Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2023-0153_eq_001.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>X</m:mi> </m:math> <jats:tex-math>X</jats:tex-math> </jats:alternatives> </jats:inline-formula> be a compact Riemann surface of genus <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2023-0153_eq_002.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>g</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:math> <jats:tex-math>gge 2</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2023-0153_eq_003.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>G</m:mi> </m:math> <jats:tex-math>G</jats:tex-math> </jats:alternatives> </jats:inline-formula> be a semisimple complex Lie group and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2023-0153_eq_004.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ρ</m:mi> <m:mo>:</m:mo> <m:mi>G</m:mi> <m:mo>→</m:mo> <m:mi mathvariant="normal">GL</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>V</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>rho :Gto {rm{GL}}left(V)</jats:tex-math> </jats:alternatives> </jats:inline-formula> be a complex representation of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2023-0153_eq_005.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>G</m:mi> </m:math> <jats:tex-math>G</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Given a principal <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2023-0153_eq_006.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>G</m:mi> </m:math> <jats:tex-math>G</jats:tex-math> </jats:alternatives> </jats:inline-formula>-bundle <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2023-0153_eq_007.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>E</m:mi> </m:math> <jats:tex-math>E</jats:tex-math> </jats:alternatives> </jats:inline-formula> over <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="graphic/j_math-2023-0153_eq_008.png" /> <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>X</m:mi> </m:math> <jats:tex-math>X</jats:tex-math> </jats:alternatives> </jats:inline-formula>, a vector bundle <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink="http://www.w3.org/1999/xlink"
设X X是g≥2g的紧致黎曼曲面ge 2, g g是半简单复李群，ρ: g→GL (V) rho: g to{rm{GL}}left (V)是g g的复表示。给定一个主G G -束E E / X X，诱导出一个矢量束E (V) E left (V)，其典型纤维是V V的复制品。A (G， ρ) left (G, rho) -希格斯对是一对(E， φ) left (E, varphi)，其中E E是X X上的一个主G G -束，φ varphi是E (V)⊗L E left (V) otimes L的全纯全局截面，L L是X X上的一个固定线束。在这项工作中，考虑了G= Spin (8, C) G= {rm{Spin}}left (8, {mathbb{C}})和Spin (8, C) {rm{Spin}}left (8, {mathbb{C}})承认的三种不可约的八维复表示的这种希格斯对。特别地，给出了这些特定希格斯对的稳定性、半稳定性和多稳定性的简化概念，并证明了相应的模空间是同构的，并描述了与自旋(8,C) {rm{Spin}}left (8, {mathbb{C}})的三种表述之一相关的稳定和非简单希格斯对的精确表达式。
{"title":"Spin(8,C)-Higgs pairs over a compact Riemann surface","authors":"Álvaro Antón-Sancho","doi":"10.1515/math-2023-0153","DOIUrl":"https://doi.org/10.1515/math-2023-0153","url":null,"abstract":"Let <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0153_eq_001.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>X</m:mi> </m:math> <jats:tex-math>X</jats:tex-math> </jats:alternatives> </jats:inline-formula> be a compact Riemann surface of genus <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0153_eq_002.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>g</m:mi> <m:mo>≥</m:mo> <m:mn>2</m:mn> </m:math> <jats:tex-math>gge 2</jats:tex-math> </jats:alternatives> </jats:inline-formula>, <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0153_eq_003.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>G</m:mi> </m:math> <jats:tex-math>G</jats:tex-math> </jats:alternatives> </jats:inline-formula> be a semisimple complex Lie group and <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0153_eq_004.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>ρ</m:mi> <m:mo>:</m:mo> <m:mi>G</m:mi> <m:mo>→</m:mo> <m:mi mathvariant=\"normal\">GL</m:mi> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>V</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> <jats:tex-math>rho :Gto {rm{GL}}left(V)</jats:tex-math> </jats:alternatives> </jats:inline-formula> be a complex representation of <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0153_eq_005.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>G</m:mi> </m:math> <jats:tex-math>G</jats:tex-math> </jats:alternatives> </jats:inline-formula>. Given a principal <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0153_eq_006.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>G</m:mi> </m:math> <jats:tex-math>G</jats:tex-math> </jats:alternatives> </jats:inline-formula>-bundle <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0153_eq_007.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>E</m:mi> </m:math> <jats:tex-math>E</jats:tex-math> </jats:alternatives> </jats:inline-formula> over <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" xlink:href=\"graphic/j_math-2023-0153_eq_008.png\" /> <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>X</m:mi> </m:math> <jats:tex-math>X</jats:tex-math> </jats:alternatives> </jats:inline-formula>, a vector bundle <jats:inline-formula> <jats:alternatives> <jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" ","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"24 11","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138527116","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}
Pub Date : 2023-08-11DOI: 10.1142/s2811007223500086
Rajat Kaushik, Sandip Banerjee
{"title":"Predator-prey ecological system with group defense and anti-predator traits of the preys: Synergies between two important ecological actions","authors":"Rajat Kaushik, Sandip Banerjee","doi":"10.1142/s2811007223500086","DOIUrl":"https://doi.org/10.1142/s2811007223500086","url":null,"abstract":"","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"41 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-08-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"81412238","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}
Pub Date : 2023-07-28DOI: 10.1142/s2811007223500062
Pinghua Yang, Rongling Yang
{"title":"The explicit solution of a class of a higher order impulsive fractional differential equation involving Hadamard fractional derivative","authors":"Pinghua Yang, Rongling Yang","doi":"10.1142/s2811007223500062","DOIUrl":"https://doi.org/10.1142/s2811007223500062","url":null,"abstract":"","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"8 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-07-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89835659","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}
Pub Date : 2023-06-01Epub Date: 2023-04-21DOI: 10.1097/JTE.0000000000000282
Richard C Armitage, Eric Williamson
{"title":"The Impact of Industrial Action on Physical Therapy Education in the United Kingdom.","authors":"Richard C Armitage, Eric Williamson","doi":"10.1097/JTE.0000000000000282","DOIUrl":"10.1097/JTE.0000000000000282","url":null,"abstract":"","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"18 1","pages":"85-86"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"61625351","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}
Pub Date : 2023-05-26DOI: 10.1142/s2811007223500037
Samara Fatima, Naseem Abbas, M. Munawar, S. M. Eldin
{"title":"Ion-acoustic wave dynamics and sensitivity study in a magnetized Auroral phase plasma","authors":"Samara Fatima, Naseem Abbas, M. Munawar, S. M. Eldin","doi":"10.1142/s2811007223500037","DOIUrl":"https://doi.org/10.1142/s2811007223500037","url":null,"abstract":"","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"15 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-05-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87415706","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}
Pub Date : 2023-02-10DOI: 10.1142/s2811007223500013
Ying Song
{"title":"Global existence and decay rates of solutions for Vlasov-Navier-Stokes-Fokker-Planck equations with magnetic field","authors":"Ying Song","doi":"10.1142/s2811007223500013","DOIUrl":"https://doi.org/10.1142/s2811007223500013","url":null,"abstract":"","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"22 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-02-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89876605","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}
Abstract The prime objective of the approach is to give geometric classifications of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>k</m:mi> </m:math> k -almost Ricci solitons associated with paracontact manifolds. Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mrow> <m:mi>M</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>φ</m:mi> <m:mo>,</m:mo> <m:mi>ξ</m:mi> <m:mo>,</m:mo> <m:mi>η</m:mi> <m:mo>,</m:mo> <m:mi>g</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {M}^{2n+1}left(varphi ,xi ,eta ,g) be a paracontact metric manifold, and if a <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>K</m:mi> </m:math> K -paracontact metric <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>g</m:mi> </m:math> g represents a <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>k</m:mi> </m:math> k -almost Ricci soliton <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>g</m:mi> <m:mo>,</m:mo> <m:mi>V</m:mi> <m:mo>,</m:mo> <m:mi>k</m:mi> <m:mo>,</m:mo> <m:mi>λ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> left(g,V,k,lambda ) and the potential vector field <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>V</m:mi> </m:math> V is Jacobi field along the Reeb vector field <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ξ</m:mi> </m:math> xi , then either <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>k</m:mi> <m:mo>=</m:mo> <m:mi>λ</m:mi> <m:mo>−</m:mo> <m:mn>2</m:mn> <m:mi>n</m:mi> </m:math> k=lambda -2n , or <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>g</m:mi> </m:math> g is a <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>k</m:mi> </m:math> k -Ricci soliton. Next, we consider <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>K</m:mi> </m:math> K -paracontact manifold as a <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>k</m:mi> </m:math> k -almost Ricci soliton with the potential vector field <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>V</m:mi> </m:math> V is infinitesimal paracontact transformation or collinear with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ξ</m:mi> </m:math> xi . We have proved that if a paracontact metric as a <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>k</m:mi> </m:math> k -almost Ricci soliton associated with the non-zero potential vector field <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>V</m:mi> </m:math> V is collinear with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>ξ</m:mi> </m:math> xi and the Ricci operator <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>Q</m:mi> </m:math> Q commutes with paracontact structure <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>φ</m:mi> </m:math> varphi , then it is Einstein of constant scalar curvature equals to
该方法的主要目的是给出与副接触流形相关的k - k -几乎Ricci孤子的几何分类。设m2n +1 (φ， ξ， η，g) {M}^{2n+1}left (varphi, xi, eta,g)是一个副接触度量流形，如果一个K K -副接触度量g g表示一个K K -几乎里奇孤子(g,V, K， λ) left (g,V, K, lambda)并且势向量场V V是沿Reeb向量场ξ xi的Jacobi场，那么K = λ−2n K = lambda -2n，或者g是k k -里奇孤子。其次，我们将K K -副接触流形视为具有势向量场V V的K K -几乎Ricci孤子，V V是无穷小的副接触变换或与ξ xi共线。我们证明了如果一个与非零势向量场V V相关的k k -几乎Ricci孤子与ξ xi共线并且Ricci算子Q Q与副接触结构φ varphi交换，那么它就是常数曲率等于-2n (2n+1) -2n left (2n+1)的爱因斯坦。最后，我们推导出具有k - k -几乎Ricci孤子的准sasakian流形是常数曲率为-2n (2n+1) -2n left (2n+1)的Einstein。
{"title":"Geometric classifications of <i>k</i>-almost Ricci solitons admitting paracontact metrices","authors":"Yanlin Li, Dhriti Sundar Patra, Nadia Alluhaibi, Fatemah Mofarreh, Akram Ali","doi":"10.1515/math-2022-0610","DOIUrl":"https://doi.org/10.1515/math-2022-0610","url":null,"abstract":"Abstract The prime objective of the approach is to give geometric classifications of <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>k</m:mi> </m:math> k -almost Ricci solitons associated with paracontact manifolds. Let <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:msup> <m:mrow> <m:mi>M</m:mi> </m:mrow> <m:mrow> <m:mn>2</m:mn> <m:mi>n</m:mi> <m:mo>+</m:mo> <m:mn>1</m:mn> </m:mrow> </m:msup> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>φ</m:mi> <m:mo>,</m:mo> <m:mi>ξ</m:mi> <m:mo>,</m:mo> <m:mi>η</m:mi> <m:mo>,</m:mo> <m:mi>g</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> {M}^{2n+1}left(varphi ,xi ,eta ,g) be a paracontact metric manifold, and if a <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>K</m:mi> </m:math> K -paracontact metric <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>g</m:mi> </m:math> g represents a <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>k</m:mi> </m:math> k -almost Ricci soliton <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mrow> <m:mo>(</m:mo> <m:mrow> <m:mi>g</m:mi> <m:mo>,</m:mo> <m:mi>V</m:mi> <m:mo>,</m:mo> <m:mi>k</m:mi> <m:mo>,</m:mo> <m:mi>λ</m:mi> </m:mrow> <m:mo>)</m:mo> </m:mrow> </m:math> left(g,V,k,lambda ) and the potential vector field <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>V</m:mi> </m:math> V is Jacobi field along the Reeb vector field <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>ξ</m:mi> </m:math> xi , then either <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>k</m:mi> <m:mo>=</m:mo> <m:mi>λ</m:mi> <m:mo>−</m:mo> <m:mn>2</m:mn> <m:mi>n</m:mi> </m:math> k=lambda -2n , or <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>g</m:mi> </m:math> g is a <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>k</m:mi> </m:math> k -Ricci soliton. Next, we consider <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>K</m:mi> </m:math> K -paracontact manifold as a <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>k</m:mi> </m:math> k -almost Ricci soliton with the potential vector field <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>V</m:mi> </m:math> V is infinitesimal paracontact transformation or collinear with <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>ξ</m:mi> </m:math> xi . We have proved that if a paracontact metric as a <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>k</m:mi> </m:math> k -almost Ricci soliton associated with the non-zero potential vector field <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>V</m:mi> </m:math> V is collinear with <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>ξ</m:mi> </m:math> xi and the Ricci operator <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>Q</m:mi> </m:math> Q commutes with paracontact structure <m:math xmlns:m=\"http://www.w3.org/1998/Math/MathML\"> <m:mi>φ</m:mi> </m:math> varphi , then it is Einstein of constant scalar curvature equals to","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"235 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136092710","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}
Abstract In this article, we obtain some sufficient conditions for the boundedness of commutators of p p -adic Hardy-Littlewood operators with symbols in central bounded mean oscillation space and Lipschitz space on the p p -adic function spaces with variable exponents such as the p p -adic local central Morrey, p p -adic Morrey-Herz, and p p -adic local block spaces with variable exponents.
{"title":"Commutators of Hardy-Littlewood operators on p-adic function spaces with variable exponents","authors":"K. Dung, P. T. Thuy","doi":"10.1515/math-2022-0579","DOIUrl":"https://doi.org/10.1515/math-2022-0579","url":null,"abstract":"Abstract In this article, we obtain some sufficient conditions for the boundedness of commutators of p p -adic Hardy-Littlewood operators with symbols in central bounded mean oscillation space and Lipschitz space on the p p -adic function spaces with variable exponents such as the p p -adic local central Morrey, p p -adic Morrey-Herz, and p p -adic local block spaces with variable exponents.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":" ","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49274183","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}
Pub Date : 2023-01-01DOI: 10.1142/s2811007223500049
S. Zawka, Temesgen T. Melese
This paper is concerned with the dynamics and optimal harvesting of a prey–predator system in a polluted environment in the presence of scavengers and pollution control. Toxicants, released from external sources and the dead bodies of prey and predators, pollute the environment, which affects the growth of both prey and predators, resulting in a decline in the economic revenue from harvest. We assume that scavengers reduce pollution by consuming dead bodies. Further, we consider pollution reduction through depollution efforts as an alternative to enhancing revenue. We propose and analyze a prey–predator–pollutant model and study the optimal harvesting problem. We investigate the persistence of the ecosystem, and we solve the optimal harvest problem using Pontryagin’s maximum principle. The results indicate that uncontrolled prey harvesting and a high rate of pollution drive the system toward the extinction of both species. A moderate amount of pollution and the reasonable harvest efforts allow the system to persist. The optimal harvest strategy highlights that investing in pollution reduction enhances the persistence of the system as well as economic revenue. Numerical examples demonstrate the significant outcomes of the study.
{"title":"Dynamics and optimal harvesting of prey–predator in a polluted environment in the presence of scavenger and pollution control","authors":"S. Zawka, Temesgen T. Melese","doi":"10.1142/s2811007223500049","DOIUrl":"https://doi.org/10.1142/s2811007223500049","url":null,"abstract":"This paper is concerned with the dynamics and optimal harvesting of a prey–predator system in a polluted environment in the presence of scavengers and pollution control. Toxicants, released from external sources and the dead bodies of prey and predators, pollute the environment, which affects the growth of both prey and predators, resulting in a decline in the economic revenue from harvest. We assume that scavengers reduce pollution by consuming dead bodies. Further, we consider pollution reduction through depollution efforts as an alternative to enhancing revenue. We propose and analyze a prey–predator–pollutant model and study the optimal harvesting problem. We investigate the persistence of the ecosystem, and we solve the optimal harvest problem using Pontryagin’s maximum principle. The results indicate that uncontrolled prey harvesting and a high rate of pollution drive the system toward the extinction of both species. A moderate amount of pollution and the reasonable harvest efforts allow the system to persist. The optimal harvest strategy highlights that investing in pollution reduction enhances the persistence of the system as well as economic revenue. Numerical examples demonstrate the significant outcomes of the study.","PeriodicalId":48713,"journal":{"name":"Open Mathematics","volume":"3 1","pages":""},"PeriodicalIF":1.7,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"72681321","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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