicalId":49251,"journal":{"name":"Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43899862","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}
The purpose of this paper is to investigate the valuation of equity-linked death benefit contracts and the multiple life insurance on two heads based on a joint survival model. Using the exponential Wiener process assumption for the stock price process and a K n distribution for the time until death, we provide explicit formulas for the expectation of the discounted payment of the guaranteed minimum death benefit products, and we derive closed expressions for some options and numerical illustrations. We investigate multiple life insurance based on a joint survival using the bivariate Sarmanov distribution with K n (i.e., the Laplace transform of their density function is a ratio of two polynomials of degree at most) marginal distributions. We present analytical results of the joint-life status.
{"title":"Valuing Equity-Linked Death Benefits on Multiple Life with Time until Death following a <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M1\">\u0000 <msub>\u0000 <mrow>\u0000 <mi>K</mi>\u0000 </mrow>\u0000 <","authors":"Franck Adékambi, E. Konzou","doi":"10.1155/2023/9984786","DOIUrl":"https://doi.org/10.1155/2023/9984786","url":null,"abstract":"The purpose of this paper is to investigate the valuation of equity-linked death benefit contracts and the multiple life insurance on two heads based on a joint survival model. Using the exponential Wiener process assumption for the stock price process and a \u0000 \u0000 \u0000 \u0000 K\u0000 \u0000 \u0000 n\u0000 \u0000 \u0000 \u0000 distribution for the time until death, we provide explicit formulas for the expectation of the discounted payment of the guaranteed minimum death benefit products, and we derive closed expressions for some options and numerical illustrations. We investigate multiple life insurance based on a joint survival using the bivariate Sarmanov distribution with \u0000 \u0000 \u0000 \u0000 K\u0000 \u0000 \u0000 n\u0000 \u0000 \u0000 \u0000 (i.e., the Laplace transform of their density function is a ratio of two polynomials of degree at most) marginal distributions. We present analytical results of the joint-life status.","PeriodicalId":49251,"journal":{"name":"Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46475882","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}
M. E. O. E. El Mokhtar, S. Benmansour, A. Matallah
In this article, we deal with the nonlocal elliptic problems of the Kirchhoff type involving the Hardy potential and critical nonlinearity on a bounded domain in R 3 . Under an appropriate condition on the nonhomogeneous term and using variational methods, we obtain two distinct solutions.
{"title":"On Nonlocal Elliptic Problems of the Kirchhoff Type Involving the Hardy Potential and Critical Nonlinearity","authors":"M. E. O. E. El Mokhtar, S. Benmansour, A. Matallah","doi":"10.1155/2023/2997093","DOIUrl":"https://doi.org/10.1155/2023/2997093","url":null,"abstract":"In this article, we deal with the nonlocal elliptic problems of the Kirchhoff type involving the Hardy potential and critical nonlinearity on a bounded domain in \u0000 \u0000 \u0000 \u0000 R\u0000 \u0000 \u0000 3\u0000 \u0000 \u0000 \u0000 . Under an appropriate condition on the nonhomogeneous term and using variational methods, we obtain two distinct solutions.","PeriodicalId":49251,"journal":{"name":"Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49666003","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}
The purpose of this study is to analyze the impact of control strategies, namely, insecticide spray, roguing of a diseased tomato plant, and protective netting to protect tomato plant from tomato yellow leaf curl virus disease (TYLCVD). For this, we formulate and analyze a simple deterministic model for the transmission dynamics of TYLCVD that incorporates these control strategies. We initially take into account the constant control case, we calculate the basic reproduction number, and we investigate the existence and stability of the disease-free and endemic equilibria. We use the Kamgang-Sallet stability to ensure that the disease-free equilibrium point is globally asymptotically stable when the reproduction number is less than one. This indicates that TYLCVD dies out independent of the initial size of the tomato population. For , the disease-free equilibrium becomes unstable, and the endemic equilibrium is globally asymptotically stable. This indicates that TYLCVD propagates. In the nonconstant control case, we use Pontryagin’s maximum principle to derive the necessary conditions for the optimal control of the disease. Our findings show that all the combined efforts of two out of three strategies can significantly reduce the power of infectivity of the disease except the combination of the use of insecticide spray and rouging infected tomato plants. Besides our numerical simulations show, the implementation of the combination of roguing diseased plants and protective netting has a similar effect in minimizing or eliminating TYLCV as the use of all strategies. Hence, as resources are always in scarce, we recommend policymakers to adapt the combination of the use of roguing diseased tomato plants and protective netting to eradicate the disease.
{"title":"Ecoepidemiological Model and Optimal Control Analysis of Tomato Yellow Leaf Curl Virus Disease in Tomato Plant","authors":"Berhe Nerea Kahsay, Oluwole D. Makinde","doi":"10.1155/2023/4066236","DOIUrl":"https://doi.org/10.1155/2023/4066236","url":null,"abstract":"The purpose of this study is to analyze the impact of control strategies, namely, insecticide spray, roguing of a diseased tomato plant, and protective netting to protect tomato plant from tomato yellow leaf curl virus disease (TYLCVD). For this, we formulate and analyze a simple deterministic model for the transmission dynamics of TYLCVD that incorporates these control strategies. We initially take into account the constant control case, we calculate the basic reproduction number, and we investigate the existence and stability of the disease-free and endemic equilibria. We use the Kamgang-Sallet stability to ensure that the disease-free equilibrium point is globally asymptotically stable when the reproduction number <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M1\"> <msub> <mrow> <mi mathvariant=\"script\">R</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> </math> is less than one. This indicates that TYLCVD dies out independent of the initial size of the tomato population. For <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M2\"> <msub> <mrow> <mi mathvariant=\"script\">R</mi> </mrow> <mrow> <mn>0</mn> </mrow> </msub> <mo><</mo> <mn>1</mn> </math> , the disease-free equilibrium becomes unstable, and the endemic equilibrium is globally asymptotically stable. This indicates that TYLCVD propagates. In the nonconstant control case, we use Pontryagin’s maximum principle to derive the necessary conditions for the optimal control of the disease. Our findings show that all the combined efforts of two out of three strategies can significantly reduce the power of infectivity of the disease except the combination of the use of insecticide spray and rouging infected tomato plants. Besides our numerical simulations show, the implementation of the combination of roguing diseased plants and protective netting has a similar effect in minimizing or eliminating TYLCV as the use of all strategies. Hence, as resources are always in scarce, we recommend policymakers to adapt the combination of the use of roguing diseased tomato plants and protective netting to eradicate the disease.","PeriodicalId":49251,"journal":{"name":"Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135017074","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}
This paper is mainly concerned with the existence of mild solutions and exact controllability for a class of fractional semilinear system of order q ∈ 1 , 2 with instantaneous and noninstantaneous impulses. First, combining the Kuratowski measure of noncompactness and the Mönch fixed point theorem, we investigated the existence result for the considered system. It is remarkable that our assumptions for impulses and the nonlinear term are weaker than the Lipschitz conditions. Next, on this basis, the exact controllability for the considered system is determined. In the end, an example is provided to support the main findings.
{"title":"Exact Controllability for a Class of Fractional Semilinear System of Order \u0000 1\u0000 <</mo>\u0000 q\u0000 <</mo>\u0000 2\u0000 with Instantaneous and Noninstantaneous Impulses","authors":"Yunhao Chu, Yansheng Liu","doi":"10.1155/2023/8300785","DOIUrl":"https://doi.org/10.1155/2023/8300785","url":null,"abstract":"This paper is mainly concerned with the existence of mild solutions and exact controllability for a class of fractional semilinear system of order \u0000 \u0000 q\u0000 ∈\u0000 \u0000 \u0000 1\u0000 ,\u0000 2\u0000 \u0000 \u0000 \u0000 with instantaneous and noninstantaneous impulses. First, combining the Kuratowski measure of noncompactness and the Mönch fixed point theorem, we investigated the existence result for the considered system. It is remarkable that our assumptions for impulses and the nonlinear term are weaker than the Lipschitz conditions. Next, on this basis, the exact controllability for the considered system is determined. In the end, an example is provided to support the main findings.","PeriodicalId":49251,"journal":{"name":"Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-08-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41457150","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}
Topological polynomial and indices based on the distance between the vertices of a connected graph are widely used in the chemistry to establish relation between the structure and the properties of molecules. In a similar way, chromatic versions of certain topological indices and the related polynomial have also been discussed in the recent literature. In this paper, we present the chromatic Schultz and Gutman polynomials and the expanded form of the Hosoya polynomial and chromatic Schultz and Gutman polynomials, and then we derive these polynomials for special cases of Jahangir graphs.
{"title":"Chromatic Schultz and Gutman Polynomials of Jahangir Graphs <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M1\">\u0000 <msub>\u0000 <mrow>\u0000 <mi>J</mi>\u0000 </mrow>\u0000 <mrow>\u0000 ","authors":"Ramy S. Shaheen, Suhail Mahfud, Qays Alhawat","doi":"10.1155/2023/4891083","DOIUrl":"https://doi.org/10.1155/2023/4891083","url":null,"abstract":"Topological polynomial and indices based on the distance between the vertices of a connected graph are widely used in the chemistry to establish relation between the structure and the properties of molecules. In a similar way, chromatic versions of certain topological indices and the related polynomial have also been discussed in the recent literature. In this paper, we present the chromatic Schultz and Gutman polynomials and the expanded form of the Hosoya polynomial and chromatic Schultz and Gutman polynomials, and then we derive these polynomials for special cases of Jahangir graphs.","PeriodicalId":49251,"journal":{"name":"Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2023-01-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64796179","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}
In this paper, we propose a continuous mathematical model that describes the spread of multistrains COVID-19 virus among humans: susceptible, exposed, infected, quarantined, hospitalized, and recovered individuals. The positivity and boundedness of the system solution are provided in order to get the well posedness of the proposed model. Secondly, three controls are considered in our model to minimize the multistrain spread of the disease, namely, vaccination, security campaigns, social distancing measures, and diagnosis. Furthermore, the optimal control problem and related optimality conditions of the Pontryagin type are discussed with the objective to minimize the number of infected individuals. Finally, numerical simulations are performed in the case of two strains of COVID-19 and with four control strategies. By using the incremental cost-effectiveness ratio (ICER) method, we show that combining vaccination with diagnosis provides the most cost-effective strategy.
{"title":"Mathematical Modelling and Optimal Control Strategies of a Multistrain COVID-19 Spread","authors":"B. Khajji, L. Boujallal, O. Balatif, M. Rachik","doi":"10.1155/2022/9071890","DOIUrl":"https://doi.org/10.1155/2022/9071890","url":null,"abstract":"In this paper, we propose a continuous mathematical model that describes the spread of multistrains COVID-19 virus among humans: susceptible, exposed, infected, quarantined, hospitalized, and recovered individuals. The positivity and boundedness of the system solution are provided in order to get the well posedness of the proposed model. Secondly, three controls are considered in our model to minimize the multistrain spread of the disease, namely, vaccination, security campaigns, social distancing measures, and diagnosis. Furthermore, the optimal control problem and related optimality conditions of the Pontryagin type are discussed with the objective to minimize the number of infected individuals. Finally, numerical simulations are performed in the case of two strains of COVID-19 and with four control strategies. By using the incremental cost-effectiveness ratio (ICER) method, we show that combining vaccination with diagnosis provides the most cost-effective strategy.","PeriodicalId":49251,"journal":{"name":"Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"64788232","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}
Let and be two timelike surfaces in Minkowski 3-space . If there exists a spacelike (timelike) Darboux line congruence between each point of and , then the surfaces are timelike Weingarten surfaces. It turns out their Tschebyscheff angles are solutions of the Sinh-Gordon equation, and the surfaces are related to each other by Backlund’s transformation. Finally, a method to construct new timelike Weingarten surface has been found.
{"title":"Timelike <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M1\">\u0000 <mi>W</mi>\u0000 </math>-Surfaces in Minkowski 3-Space <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M2\">\u0000 <msubsup>\u0000 <mrow>\u0000 ","authors":"N. Alluhaibi, R. Abdel-Baky","doi":"10.1155/2022/7998748","DOIUrl":"https://doi.org/10.1155/2022/7998748","url":null,"abstract":"<jats:p>Let <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M3\">\u0000 <mi>M</mi>\u0000 </math>\u0000 </jats:inline-formula> and <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M4\">\u0000 <msup>\u0000 <mrow>\u0000 <mi>M</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mo>∗</mo>\u0000 </mrow>\u0000 </msup>\u0000 </math>\u0000 </jats:inline-formula> be two timelike surfaces in Minkowski 3-space <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M5\">\u0000 <msubsup>\u0000 <mrow>\u0000 <mi>ℝ</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mn>1</mn>\u0000 </mrow>\u0000 <mrow>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 </msubsup>\u0000 </math>\u0000 </jats:inline-formula>. If there exists a spacelike (timelike) Darboux line congruence between each point of <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M6\">\u0000 <mi>M</mi>\u0000 </math>\u0000 </jats:inline-formula> and <jats:inline-formula>\u0000 <math xmlns=\"http://www.w3.org/1998/Math/MathML\" id=\"M7\">\u0000 <msup>\u0000 <mrow>\u0000 <mi>M</mi>\u0000 </mrow>\u0000 <mrow>\u0000 <mo>∗</mo>\u0000 </mrow>\u0000 </msup>\u0000 </math>\u0000 </jats:inline-formula>, then the surfaces are timelike Weingarten surfaces. It turns out their Tschebyscheff angles are solutions of the Sinh-Gordon equation, and the surfaces are related to each other by Backlund’s transformation. Finally, a method to construct new timelike Weingarten surface has been found.</jats:p>","PeriodicalId":49251,"journal":{"name":"Journal of Applied Mathematics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2022-03-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43456484","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}
An irreversible conversion process is a dynamic process on a graph where a one-way change of state (from state 0 to state 1) is applied on the vertices if they satisfy a conversion rule that is determined at the beginning of the study. The irreversible k -threshold conversion process on a graph G = ð V , E Þ is an iterative process which begins by choosing a set S 0 ⊆ V , and for each step t ð t = 1, 2, ⋯ , Þ , S t is obtained from S t − 1 by adjoining all vertices that have at least k neighbors in S t − 1 . S 0 is called the seed set of the k -threshold conversion process, and if S t = V ð G Þ for some t ≥ 0 , then S 0 is an irreversible k -threshold conversion set (IkCS) of G . The k -threshold conversion number of G (denoted by ( C k ð G Þ ) is the minimum cardinality of all the IkCSs of G : In this paper, we determine C 2 ð G Þ for the circulant graph C n ðf 1, r gÞ when r is arbitrary; we also fi nd C 3 ð C n ðf 1, r gÞÞ when r = 2, 3 . We also introduce an upper bound for C 3 ð C n ðf 1, 4 gÞÞ . Finally, we suggest an upper bound for C 3 ð C n ðf 1, r gÞÞ if n ≥ 2 ð r + 1 Þ and n ≡ 0 ð mod2 ð r + 1 ÞÞ .
不可逆转换过程是图上的动态过程，如果顶点满足研究开始时确定的转换规则，则对顶点进行单向状态变化(从状态0到状态1)。图G = ð V, E Þ上的不可逆k阈值转换过程是一个迭代过程，该过程首先选择一个集s0≥≥V，对于每一步t ð t = 1,2，⋯Þ，通过相邻S t−1中至少有k个邻居的所有顶点，从S t−1中得到S t。S 0称为k阈值转换过程的种子集，如果S t = V ð G Þ对于某些t≥0，则S 0是G的不可逆k阈值转换集(IkCS)。G的k阈值转换数(记为(C k ð G Þ)是G的所有ikcs的最小基数:在本文中，当r为任意时，我们确定了循环图C n ðf 1, r gÞ的C 2 ð G Þ;当r = 2,3时，我们还发现C 3 ð C n ð f1, r gÞÞ。我们还引入了C 3 ð C n ðf 1,4 gÞÞ的上界。最后，我们提出了C 3 ð C n ðf 1, r gÞÞ如果n≥2 ð r + 1 Þ和n≡0 ð mod2 ð r + 1 ÞÞ的上界。
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