Pub Date : 2023-12-15DOI: 10.1007/s11587-023-00829-2
Ines Ben Omrane, Mourad Ben Slimane, Sadek Gala, Maria Alessandra Ragusa
This paper is devoted to investigating regularity criteria for the 3D micropolar fluid equations in terms of pressure in weak Lebesgue space. More precisely, we mainly proved that the weak solution is regular on (0, T] provided that either the norm (left| pi right| _{L^{alpha ,infty }(0,T;L^{beta ,infty }(mathbb {R}^{3}))}) with (frac{2}{alpha }+ frac{3}{beta }=2) and (frac{3}{2}<beta <infty ) or (left| nabla pi right| _{L^{alpha ,infty }(0,T;L^{beta ,infty }(mathbb {R} ^{3}))}) with (frac{2}{alpha }+frac{3}{beta }=3) and (1<beta <infty ) is sufficiently small.
本文致力于研究三维微波流体方程在弱 Lebesgue 空间中的正则性准则。更确切地说，我们主要证明了弱解在 (0, T] 上是正则的，条件是规范 (left| pi right| _{L^{alpha ,infty }(0,T.)；L^{beta ,infty }(mathbb {R}^{3}))}) with (frac{2}{alpha }+ frac{3}{beta }=2) and(frac{3}{2}<；beta <infty ) or(left| nabla pi right| _{L^{alpha ,infty }(0,T.)) 和L^{beta ,infty }(mathbb {R} ^{3}))}) with (frac{2}{alpha }+frac{3}{beta }=3) and(1<beta <infty) is sufficiently small.
{"title":"A weak- $$L^{p}$$ Prodi–Serrin type regularity criterion for the micropolar fluid equations in terms of the pressure","authors":"Ines Ben Omrane, Mourad Ben Slimane, Sadek Gala, Maria Alessandra Ragusa","doi":"10.1007/s11587-023-00829-2","DOIUrl":"https://doi.org/10.1007/s11587-023-00829-2","url":null,"abstract":"<p>This paper is devoted to investigating regularity criteria for the 3D micropolar fluid equations in terms of pressure in weak Lebesgue space. More precisely, we mainly proved that the weak solution is regular on (0, <i>T</i>] provided that either the norm <span>(left| pi right| _{L^{alpha ,infty }(0,T;L^{beta ,infty }(mathbb {R}^{3}))})</span> with <span>(frac{2}{alpha }+ frac{3}{beta }=2)</span> and <span>(frac{3}{2}<beta <infty )</span> or <span>(left| nabla pi right| _{L^{alpha ,infty }(0,T;L^{beta ,infty }(mathbb {R} ^{3}))})</span> with <span>(frac{2}{alpha }+frac{3}{beta }=3)</span> and <span>(1<beta <infty )</span> is sufficiently small.\u0000</p>","PeriodicalId":21373,"journal":{"name":"Ricerche di Matematica","volume":"5 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138691173","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}
is considered in a bounded domain (Omega subset {mathbb {R}}^3). Uniform energy decay was discussed which depends on the relaxation function (-k'(s)) in the previous work (Guo et al., Z Angew Math Phys 69:65, 2018) for (1le mle 5). Depending on a key integral inequality obtained by Martinez (ESAIM Control Optim Calc Var 4:419–444, 1999), we establish the decay estimate of the total energy for (m>5). Our results improve and complement the previous one. As an example, a logarithmic energy decay is also presented.
本文考虑了有界域 (Omega subset {mathbb {R}}^3) 中的一类阻尼粘弹性波方程 $$begin{aligned}u_{tt}-k(0)Delta u-int _0^infty k'(s)Delta u(t-s)ds+|u_t|^{m-1}u_t=|u|^{p-1}u end{aligned}$$。在之前的工作（Guo et al., Z Angew Math Phys 69:65, 2018）中，针对(1le mle 5)讨论了取决于弛豫函数(-k'(s))的均匀能量衰减。根据马丁内斯（ESAIM Control Optim Calc Var 4:419-444，1999）获得的关键积分不等式，我们建立了对（m>5 ）总能量的衰变估计。我们的结果改进并补充了之前的结果。作为一个例子，我们还提出了对数能量衰减。
{"title":"Energy decay of wave equations with infinite memory effects versus supercritical frictional dampings","authors":"Menglan Liao","doi":"10.1007/s11587-023-00832-7","DOIUrl":"https://doi.org/10.1007/s11587-023-00832-7","url":null,"abstract":"<p>In this paper, a class of damped viscoelastic wave equations </p><span>$$begin{aligned} u_{tt}-k(0)Delta u-int _0^infty k'(s)Delta u(t-s)ds+|u_t|^{m-1}u_t=|u|^{p-1}u end{aligned}$$</span><p>is considered in a bounded domain <span>(Omega subset {mathbb {R}}^3)</span>. Uniform energy decay was discussed which depends on the relaxation function <span>(-k'(s))</span> in the previous work (Guo et al., Z Angew Math Phys 69:65, 2018) for <span>(1le mle 5)</span>. Depending on a key integral inequality obtained by Martinez (ESAIM Control Optim Calc Var 4:419–444, 1999), we establish the decay estimate of the total energy for <span>(m>5)</span>. Our results improve and complement the previous one. As an example, a logarithmic energy decay is also presented.</p>","PeriodicalId":21373,"journal":{"name":"Ricerche di Matematica","volume":"56 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-12-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138565956","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 : 2023-12-09DOI: 10.1007/s11587-023-00831-8
Elvira Barbera, Annamaria Pollino
A hyperbolic reaction–diffusion model is developed in the framework of Extended Thermodynamics in order to describe the spatio-temporal dynamics of populations afflicted by chronic wasting diseases. The hyperbolic structure of the system guarantees that the wave processes occur at finite velocity, so that the paradox of instantaneous diffusion, typical of parabolic systems, is removed. The character of steady states, together with the Hopf bifurcation, are investigated through linear stability analysis. The model is integrated numerically to valuate the behavior of the populations. Finally, the propagation of acceleration waves is analyzed.
{"title":"A hyperbolic reaction–diffusion model of chronic wasting disease","authors":"Elvira Barbera, Annamaria Pollino","doi":"10.1007/s11587-023-00831-8","DOIUrl":"https://doi.org/10.1007/s11587-023-00831-8","url":null,"abstract":"<p>A hyperbolic reaction–diffusion model is developed in the framework of Extended Thermodynamics in order to describe the spatio-temporal dynamics of populations afflicted by chronic wasting diseases. The hyperbolic structure of the system guarantees that the wave processes occur at finite velocity, so that the paradox of instantaneous diffusion, typical of parabolic systems, is removed. The character of steady states, together with the Hopf bifurcation, are investigated through linear stability analysis. The model is integrated numerically to valuate the behavior of the populations. Finally, the propagation of acceleration waves is analyzed.\u0000</p>","PeriodicalId":21373,"journal":{"name":"Ricerche di Matematica","volume":"17 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-12-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138563906","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 : 2023-12-02DOI: 10.1007/s11587-023-00830-9
Parveen, Jitender Kumar
The power graph of a finite group G is a simple undirected graph with vertex set G and two vertices are adjacent if one is a power of the other. The enhanced power graph of a finite group G is a simple undirected graph whose vertex set is the group G and two vertices a, b are adjacent if there exists (c in G) such that both a and b are powers of c. In this paper, we study the difference graph (mathcal {D}(G)) of a finite group G which is the difference of the enhanced power graph and the power graph of G with all isolated vertices removed. We characterize all the finite nilpotent groups G such that the genus (or cross-cap) of the difference graph (mathcal {D}(G)) is at most 2.
有限群G的幂图是一个简单的无向图，其顶点集G和两个顶点相邻，如果其中一个是另一个的幂。有限群G的增强幂图是一个简单无向图，其顶点集为群G，且两个顶点a, b相邻，如果存在(c in G)使得a和b都是c的幂。本文研究了有限群G的差分图(mathcal {D}(G))，该差分图是所有孤立顶点被去掉后增强幂图与G的幂图之差。我们刻画了所有有限幂零群G，使得差分图(mathcal {D}(G))的属(或交叉帽)不超过2。
{"title":"Nilpotent groups whose difference graphs have positive genus","authors":"Parveen, Jitender Kumar","doi":"10.1007/s11587-023-00830-9","DOIUrl":"https://doi.org/10.1007/s11587-023-00830-9","url":null,"abstract":"<p>The power graph of a finite group <i>G</i> is a simple undirected graph with vertex set <i>G</i> and two vertices are adjacent if one is a power of the other. The enhanced power graph of a finite group <i>G</i> is a simple undirected graph whose vertex set is the group <i>G</i> and two vertices <i>a</i>, <i>b</i> are adjacent if there exists <span>(c in G)</span> such that both <i>a</i> and <i>b</i> are powers of <i>c</i>. In this paper, we study the difference graph <span>(mathcal {D}(G))</span> of a finite group <i>G</i> which is the difference of the enhanced power graph and the power graph of <i>G</i> with all isolated vertices removed. We characterize all the finite nilpotent groups <i>G</i> such that the genus (or cross-cap) of the difference graph <span>(mathcal {D}(G))</span> is at most 2.</p>","PeriodicalId":21373,"journal":{"name":"Ricerche di Matematica","volume":"64 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138514792","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 : 2023-11-27DOI: 10.1007/s11587-023-00826-5
Gessica Alecci, Stefano Barbero, Nadir Murru
Several operations can be defined on the set of all linear recurrent sequences, such as the binomial convolution (Hurwitz product) or the multinomial convolution (Newton product). Using elementary techniques, we prove that this set equipped with the termwise sum and the aforementioned products is an R-algebra, given any commutative ring R with identity. Moreover, we provide explicitly a characteristic polynomial of the Hurwitz product and Newton product of any two linear recurrent sequences. Finally, we also investigate whether these R-algebras are isomorphic, considering also the R-algebras obtained using the Hadamard product and the convolution product.
{"title":"Some notes on the algebraic structure of linear recurrent sequences","authors":"Gessica Alecci, Stefano Barbero, Nadir Murru","doi":"10.1007/s11587-023-00826-5","DOIUrl":"https://doi.org/10.1007/s11587-023-00826-5","url":null,"abstract":"<p>Several operations can be defined on the set of all linear recurrent sequences, such as the binomial convolution (Hurwitz product) or the multinomial convolution (Newton product). Using elementary techniques, we prove that this set equipped with the termwise sum and the aforementioned products is an <i>R</i>-algebra, given any commutative ring <i>R</i> with identity. Moreover, we provide explicitly a characteristic polynomial of the Hurwitz product and Newton product of any two linear recurrent sequences. Finally, we also investigate whether these <i>R</i>-algebras are isomorphic, considering also the <i>R</i>-algebras obtained using the Hadamard product and the convolution product.\u0000</p>","PeriodicalId":21373,"journal":{"name":"Ricerche di Matematica","volume":"84 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138514793","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 : 2023-11-26DOI: 10.1007/s11587-023-00825-6
Giovanni Cimatti
The Galerkin’s method is applied to prove the existence of at least one solution of the initial boundary value problem for the nonlinear system of partial differential equations modelling the electromagnetic heating of materials. In addition a criteria of stability and instability based on a Liapunoff function is presented.
{"title":"Existence of solutions for the problem of microwave heating and a Liapunoff criteria for stability and instability","authors":"Giovanni Cimatti","doi":"10.1007/s11587-023-00825-6","DOIUrl":"https://doi.org/10.1007/s11587-023-00825-6","url":null,"abstract":"<p>The Galerkin’s method is applied to prove the existence of at least one solution of the initial boundary value problem for the nonlinear system of partial differential equations modelling the electromagnetic heating of materials. In addition a criteria of stability and instability based on a Liapunoff function is presented.</p>","PeriodicalId":21373,"journal":{"name":"Ricerche di Matematica","volume":"62 1","pages":""},"PeriodicalIF":1.2,"publicationDate":"2023-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138514766","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 : 2023-11-04DOI: 10.1007/s11587-023-00822-9
C. Accarino, F. Capone, R. De Luca, G. Massa
Abstract In this paper, a predator–prey model with intraguild predation describing the evolution between three interacting species—namely prey, mesopredator and top predator—is investigated, with the aim to model a complete food web. In particular, the longtime behaviour of the solutions is analysed, proving the existence of an absorbing set, and the linear and nonlinear stability analyses of the coexistence equilibrium are performed.
{"title":"On the dynamics of a Leslie–Gower predator–prey ternary model with intraguild","authors":"C. Accarino, F. Capone, R. De Luca, G. Massa","doi":"10.1007/s11587-023-00822-9","DOIUrl":"https://doi.org/10.1007/s11587-023-00822-9","url":null,"abstract":"Abstract In this paper, a predator–prey model with intraguild predation describing the evolution between three interacting species—namely prey, mesopredator and top predator—is investigated, with the aim to model a complete food web. In particular, the longtime behaviour of the solutions is analysed, proving the existence of an absorbing set, and the linear and nonlinear stability analyses of the coexistence equilibrium are performed.","PeriodicalId":21373,"journal":{"name":"Ricerche di Matematica","volume":"12 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135773372","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 : 2023-11-03DOI: 10.1007/s11587-023-00824-7
Wenbin Yang, Xin Chang
{"title":"Hopf bifurcation and Turing patterns for a diffusive predator–prey system with weak Allee effect","authors":"Wenbin Yang, Xin Chang","doi":"10.1007/s11587-023-00824-7","DOIUrl":"https://doi.org/10.1007/s11587-023-00824-7","url":null,"abstract":"","PeriodicalId":21373,"journal":{"name":"Ricerche di Matematica","volume":"36 4","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135818302","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 : 2023-10-25DOI: 10.1007/s11587-023-00809-6
Alexander Holguín-Villa, John H. Castillo
Abstract Let $$mathbb {F}G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> denote the group algebra of a locally finite group G over the infinite field $$mathbb {F}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>F</mml:mi> </mml:math> with $$mathop {textrm{char}}nolimits (mathbb {F})ne 2$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mtext>char</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>F</mml:mi> <mml:mo>)</mml:mo> <mml:mo>≠</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , and let $$circledast :mathbb {F}Grightarrow mathbb {F}G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>⊛</mml:mo> <mml:mo>:</mml:mo> <mml:mi>F</mml:mi> <mml:mi>G</mml:mi> <mml:mo>→</mml:mo> <mml:mi>F</mml:mi> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> denote the involution defined by $$alpha =Sigma alpha _{g}g mapsto alpha ^circledast =Sigma alpha _{g}sigma (g)g^{*}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>=</mml:mo> <mml:mi>Σ</mml:mi> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mi>g</mml:mi> <mml:mo>↦</mml:mo> <mml:msup> <mml:mi>α</mml:mi> <mml:mo>⊛</mml:mo> </mml:msup> <mml:mo>=</mml:mo> <mml:mi>Σ</mml:mi> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mi>σ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>g</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:msup> <mml:mi>g</mml:mi> <mml:mrow> <mml:mrow /> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , where $$sigma :Grightarrow {pm 1}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>σ</mml:mi> <mml:mo>:</mml:mo> <mml:mi>G</mml:mi> <mml:mo>→</mml:mo> <mml:mo>{</mml:mo> <mml:mo>±</mml:mo> <mml:mn>1</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> is a group homomorphism (called an orientation) and $$*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mrow /> <mml:mo>∗</mml:mo> </mml:mrow> </mml:math> is an involution of the group G . In this paper we prove, under some assumptions, that if the $$circledast $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mo>⊛</mml:mo> </mml:math> -symmetric units of $$mathbb {F}G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> satisfies a group identity then $$mathbb {F}G$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> satisfies a polynomial identity, i.e., we give an affirmative answer to a Conjecture of B. Hartley in this setting. Moreover, in the case when the prime radical $$eta (mathbb {F}G)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>η</mml:mi> <mml:mo>(</mml:mo> <mml:mi>F</mml:mi> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:m
{"title":"Group identities on symmetric units under oriented involutions in group algebras","authors":"Alexander Holguín-Villa, John H. Castillo","doi":"10.1007/s11587-023-00809-6","DOIUrl":"https://doi.org/10.1007/s11587-023-00809-6","url":null,"abstract":"Abstract Let $$mathbb {F}G$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> denote the group algebra of a locally finite group G over the infinite field $$mathbb {F}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>F</mml:mi> </mml:math> with $$mathop {textrm{char}}nolimits (mathbb {F})ne 2$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mtext>char</mml:mtext> <mml:mo>(</mml:mo> <mml:mi>F</mml:mi> <mml:mo>)</mml:mo> <mml:mo>≠</mml:mo> <mml:mn>2</mml:mn> </mml:mrow> </mml:math> , and let $$circledast :mathbb {F}Grightarrow mathbb {F}G$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mo>⊛</mml:mo> <mml:mo>:</mml:mo> <mml:mi>F</mml:mi> <mml:mi>G</mml:mi> <mml:mo>→</mml:mo> <mml:mi>F</mml:mi> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> denote the involution defined by $$alpha =Sigma alpha _{g}g mapsto alpha ^circledast =Sigma alpha _{g}sigma (g)g^{*}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>α</mml:mi> <mml:mo>=</mml:mo> <mml:mi>Σ</mml:mi> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mi>g</mml:mi> <mml:mo>↦</mml:mo> <mml:msup> <mml:mi>α</mml:mi> <mml:mo>⊛</mml:mo> </mml:msup> <mml:mo>=</mml:mo> <mml:mi>Σ</mml:mi> <mml:msub> <mml:mi>α</mml:mi> <mml:mi>g</mml:mi> </mml:msub> <mml:mi>σ</mml:mi> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>g</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> <mml:msup> <mml:mi>g</mml:mi> <mml:mrow> <mml:mrow /> <mml:mo>∗</mml:mo> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> , where $$sigma :Grightarrow {pm 1}$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>σ</mml:mi> <mml:mo>:</mml:mo> <mml:mi>G</mml:mi> <mml:mo>→</mml:mo> <mml:mo>{</mml:mo> <mml:mo>±</mml:mo> <mml:mn>1</mml:mn> <mml:mo>}</mml:mo> </mml:mrow> </mml:math> is a group homomorphism (called an orientation) and $$*$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mrow /> <mml:mo>∗</mml:mo> </mml:mrow> </mml:math> is an involution of the group G . In this paper we prove, under some assumptions, that if the $$circledast $$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mo>⊛</mml:mo> </mml:math> -symmetric units of $$mathbb {F}G$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> satisfies a group identity then $$mathbb {F}G$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>F</mml:mi> <mml:mi>G</mml:mi> </mml:mrow> </mml:math> satisfies a polynomial identity, i.e., we give an affirmative answer to a Conjecture of B. Hartley in this setting. Moreover, in the case when the prime radical $$eta (mathbb {F}G)$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mrow> <mml:mi>η</mml:mi> <mml:mo>(</mml:mo> <mml:mi>F</mml:mi> <mml:mi>G</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:m","PeriodicalId":21373,"journal":{"name":"Ricerche di Matematica","volume":"40 9","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"134972395","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 : 2023-10-11DOI: 10.1007/s11587-023-00819-4
F. Capone, R. De Luca, L. Fiorentino, V. Luongo, G. Massa
Abstract The aim of this paper is to investigate a reaction-diffusion Leslie–Gower predator–prey model, incorporating the intraguild predation and both self and cross-diffusion. The longtime behaviour of the solutions is analysed, proving the existence of an absorbing set. The existence of patterns is investigated by looking for conditions guaranteeing that an equilibrium, stable in the absence of diffusion, becomes unstable when diffusion is allowed.
{"title":"Turing instability for a Leslie–Gower model","authors":"F. Capone, R. De Luca, L. Fiorentino, V. Luongo, G. Massa","doi":"10.1007/s11587-023-00819-4","DOIUrl":"https://doi.org/10.1007/s11587-023-00819-4","url":null,"abstract":"Abstract The aim of this paper is to investigate a reaction-diffusion Leslie–Gower predator–prey model, incorporating the intraguild predation and both self and cross-diffusion. The longtime behaviour of the solutions is analysed, proving the existence of an absorbing set. The existence of patterns is investigated by looking for conditions guaranteeing that an equilibrium, stable in the absence of diffusion, becomes unstable when diffusion is allowed.","PeriodicalId":21373,"journal":{"name":"Ricerche di Matematica","volume":"198 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136209027","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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