Pub Date : 2024-11-26DOI: 10.1016/j.jcta.2024.105983
Hongxue Liang , Alessandro Montinaro
In this paper, we provide a complete classification of 2- designs admitting a flag-transitive automorphism group of affine type with the only exception of the semilinear 1-dimensional group. Alongside this analysis, we provide a construction of seven new families of such flag-transitive 2-designs, one of them infinite, and some of them involving remarkable objects such as t-spreads, translation planes, quadrics and Segre varieties.
Our result together with those of Alavi et al. [1], [2], Praeger et al. [17], Zhou and the first author [39], [40] provides a complete classification of 2- design admitting a flag-transitive automorphism group with the only exception of the semilinear 1-dimensional case.
{"title":"A classification of the flag-transitive 2-(v,k,2) designs","authors":"Hongxue Liang , Alessandro Montinaro","doi":"10.1016/j.jcta.2024.105983","DOIUrl":"10.1016/j.jcta.2024.105983","url":null,"abstract":"<div><div>In this paper, we provide a complete classification of 2-<span><math><mo>(</mo><mi>v</mi><mo>,</mo><mi>k</mi><mo>,</mo><mn>2</mn><mo>)</mo></math></span> designs admitting a flag-transitive automorphism group of affine type with the only exception of the semilinear 1-dimensional group. Alongside this analysis, we provide a construction of seven new families of such flag-transitive 2-designs, one of them infinite, and some of them involving remarkable objects such as <em>t</em>-spreads, translation planes, quadrics and Segre varieties.</div><div>Our result together with those of Alavi et al. <span><span>[1]</span></span>, <span><span>[2]</span></span>, Praeger et al. <span><span>[17]</span></span>, Zhou and the first author <span><span>[39]</span></span>, <span><span>[40]</span></span> provides a complete classification of 2-<span><math><mo>(</mo><mi>v</mi><mo>,</mo><mi>k</mi><mo>,</mo><mn>2</mn><mo>)</mo></math></span> design admitting a flag-transitive automorphism group with the only exception of the semilinear 1-dimensional case.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"211 ","pages":"Article 105983"},"PeriodicalIF":0.9,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142701472","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-26DOI: 10.1016/j.camwa.2024.11.024
Jun Liu , Chenxi Ji , Wenbin Ye , Lei Gan , Lei Qin , Quansheng Zang , Haibo Wang
In this paper, a modified scaled boundary finite element method (SBFEM) is developed to study static and vibration behaviors of laminated conical shells under the conical coordinate system. In the modified SBFEM, the geometry of the conical shell is defined entirely by scaling the internal surface of the structure. This approach eliminates geometric errors caused by discretization, thereby enhancing modeling accuracy. The three-dimensional problem is simplified to a two-dimensional analysis since discretization is only applied to the boundary of the computational domain. Additionally, the semi-analytic property of the SBFEM allows for the derivation of a linear analytical solution for the laminated conical shell in the radial direction. First, a scaled boundary coordinate system for the scaling surface is established, and a second-order scaled boundary finite element governing equation with variable coefficients is derived for a single layer of the conical shell using the principle of virtual work. Next, the governing equation is transformed into a first-order system by introducing a combined vector of displacement and nodal force, and the stiffness matrices for each layer of the laminated conical shell are obtained using the precise integration method. Finally, an overall analysis of the laminated structure is conducted by assembling each single-layer structure while considering the continuity boundary condition at interfaces. Static and vibration analyses of laminated conical shells are conducted, and the results are compared with those from the literature to demonstrate the adaptability and convergence of the proposed method. Several numerical examples are presented to examine the effects of various geometric parameters, such as thickness, length, semi-vertex angles, layup directions, and stacking sequences, on the responses of the structure.
{"title":"Static and vibration analyses of laminated conical shells under various boundary conditions using a modified scaled boundary finite element method","authors":"Jun Liu , Chenxi Ji , Wenbin Ye , Lei Gan , Lei Qin , Quansheng Zang , Haibo Wang","doi":"10.1016/j.camwa.2024.11.024","DOIUrl":"10.1016/j.camwa.2024.11.024","url":null,"abstract":"<div><div>In this paper, a modified scaled boundary finite element method (SBFEM) is developed to study static and vibration behaviors of laminated conical shells under the conical coordinate system. In the modified SBFEM, the geometry of the conical shell is defined entirely by scaling the internal surface of the structure. This approach eliminates geometric errors caused by discretization, thereby enhancing modeling accuracy. The three-dimensional problem is simplified to a two-dimensional analysis since discretization is only applied to the boundary of the computational domain. Additionally, the semi-analytic property of the SBFEM allows for the derivation of a linear analytical solution for the laminated conical shell in the radial direction. First, a scaled boundary coordinate system for the scaling surface is established, and a second-order scaled boundary finite element governing equation with variable coefficients is derived for a single layer of the conical shell using the principle of virtual work. Next, the governing equation is transformed into a first-order system by introducing a combined vector of displacement and nodal force, and the stiffness matrices for each layer of the laminated conical shell are obtained using the precise integration method. Finally, an overall analysis of the laminated structure is conducted by assembling each single-layer structure while considering the continuity boundary condition at interfaces. Static and vibration analyses of laminated conical shells are conducted, and the results are compared with those from the literature to demonstrate the adaptability and convergence of the proposed method. Several numerical examples are presented to examine the effects of various geometric parameters, such as thickness, length, semi-vertex angles, layup directions, and stacking sequences, on the responses of the structure.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"177 ","pages":"Pages 147-166"},"PeriodicalIF":2.9,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142698885","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-26DOI: 10.1007/s11040-024-09496-7
Peter H. van der Kamp, G. R. W. Quispel, David I. McLaren
To any tree on n vertices we associate an n-dimensional Lotka–Volterra system with (3n-2) parameters and, for generic values of the parameters, prove it is superintegrable, i.e. it admits (n-1) functionally independent integrals. We also show how each system can be reduced to an ((n-1))-dimensional system which is superintegrable and solvable by quadratures.
{"title":"Trees and Superintegrable Lotka–Volterra Families","authors":"Peter H. van der Kamp, G. R. W. Quispel, David I. McLaren","doi":"10.1007/s11040-024-09496-7","DOIUrl":"10.1007/s11040-024-09496-7","url":null,"abstract":"<div><p>To any tree on <i>n</i> vertices we associate an <i>n</i>-dimensional Lotka–Volterra system with <span>(3n-2)</span> parameters and, for generic values of the parameters, prove it is superintegrable, i.e. it admits <span>(n-1)</span> functionally independent integrals. We also show how each system can be reduced to an (<span>(n-1)</span>)-dimensional system which is superintegrable and solvable by quadratures.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"27 4","pages":""},"PeriodicalIF":0.9,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142714575","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-26DOI: 10.1007/s10455-024-09981-w
Izar Alonso
We consider two different (text {SU}(2)^2)-invariant cohomogeneity one manifolds, one non-compact (M=mathbb {R}^4 times S^3) and one compact (M=S^4 times S^3), and study the existence of coclosed (text {SU}(2)^2)-invariant (G_2)-structures constructed from half-flat (text {SU}(3))-structures. For (mathbb {R}^4 times S^3), we prove the existence of a family of coclosed (but not necessarily torsion-free) (G_2)-structures which is given by three smooth functions satisfying certain boundary conditions around the singular orbit and a non-zero parameter. Moreover, any coclosed (G_2)-structure constructed from a half-flat (text {SU}(3))-structure is in this family. For (S^4 times S^3), we prove that there are no (text {SU}(2)^2)-invariant coclosed (G_2)-structures constructed from half-flat (text {SU}(3))-structures.
{"title":"Coclosed (G_2)-structures on (text {SU}(2)^2)-invariant cohomogeneity one manifolds","authors":"Izar Alonso","doi":"10.1007/s10455-024-09981-w","DOIUrl":"10.1007/s10455-024-09981-w","url":null,"abstract":"<div><p>We consider two different <span>(text {SU}(2)^2)</span>-invariant cohomogeneity one manifolds, one non-compact <span>(M=mathbb {R}^4 times S^3)</span> and one compact <span>(M=S^4 times S^3)</span>, and study the existence of coclosed <span>(text {SU}(2)^2)</span>-invariant <span>(G_2)</span>-structures constructed from half-flat <span>(text {SU}(3))</span>-structures. For <span>(mathbb {R}^4 times S^3)</span>, we prove the existence of a family of coclosed (but not necessarily torsion-free) <span>(G_2)</span>-structures which is given by three smooth functions satisfying certain boundary conditions around the singular orbit and a non-zero parameter. Moreover, any coclosed <span>(G_2)</span>-structure constructed from a half-flat <span>(text {SU}(3))</span>-structure is in this family. For <span>(S^4 times S^3)</span>, we prove that there are no <span>(text {SU}(2)^2)</span>-invariant coclosed <span>(G_2)</span>-structures constructed from half-flat <span>(text {SU}(3))</span>-structures.</p></div>","PeriodicalId":8268,"journal":{"name":"Annals of Global Analysis and Geometry","volume":"67 1","pages":""},"PeriodicalIF":0.6,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10455-024-09981-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142714428","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-26DOI: 10.1016/j.dam.2024.11.022
Xiaofeng Gu , Jessica Li , Eric H. Yang , William Y. Zhang
Let be a connected graph. A cyclic base ordering of is a cyclic ordering of elements in such that every cyclically consecutive edges form a spanning tree of . The density of is defined to be ; and is uniformly dense if for every connected subgraph of . It was conjectured by Kajitani, Ueno and Miyano that has a cyclic base ordering if and only if is uniformly dense. We show that the conjecture holds for maximal 2-degenerate graphs and graphs with uniform ear decompositions. As applications, book graphs, broken fan and broken wheel graphs have cyclic base ordering. We also study cyclic base ordering of double wheel graphs and the square of cycles.
设 G 是一个连通图。G 的循环基序是 E(G) 中元素的循环排序，使得每个循环连续的 |V(G)|-1 边构成 G 的生成树。G 的密度定义为 d(G)=|E(G)|/(|V(G)|-1)；如果对于 G 的每个连通子图 H，d(H)≤d(G)，则 G 是均匀致密的。我们证明了这一猜想在最大 2-degenerate 图和具有均匀耳分解的图中成立。作为应用，书图、破扇图和破轮图都具有循环基序。我们还研究了双轮图的循环基序和循环平方。
{"title":"Cyclic base ordering of certain degenerate graphs","authors":"Xiaofeng Gu , Jessica Li , Eric H. Yang , William Y. Zhang","doi":"10.1016/j.dam.2024.11.022","DOIUrl":"10.1016/j.dam.2024.11.022","url":null,"abstract":"<div><div>Let <span><math><mi>G</mi></math></span> be a connected graph. A cyclic base ordering of <span><math><mi>G</mi></math></span> is a cyclic ordering of elements in <span><math><mrow><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> such that every cyclically consecutive <span><math><mrow><mrow><mo>|</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>−</mo><mn>1</mn></mrow></math></span> edges form a spanning tree of <span><math><mi>G</mi></math></span>. The density of <span><math><mi>G</mi></math></span> is defined to be <span><math><mrow><mi>d</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>=</mo><mrow><mo>|</mo><mi>E</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>/</mo><mrow><mo>(</mo><mrow><mo>|</mo><mi>V</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>|</mo></mrow><mo>−</mo><mn>1</mn><mo>)</mo></mrow></mrow></math></span>; and <span><math><mi>G</mi></math></span> is uniformly dense if <span><math><mrow><mi>d</mi><mrow><mo>(</mo><mi>H</mi><mo>)</mo></mrow><mo>≤</mo><mi>d</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> for every connected subgraph <span><math><mi>H</mi></math></span> of <span><math><mi>G</mi></math></span>. It was conjectured by Kajitani, Ueno and Miyano that <span><math><mi>G</mi></math></span> has a cyclic base ordering if and only if <span><math><mi>G</mi></math></span> is uniformly dense. We show that the conjecture holds for maximal 2-degenerate graphs and graphs with uniform ear decompositions. As applications, book graphs, broken fan and broken wheel graphs have cyclic base ordering. We also study cyclic base ordering of double wheel graphs and the square of cycles.</div></div>","PeriodicalId":50573,"journal":{"name":"Discrete Applied Mathematics","volume":"362 ","pages":"Pages 148-156"},"PeriodicalIF":1.0,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142706126","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
as an internal water wave model on the physical time scale. Here, ({varepsilon }) is a small parameter measuring the weak nonlinearity of the waves, (mu ) is the shallowness parameter, and (gamma in (0,1)) is the ratio between the densities of the two fluids. To be precise, we first prove the existence of a solution to the internal water wave equations for a two-layer fluid with surface tension, where one layer is of shallow depth and the other is of infinite depth. The existence time is of order ({mathcal {O}}(frac{1}{{varepsilon }})) for a small amount of surface tension such that ({varepsilon }^2 le textrm{bo}^{-1} ) where (textrm{bo}) is the Bond number. Then, we show that these solutions are close, on the same time scale, to the solutions of the BO equation with a precision of order ({mathcal {O}}(mu + textrm{bo}^{-1})). In addition, we provide the justification of new equations with improved dispersive properties, the Benjamin equation, and the Intermediate Long Wave (ILW) equation in the deep-water limit.
The long-time well-posedness of the two-layer fluid problem was first studied by Lannes [Arch. Ration. Mech. Anal., 208(2):481-567, 2013] in the case where both fluids have finite depth. Here, we adapt this work to the case where one of the fluid domains is of finite depth, and the other one is of infinite depth. The novelties of the proof are related to the geometry of the problem, where the difference in domains alters the functional setting for the Dirichlet-Neumann operators involved. In particular, we study the various compositions of these operators that require a refined symbolic analysis of the Dirichlet-Neumann operator on infinite depth and derive new pseudo-differential estimates that might be of independent interest.
{"title":"Justification of the Benjamin–Ono equation as an internal water waves model","authors":"Martin Oen Paulsen","doi":"10.1007/s40818-024-00190-z","DOIUrl":"10.1007/s40818-024-00190-z","url":null,"abstract":"<div><p>In this paper, we give the first rigorous justification of the Benjamin-Ono equation: </p><div><div><span>$$begin{aligned} hspace{3cm} partial _t zeta + (1 - frac{gamma }{2}sqrt{mu }|textrm{D}|)partial _x zeta + frac{3{varepsilon }}{2}zeta partial _xzeta =0, hspace{2cm} text {(BO)} end{aligned}$$</span></div></div><p>as an internal water wave model on the physical time scale. Here, <span>({varepsilon })</span> is a small parameter measuring the weak nonlinearity of the waves, <span>(mu )</span> is the shallowness parameter, and <span>(gamma in (0,1))</span> is the ratio between the densities of the two fluids. To be precise, we first prove the existence of a solution to the internal water wave equations for a two-layer fluid with surface tension, where one layer is of shallow depth and the other is of infinite depth. The existence time is of order <span>({mathcal {O}}(frac{1}{{varepsilon }}))</span> for a small amount of surface tension such that <span>({varepsilon }^2 le textrm{bo}^{-1} )</span> where <span>(textrm{bo})</span> is the Bond number. Then, we show that these solutions are close, on the same time scale, to the solutions of the BO equation with a precision of order <span>({mathcal {O}}(mu + textrm{bo}^{-1}))</span>. In addition, we provide the justification of new equations with improved dispersive properties, the Benjamin equation, and the Intermediate Long Wave (ILW) equation in the deep-water limit.</p><p>The long-time well-posedness of the two-layer fluid problem was first studied by Lannes [Arch. Ration. Mech. Anal., 208(2):481-567, 2013] in the case where both fluids have finite depth. Here, we adapt this work to the case where one of the fluid domains is of finite depth, and the other one is of infinite depth. The novelties of the proof are related to the geometry of the problem, where the difference in domains alters the functional setting for the Dirichlet-Neumann operators involved. In particular, we study the various compositions of these operators that require a refined symbolic analysis of the Dirichlet-Neumann operator on infinite depth and derive new pseudo-differential estimates that might be of independent interest.</p></div>","PeriodicalId":36382,"journal":{"name":"Annals of Pde","volume":"10 2","pages":""},"PeriodicalIF":2.4,"publicationDate":"2024-11-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s40818-024-00190-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142714572","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-25DOI: 10.1007/s10444-024-10207-7
J.K. Djoko, T. Sayah
In two dimensions, we propose and analyse an iterative a posteriori error indicator for the discontinuous Galerkin finite element approximations of the Stokes equations under boundary conditions of friction type. Two sources of error are identified here, namely; the discretisation error and the linearization error. Under a smallness assumption on data, we prove that the devised error estimator is reliable. Balancing these two errors is crucial to design an adaptive strategy for mesh refinement. We illustrate the theory with some representative numerical examples.
{"title":"Discontinuous Galerkin schemes for Stokes flow with Tresca boundary condition: iterative a posteriori error analysis","authors":"J.K. Djoko, T. Sayah","doi":"10.1007/s10444-024-10207-7","DOIUrl":"10.1007/s10444-024-10207-7","url":null,"abstract":"<div><p>In two dimensions, we propose and analyse an iterative <i>a posteriori</i> error indicator for the discontinuous Galerkin finite element approximations of the Stokes equations under boundary conditions of friction type. Two sources of error are identified here, namely; the discretisation error and the linearization error. Under a smallness assumption on data, we prove that the devised error estimator is reliable. Balancing these two errors is crucial to design an adaptive strategy for mesh refinement. We illustrate the theory with some representative numerical examples.</p></div>","PeriodicalId":50869,"journal":{"name":"Advances in Computational Mathematics","volume":"50 6","pages":""},"PeriodicalIF":1.7,"publicationDate":"2024-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10444-024-10207-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142694785","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-25DOI: 10.1016/j.ejc.2024.104088
Xiao-Chuan Liu , Danni Peng , Xu Yang
In an edge-colored graph , a rainbow clique is a complete subgraph on vertices in which all the edges have distinct colors. Let and be the number of edges and colors in , respectively. In this paper, we show that for any , if and , then for sufficiently large , the number of rainbow cliques in is .
We also characterize the extremal graphs without a rainbow clique , for , when is maximum.
Our results not only address existing questions but also complete the findings of Ehard and Mohr (2020).
在边色图 G 中，彩虹簇 Kk 是 k 个顶点上的一个完整子图，其中所有的边都有不同的颜色。设 e(G) 和 c(G) 分别为 G 中的边数和颜色数。本文将证明，对于任意ɛ>0，如果 e(G)+c(G)≥(1+k-3k-2+2ɛ)n2 且 k≥3 ，那么对于足够大的 n，G 中彩虹小群 Kk 的数目为 Ω(nk)。我们还描述了在 k=4,5 时，e(G)+c(G) 最大时没有彩虹簇 Kk 的极值图 G 的特征。我们的结果不仅解决了现有问题，还完善了 Ehard 和 Mohr (2020) 的发现。
{"title":"More on rainbow cliques in edge-colored graphs","authors":"Xiao-Chuan Liu , Danni Peng , Xu Yang","doi":"10.1016/j.ejc.2024.104088","DOIUrl":"10.1016/j.ejc.2024.104088","url":null,"abstract":"<div><div>In an edge-colored graph <span><math><mi>G</mi></math></span>, a rainbow clique <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> is a complete subgraph on <span><math><mi>k</mi></math></span> vertices in which all the edges have distinct colors. Let <span><math><mrow><mi>e</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><mi>c</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> be the number of edges and colors in <span><math><mi>G</mi></math></span>, respectively. In this paper, we show that for any <span><math><mrow><mi>ɛ</mi><mo>></mo><mn>0</mn></mrow></math></span>, if <span><math><mrow><mi>e</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mi>c</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>≥</mo><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mfrac><mrow><mi>k</mi><mo>−</mo><mn>3</mn></mrow><mrow><mi>k</mi><mo>−</mo><mn>2</mn></mrow></mfrac><mo>+</mo><mn>2</mn><mi>ɛ</mi><mo>)</mo></mrow><mfenced><mfrac><mrow><mi>n</mi></mrow><mrow><mn>2</mn></mrow></mfrac></mfenced></mrow></math></span> and <span><math><mrow><mi>k</mi><mo>≥</mo><mn>3</mn></mrow></math></span>, then for sufficiently large <span><math><mi>n</mi></math></span>, the number of rainbow cliques <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> in <span><math><mi>G</mi></math></span> is <span><math><mrow><mi>Ω</mi><mrow><mo>(</mo><msup><mrow><mi>n</mi></mrow><mrow><mi>k</mi></mrow></msup><mo>)</mo></mrow></mrow></math></span>.</div><div>We also characterize the extremal graphs <span><math><mi>G</mi></math></span> without a rainbow clique <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, for <span><math><mrow><mi>k</mi><mo>=</mo><mn>4</mn><mo>,</mo><mn>5</mn></mrow></math></span>, when <span><math><mrow><mi>e</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow><mo>+</mo><mi>c</mi><mrow><mo>(</mo><mi>G</mi><mo>)</mo></mrow></mrow></math></span> is maximum.</div><div>Our results not only address existing questions but also complete the findings of Ehard and Mohr (2020).</div></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"124 ","pages":"Article 104088"},"PeriodicalIF":1.0,"publicationDate":"2024-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142705765","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-25DOI: 10.1007/s00285-024-02162-0
Jiawei Deng, Hongying Shu, Sanyi Tang, Lin Wang, Xiang-Sheng Wang
Biological invasions significantly impact native ecosystems, altering ecological processes and community behaviors through predation and competition. The introduction of non-native species can lead to either coexistence or extinction within local habitats. Our research develops a lizard population model that integrates aspects of competition, intraguild predation, and the dispersal behavior of intraguild prey. We analyze the model to determine the existence and stability of various ecological equilibria, uncovering the potential for bistability under certain conditions. By employing the dispersal rate as a bifurcation parameter, we reveal complex bifurcation dynamics associated with the positive equilibrium. Additionally, we conduct a two-parameter bifurcation analysis to investigate the combined impact of dispersal and intraguild predation on ecological structures. Our findings indicate that intraguild predation not only influences the movement patterns of brown anoles but also plays a crucial role in sustaining the coexistence of different lizard species in diverse habitats.
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Pub Date : 2024-11-25DOI: 10.1007/s11538-024-01376-z
Sivan Leviyang
Single cell RNA-seq (scRNAseq) workflows typically start with a count matrix and end with the clustering of sampled cells. While a range of methods have been developed to cluster scRNAseq datasets, no theoretical tools exist to explain why a particular cluster exists or why a hypothesized cluster is missing. Recently, several authors have shown that eigenvalues of scRNAseq count matrices can be approximated using random matrix models. In this work, we extend these previous works to the study of a scRNAseq workflow. We model scaled count matrices using random matrices with normally distributed entries. Using these random matrix models, we quantify the differential expression of a cluster and develop predictions for the workflow, and in particular clustering, as a function of the differential expression. We also use results from random matrix theory (RMT) to develop predictive formulas for portions of the scRNAseq workflow. Using simulated and real datasets, we show that our predictions are accurate if certain conditions hold on differential expression, with our RMT based predictions requiring particularly stringent condition. We find that real datasets violate these conditions, leading to bias in our predictions, but our predictions are better than a naive estimator and we point out future work that can improve the predictions. To our knowledge, our formulas represents the first predictive results for scRNAseq workflows.
{"title":"Analysis of a Single Cell RNA-seq Workflow by Random Matrix Theory Methods.","authors":"Sivan Leviyang","doi":"10.1007/s11538-024-01376-z","DOIUrl":"https://doi.org/10.1007/s11538-024-01376-z","url":null,"abstract":"<p><p>Single cell RNA-seq (scRNAseq) workflows typically start with a count matrix and end with the clustering of sampled cells. While a range of methods have been developed to cluster scRNAseq datasets, no theoretical tools exist to explain why a particular cluster exists or why a hypothesized cluster is missing. Recently, several authors have shown that eigenvalues of scRNAseq count matrices can be approximated using random matrix models. In this work, we extend these previous works to the study of a scRNAseq workflow. We model scaled count matrices using random matrices with normally distributed entries. Using these random matrix models, we quantify the differential expression of a cluster and develop predictions for the workflow, and in particular clustering, as a function of the differential expression. We also use results from random matrix theory (RMT) to develop predictive formulas for portions of the scRNAseq workflow. Using simulated and real datasets, we show that our predictions are accurate if certain conditions hold on differential expression, with our RMT based predictions requiring particularly stringent condition. We find that real datasets violate these conditions, leading to bias in our predictions, but our predictions are better than a naive estimator and we point out future work that can improve the predictions. To our knowledge, our formulas represents the first predictive results for scRNAseq workflows.</p>","PeriodicalId":9372,"journal":{"name":"Bulletin of Mathematical Biology","volume":"87 1","pages":"4"},"PeriodicalIF":2.0,"publicationDate":"2024-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142709114","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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