SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2611-2639, December 2024. Abstract. In recent years, stochastic partial differential equations (SPDEs) have become a well-studied field in mathematics. With their increase in popularity, it becomes important to efficiently approximate their solutions. Thus, our goal is a contribution towards the development of efficient and practical time-stepping methods for SPDEs. Operator splitting schemes provide powerful, efficient, and flexible numerical methods for deterministic and stochastic differential equations. An example is given by domain decomposition schemes, where one splits the domain into subdomains and constructs the numerical approximation in a divide-and-conquer strategy. Instead of solving one expensive problem on the entire domain, one then deals with cheaper problems on the subdomains. This is particularly useful in modern computer architectures, as the subproblems may often be solved in parallel. While splitting methods have already been used to study domain decomposition methods for deterministic PDEs, this is a new approach for SPDEs. This implies that the existing convergence analysis is not directly applicable, even though the building blocks of the operator splitting domain decomposition method are standard. We provide an abstract convergence analysis of a splitting scheme for stochastic evolution equations and state a domain decomposition scheme as an application of the setting. The theoretical results are verified through numerical experiments.
{"title":"A Domain Decomposition Method for Stochastic Evolution Equations","authors":"Evelyn Buckwar, Ana Djurdjevac, Monika Eisenmann","doi":"10.1137/24m1629845","DOIUrl":"https://doi.org/10.1137/24m1629845","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 62, Issue 6, Page 2611-2639, December 2024. <br/> Abstract. In recent years, stochastic partial differential equations (SPDEs) have become a well-studied field in mathematics. With their increase in popularity, it becomes important to efficiently approximate their solutions. Thus, our goal is a contribution towards the development of efficient and practical time-stepping methods for SPDEs. Operator splitting schemes provide powerful, efficient, and flexible numerical methods for deterministic and stochastic differential equations. An example is given by domain decomposition schemes, where one splits the domain into subdomains and constructs the numerical approximation in a divide-and-conquer strategy. Instead of solving one expensive problem on the entire domain, one then deals with cheaper problems on the subdomains. This is particularly useful in modern computer architectures, as the subproblems may often be solved in parallel. While splitting methods have already been used to study domain decomposition methods for deterministic PDEs, this is a new approach for SPDEs. This implies that the existing convergence analysis is not directly applicable, even though the building blocks of the operator splitting domain decomposition method are standard. We provide an abstract convergence analysis of a splitting scheme for stochastic evolution equations and state a domain decomposition scheme as an application of the setting. The theoretical results are verified through numerical experiments.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"81 1","pages":""},"PeriodicalIF":2.9,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142679164","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 : 2024-11-20DOI: 10.1016/j.chaos.2024.115740
Mengjiao Wang, Zou Yi, Zhijun Li
Existing research indicates that discrete-time chaotic systems are more likely to achieve hyperchaotic states in lower dimensions compared to continuous-time chaotic systems. Recently, introducing discrete memristors into chaotic map to enhance system dynamics performance has become a hot topic in the field of chaos research. In this paper, a memristive Ikeda map (MIKM) based on discrete memristors is proposed and the system dynamics behavior is analyzed in depth by chaotic attractor phase diagrams, Lyapunov exponent spectrum, bifurcation diagrams, spectral entropy (SE), distributional properties and fractal dimensions. Numerical simulation results indicate that the introduction of discrete memristor enriches the dynamic characteristics of the Ikeda map, such as expanding the range of chaos, enhancing the ergodicity, and prompting the transition from chaotic to hyperchaotic states. We further studied the influence of coupling strength K on the dynamic behavior of the system. We explored the use of the discrete memristor as internal perturbations to achieve parameter-controlled symmetric attractors and the introduction of constant controllers to achieve signal polarity adjustment. At the same time, we implemented the improved Ikead map on the STM32 hardware platform and developed a pseudo-random number generator (PRNG). Finally, an image encryption algorithm was designed based on the proposed improved Ikeda map. The experimental results show that the proposed algorithm is robust.
现有研究表明，与连续时间混沌系统相比，离散时间混沌系统更有可能在较低维度上实现超混沌状态。近来，在混沌图中引入离散忆阻器以提高系统动力学性能已成为混沌研究领域的热门话题。本文提出了一种基于离散忆阻器的忆阻池田混沌图（MIKM），并通过混沌吸引子相图、Lyapunov指数谱、分岔图、谱熵（SE）、分布特性和分形维度等对系统动力学行为进行了深入分析。数值模拟结果表明，离散忆阻器的引入丰富了池田图的动态特性，如扩大了混沌范围、增强了遍历性、促使混沌状态向超混沌状态过渡等。我们进一步研究了耦合强度 K 对系统动态行为的影响。我们探索了利用离散忆阻器作为内部扰动来实现参数控制的对称吸引子，并引入常数控制器来实现信号极性调整。同时，我们在 STM32 硬件平台上实现了改进的 Ikead 地图，并开发了一个伪随机数发生器（PRNG）。最后，基于改进的池田映射设计了一种图像加密算法。实验结果表明，所提出的算法具有良好的鲁棒性。
{"title":"A memristive Ikeda map and its application in image encryption","authors":"Mengjiao Wang, Zou Yi, Zhijun Li","doi":"10.1016/j.chaos.2024.115740","DOIUrl":"https://doi.org/10.1016/j.chaos.2024.115740","url":null,"abstract":"Existing research indicates that discrete-time chaotic systems are more likely to achieve hyperchaotic states in lower dimensions compared to continuous-time chaotic systems. Recently, introducing discrete memristors into chaotic map to enhance system dynamics performance has become a hot topic in the field of chaos research. In this paper, a memristive Ikeda map (MIKM) based on discrete memristors is proposed and the system dynamics behavior is analyzed in depth by chaotic attractor phase diagrams, Lyapunov exponent spectrum, bifurcation diagrams, spectral entropy (SE), distributional properties and fractal dimensions. Numerical simulation results indicate that the introduction of discrete memristor enriches the dynamic characteristics of the Ikeda map, such as expanding the range of chaos, enhancing the ergodicity, and prompting the transition from chaotic to hyperchaotic states. We further studied the influence of coupling strength <mml:math altimg=\"si108.svg\" display=\"inline\"><mml:mi>K</mml:mi></mml:math> on the dynamic behavior of the system. We explored the use of the discrete memristor as internal perturbations to achieve parameter-controlled symmetric attractors and the introduction of constant controllers to achieve signal polarity adjustment. At the same time, we implemented the improved Ikead map on the STM32 hardware platform and developed a pseudo-random number generator (PRNG). Finally, an image encryption algorithm was designed based on the proposed improved Ikeda map. The experimental results show that the proposed algorithm is robust.","PeriodicalId":9764,"journal":{"name":"Chaos Solitons & Fractals","volume":"18 1","pages":""},"PeriodicalIF":7.8,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142679228","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-20DOI: 10.1016/j.chaos.2024.115759
I.D. Kolesnikov, A.V. Bukh, S.S. Muni, J.S. Ram
We aim to explore the features of destroying the spiral wave regime in a lattice network of Chialvo neurons by applying external noise with different statistical characteristics. Chialvo neurons are represented with a two-dimensional recurrence map. The lattice of neurons under study observed with random initial conditions and with special initial conditions for local and nonlocal coupling. We consider a detailed two-parameter plot in the plane of coupling strength — distribution width of Lévy process which revealed that the existence of spiral waves are dependent on the network and noise parameters. We examine how coupling strength and range parameters influence on the spiral wave dynamics in a coupled lattice system. Increasing the coupling range enlarges the region where spiral waves can exist. Additionally we show that the destruction of spiral waves is achievable with a certain threshold of the distribution width parameter value depending on the noise stability parameter value and the noise asymmetry parameter value. A decrease in the noise stability parameter as well as in the noise asymmetry parameter decreases the threshold value. We show that the influence of Lévy noise on spiral waves in the lattice of Chialvo neurons results in a transition to target waves that are more stable than in the case of transition for random initial conditions to target waves without noise. Finally, we have found that the noise could cause the lattice to switch between various spiral-like regimes as time passes.
{"title":"Impact of Lévy noise on spiral waves in a lattice of Chialvo neuron map","authors":"I.D. Kolesnikov, A.V. Bukh, S.S. Muni, J.S. Ram","doi":"10.1016/j.chaos.2024.115759","DOIUrl":"https://doi.org/10.1016/j.chaos.2024.115759","url":null,"abstract":"We aim to explore the features of destroying the spiral wave regime in a lattice network of Chialvo neurons by applying external noise with different statistical characteristics. Chialvo neurons are represented with a two-dimensional recurrence map. The lattice of neurons under study observed with random initial conditions and with special initial conditions for local and nonlocal coupling. We consider a detailed two-parameter plot in the plane of coupling strength — distribution width of Lévy process which revealed that the existence of spiral waves are dependent on the network and noise parameters. We examine how coupling strength and range parameters influence on the spiral wave dynamics in a coupled lattice system. Increasing the coupling range enlarges the region where spiral waves can exist. Additionally we show that the destruction of spiral waves is achievable with a certain threshold of the distribution width parameter value depending on the noise stability parameter value and the noise asymmetry parameter value. A decrease in the noise stability parameter as well as in the noise asymmetry parameter decreases the threshold value. We show that the influence of Lévy noise on spiral waves in the lattice of Chialvo neurons results in a transition to target waves that are more stable than in the case of transition for random initial conditions to target waves without noise. Finally, we have found that the noise could cause the lattice to switch between various spiral-like regimes as time passes.","PeriodicalId":9764,"journal":{"name":"Chaos Solitons & Fractals","volume":"15 1","pages":""},"PeriodicalIF":7.8,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142679223","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-20DOI: 10.1007/s00205-024-02060-1
Max Weissenbacher
We study the massless Vlasov equation on the exterior of the subextremal and extremal Reissner–Nordström spacetimes. We prove that moments decay at an exponential rate in the subextremal case and at a polynomial rate in the extremal case. This polynomial rate is shown to be sharp along the event horizon. In the extremal case we show that transversal derivatives of certain components of the energy momentum tensor do not decay along the event horizon if the solution and its first time derivative are initially supported on a neighbourhood of the event horizon. The non-decay of transversal derivatives in the extremal case is compared to the work of Aretakis on instability for the wave equation. Unlike Aretakis’ results for the wave equation, which exploit a hierarchy of conservation laws, our proof is based entirely on a quantitative analysis of the geodesic flow and conservation laws do not feature in the present work.
{"title":"Decay and non-decay for the massless Vlasov equation on subextremal and extremal Reissner–Nordström black holes","authors":"Max Weissenbacher","doi":"10.1007/s00205-024-02060-1","DOIUrl":"10.1007/s00205-024-02060-1","url":null,"abstract":"<div><p>We study the massless Vlasov equation on the exterior of the subextremal and extremal Reissner–Nordström spacetimes. We prove that moments decay at an exponential rate in the subextremal case and at a polynomial rate in the extremal case. This polynomial rate is shown to be sharp along the event horizon. In the extremal case we show that transversal derivatives of certain components of the energy momentum tensor do not decay along the event horizon if the solution and its first time derivative are initially supported on a neighbourhood of the event horizon. The non-decay of transversal derivatives in the extremal case is compared to the work of Aretakis on instability for the wave equation. Unlike Aretakis’ results for the wave equation, which exploit a hierarchy of conservation laws, our proof is based entirely on a quantitative analysis of the geodesic flow and conservation laws do not feature in the present work.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 6","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-11-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00205-024-02060-1.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142672494","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 : 2024-11-19DOI: 10.1016/j.chaos.2024.115702
Dave Cliff
I present a replication and, to some extent, a refutation of key results published by Zhong, Zhang, Li, Dai, & Yang in their 2022 paper “Species coexistence in spatial cyclic game of five species” (Chaos, Solitons and Fractals, 156: 111806), where ecosystem species coexistence was explored via simulation studies of the evolutionary spatial cyclic game (Escg) Rock–Paper–Scissors–Lizard–Spock (Rpsls) with certain predator–prey relationships removed from the game’s “interaction structure”, i.e. with specific arcs ablated in the Escg’s dominance network, and with the Escg run for 105 Monte Carlo Steps (mcs) to identify its asymptotic behaviors. I replicate the results presented by Zhong et al. for interaction structures with one, two, three, and four arcs ablated from the dominance network. I then empirically demonstrate that the dynamics of the RpslsEscg have sufficiently long time constants that the true asymptotic outcomes can often only be identified after running the ablated Escg for 107mcs or longer, and that the true long-term outcomes can be markedly less diverse than those reported by Zhong et al. as asymptotic. Finally I demonstrate that, when run for sufficiently many mcs, the original unablated Rpsls system exhibits essentially the same asymptotic outcomes as the ablated Rpsls systems, and in this sense the only causal effect of the ablations is to alter the time required for the system to converge to the long-term asymptotic states that the unablated system eventually settles to anyhow.
我介绍了 Zhong、Zhang、Li、Dai、& Yang 在 2022 年发表的论文 "Species coexistence in spatial cyclic game of five species"（《混沌、孤子与分形》，156 卷，第 111806 期）中的关键结果的复制，并在一定程度上对其进行了反驳：111806）中，通过对进化空间循环博弈（Escg）"石头-剪子-蜥蜴-麻雀"（Rpsls）的模拟研究，探讨了生态系统中的物种共存问题。即在 Escg 的优势网络中删除特定的弧，并让 Escg 运行 105 个蒙特卡罗步（mcs），以确定其渐近行为。我复制了 Zhong 等人针对支配网络中消减了一条、两条、三条和四条弧的交互结构得出的结果。然后，我通过经验证明，RpslsEscg 的动态具有足够长的时间常数，只有在消融 Escg 运行 107mcs 或更长时间后，才能识别出真正的渐近结果，而且真正的长期结果的多样性可能明显低于 Zhong 等人报告的渐近结果。最后，我证明了当运行足够多的 mcs 时，原始的未消融 Rpsls 系统与消融 Rpsls 系统表现出基本相同的渐近结果，从这个意义上说，消融的唯一因果效应是改变了系统收敛到长期渐近状态所需的时间，而未消融系统最终无论如何都会收敛到长期渐近状态。
{"title":"On long-term species coexistence in five-species evolutionary spatial cyclic games with ablated and non-ablated dominance networks","authors":"Dave Cliff","doi":"10.1016/j.chaos.2024.115702","DOIUrl":"https://doi.org/10.1016/j.chaos.2024.115702","url":null,"abstract":"I present a replication and, to some extent, a refutation of key results published by Zhong, Zhang, Li, Dai, & Yang in their 2022 paper “Species coexistence in spatial cyclic game of five species” (<ce:italic>Chaos, Solitons and Fractals</ce:italic>, 156: 111806), where ecosystem species coexistence was explored via simulation studies of the evolutionary spatial cyclic game (<ce:small-caps>Escg</ce:small-caps>) Rock–Paper–Scissors–Lizard–Spock (<ce:small-caps>Rpsls</ce:small-caps>) with certain predator–prey relationships removed from the game’s “interaction structure”, i.e. with specific arcs ablated in the <ce:small-caps>Escg</ce:small-caps>’s dominance network, and with the <ce:small-caps>Escg</ce:small-caps> run for <mml:math altimg=\"si362.svg\" display=\"inline\"><mml:mrow><mml:mn>1</mml:mn><mml:msup><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mn>5</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math> Monte Carlo Steps (<ce:small-caps>mcs</ce:small-caps>) to identify its asymptotic behaviors. I replicate the results presented by Zhong et al. for interaction structures with one, two, three, and four arcs ablated from the dominance network. I then empirically demonstrate that the dynamics of the <ce:small-caps>Rpsls</ce:small-caps><ce:small-caps>Escg</ce:small-caps> have sufficiently long time constants that the true asymptotic outcomes can often only be identified after running the ablated <ce:small-caps>Escg</ce:small-caps> for <mml:math altimg=\"si605.svg\" display=\"inline\"><mml:mrow><mml:mn>1</mml:mn><mml:msup><mml:mrow><mml:mn>0</mml:mn></mml:mrow><mml:mrow><mml:mn>7</mml:mn></mml:mrow></mml:msup></mml:mrow></mml:math><ce:small-caps>mcs</ce:small-caps> or longer, and that the true long-term outcomes can be markedly less diverse than those reported by Zhong et al. as asymptotic. Finally I demonstrate that, when run for sufficiently many <ce:small-caps>mcs</ce:small-caps>, the original unablated <ce:small-caps>Rpsls</ce:small-caps> system exhibits essentially the same asymptotic outcomes as the ablated <ce:small-caps>Rpsls</ce:small-caps> systems, and in this sense the only causal effect of the ablations is to alter the time required for the system to converge to the long-term asymptotic states that the unablated system eventually settles to anyhow.","PeriodicalId":9764,"journal":{"name":"Chaos Solitons & Fractals","volume":"231 1","pages":""},"PeriodicalIF":7.8,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142679234","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-11-19DOI: 10.1016/j.cnsns.2024.108458
Jiaqi Fan, Yuanyuan Li
In this paper, we consider resistance distances in stretched Cantor product networks, a family of non-self-similar networks. By constructing the networks in a iterated way, we give an approach to encode every node in their vertex set. And then we simplify the complex resistor networks by induction on the basic network pattern. Using classical results of circuit theory, we obtain the exact formulae of the resistance distances of some pairs of nodes in stretched Cantor product networks.
{"title":"Resistance distances in stretched Cantor product networks","authors":"Jiaqi Fan, Yuanyuan Li","doi":"10.1016/j.cnsns.2024.108458","DOIUrl":"https://doi.org/10.1016/j.cnsns.2024.108458","url":null,"abstract":"In this paper, we consider resistance distances in stretched Cantor product networks, a family of non-self-similar networks. By constructing the networks in a iterated way, we give an approach to encode every node in their vertex set. And then we simplify the complex resistor networks by induction on the basic network pattern. Using classical results of circuit theory, we obtain the exact formulae of the resistance distances of some pairs of nodes in stretched Cantor product networks.","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":"19 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142679154","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 : 2024-11-19DOI: 10.1007/s00205-024-02058-9
Gengsheng Wang, Yubiao Zhang, Enrique Zuazua
This paper presents a complete analysis of the observability property of heat equations with time-dependent real analytic memory kernels. More precisely, we characterize the geometry of the space-time measurable observation sets ensuring sharp observability inequalities, which are relevant both for control and inverse problems purposes. Despite the abundant literature on the observation of heat-like equations, existing methods do not apply to models involving memory terms. We present a new methodology and observation strategy, relying on the decomposition of the flow, the time-analyticity of solutions and the propagation of singularities. This allows us to obtain a sufficient and necessary geometric condition on the measurable observation sets for sharp two-sided observability inequalities. In addition, some applications to control and relevant open problems are presented.
{"title":"Observability for Heat Equations with Time-Dependent Analytic Memory","authors":"Gengsheng Wang, Yubiao Zhang, Enrique Zuazua","doi":"10.1007/s00205-024-02058-9","DOIUrl":"10.1007/s00205-024-02058-9","url":null,"abstract":"<div><p>This paper presents a complete analysis of the observability property of heat equations with time-dependent real analytic memory kernels. More precisely, we characterize the geometry of the space-time measurable observation sets ensuring sharp observability inequalities, which are relevant both for control and inverse problems purposes. Despite the abundant literature on the observation of heat-like equations, existing methods do not apply to models involving memory terms. We present a new methodology and observation strategy, relying on the decomposition of the flow, the time-analyticity of solutions and the propagation of singularities. This allows us to obtain a sufficient and necessary geometric condition on the measurable observation sets for sharp two-sided observability inequalities. In addition, some applications to control and relevant open problems are presented.</p></div>","PeriodicalId":55484,"journal":{"name":"Archive for Rational Mechanics and Analysis","volume":"248 6","pages":""},"PeriodicalIF":2.6,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142672511","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":1,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Purnaprajna Bangere, Francisco Javier Gallego, Jayan Mukherjee, Debaditya Raychaudhury
<p>We develop a new way of systematically constructing infinitely many families of smooth subvarieties <span></span><math> <semantics> <mi>X</mi> <annotation>$X$</annotation> </semantics></math> of any given dimension <span></span><math> <semantics> <mi>m</mi> <annotation>$m$</annotation> </semantics></math>, <span></span><math> <semantics> <mrow> <mi>m</mi> <mo>⩾</mo> <mn>3</mn> </mrow> <annotation>$m geqslant 3$</annotation> </semantics></math>, and any given codimension in <span></span><math> <semantics> <msup> <mi>P</mi> <mi>N</mi> </msup> <annotation>$mathbb {P}^N$</annotation> </semantics></math>, embedded by complete subcanonical linear series, and, in particular, in the range of Hartshorne's conjecture. We accomplish this by showing the existence of everywhere non-reduced schemes called ropes, embedded in <span></span><math> <semantics> <msup> <mi>P</mi> <mi>N</mi> </msup> <annotation>$mathbb {P}^N$</annotation> </semantics></math>, and by smoothing them. In the range <span></span><math> <semantics> <mrow> <mn>3</mn> <mo>⩽</mo> <mi>m</mi> <mo><</mo> <mrow> <mi>N</mi> <mo>/</mo> <mn>2</mn> </mrow> </mrow> <annotation>$3 leqslant m < {{N/2}}$</annotation> </semantics></math>, we construct smooth subvarieties, embedded by complete subcanonical linear series, that are not complete intersections. We also go beyond a question of Enriques on constructing simple canonical surfaces in projective spaces, and construct simple canonical varieties in all dimensions. The canonical map of infinitely many of these simple canonical varieties is finite birational but not an embedding. Finally, we show the existence of components of moduli spaces of varieties of general type (in all dimensions <span></span><math> <semantics> <mi>m</mi> <annotation>$m$</annotation> </semantics></math>, <span></span><math> <semantics> <mrow> <mi>m</mi> <mo>⩾</mo> <mn>3</mn> </mrow> <annotation>$m geqslant 3$</annotation> </semantics></math>) that are analogues of the moduli space of curves of genus <span></span><math> <semantics> <mrow> <mi>g</mi> <mo>></mo>
我们开发了一种新方法，可以系统地构造任意给定维数 m $m$ , m ⩾ 3 $m geqslant 3$ , 以及 P N $mathbb {P}^N$ 中任意给定编码维数的无限多光滑子域 X $X$ 族，这些子域由完全次经典线性数列嵌入，尤其是在哈特肖恩猜想的范围内。为此，我们证明了嵌入 P N $mathbb {P}^N$ 的无处不还原的方案（称为绳索）的存在，并对其进行平滑处理。在 3 ⩽ m < N / 2 $3 leqslant m < {{N/2}}$ 的范围内，我们通过完整的次经典线性数列嵌入，构造了不是完全交集的光滑子域。我们还超越了恩里克斯提出的关于在投影空间中构造简单典型面的问题，构造了所有维度中的简单典型面。无限多的这些简单典型面的典型映射是有限双向的，但不是嵌入。最后，我们证明了一般类型（在所有维度上 m $m$ , m ⩾ 3 $m geqslant 3$）曲线的模空间存在类似于g > 2 $g > 2$属曲线的模空间的成分，这些成分与典型映射及其变形的行为有关。在许多情况下，这些分量的一般元素都是典型嵌入的，而且它们的标度都在哈特肖恩猜想的范围内。
{"title":"Construction of varieties of low codimension with applications to moduli spaces of varieties of general type","authors":"Purnaprajna Bangere, Francisco Javier Gallego, Jayan Mukherjee, Debaditya Raychaudhury","doi":"10.1112/jlms.70030","DOIUrl":"https://doi.org/10.1112/jlms.70030","url":null,"abstract":"<p>We develop a new way of systematically constructing infinitely many families of smooth subvarieties <span></span><math>\u0000 <semantics>\u0000 <mi>X</mi>\u0000 <annotation>$X$</annotation>\u0000 </semantics></math> of any given dimension <span></span><math>\u0000 <semantics>\u0000 <mi>m</mi>\u0000 <annotation>$m$</annotation>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 <mo>⩾</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation>$m geqslant 3$</annotation>\u0000 </semantics></math>, and any given codimension in <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>P</mi>\u0000 <mi>N</mi>\u0000 </msup>\u0000 <annotation>$mathbb {P}^N$</annotation>\u0000 </semantics></math>, embedded by complete subcanonical linear series, and, in particular, in the range of Hartshorne's conjecture. We accomplish this by showing the existence of everywhere non-reduced schemes called ropes, embedded in <span></span><math>\u0000 <semantics>\u0000 <msup>\u0000 <mi>P</mi>\u0000 <mi>N</mi>\u0000 </msup>\u0000 <annotation>$mathbb {P}^N$</annotation>\u0000 </semantics></math>, and by smoothing them. In the range <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mn>3</mn>\u0000 <mo>⩽</mo>\u0000 <mi>m</mi>\u0000 <mo><</mo>\u0000 <mrow>\u0000 <mi>N</mi>\u0000 <mo>/</mo>\u0000 <mn>2</mn>\u0000 </mrow>\u0000 </mrow>\u0000 <annotation>$3 leqslant m &lt; {{N/2}}$</annotation>\u0000 </semantics></math>, we construct smooth subvarieties, embedded by complete subcanonical linear series, that are not complete intersections. We also go beyond a question of Enriques on constructing simple canonical surfaces in projective spaces, and construct simple canonical varieties in all dimensions. The canonical map of infinitely many of these simple canonical varieties is finite birational but not an embedding. Finally, we show the existence of components of moduli spaces of varieties of general type (in all dimensions <span></span><math>\u0000 <semantics>\u0000 <mi>m</mi>\u0000 <annotation>$m$</annotation>\u0000 </semantics></math>, <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>m</mi>\u0000 <mo>⩾</mo>\u0000 <mn>3</mn>\u0000 </mrow>\u0000 <annotation>$m geqslant 3$</annotation>\u0000 </semantics></math>) that are analogues of the moduli space of curves of genus <span></span><math>\u0000 <semantics>\u0000 <mrow>\u0000 <mi>g</mi>\u0000 <mo>></mo>\u0000 ","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"110 6","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142674089","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 : 2024-11-19DOI: 10.1016/j.cnsns.2024.108453
Noureddine Moujane, Mohamed El Ouaarabi
In this paper, we investigate the existence of solutions for double-phase problems with variable exponents of the Kirchhoff–Schrödinger type, incorporating a convection term. By imposing certain assumptions and utilizing the topological degree for a class of (S+)-demicontinuous operators, along with the Galerkin method within the framework of Musielak–Orlicz–Sobolev spaces, we establish the existence of strong generalized solutions and weak solutions for the problems under consideration.
{"title":"On a class of Schrödinger–Kirchhoff-double phase problems with convection term and variable exponents","authors":"Noureddine Moujane, Mohamed El Ouaarabi","doi":"10.1016/j.cnsns.2024.108453","DOIUrl":"https://doi.org/10.1016/j.cnsns.2024.108453","url":null,"abstract":"In this paper, we investigate the existence of solutions for double-phase problems with variable exponents of the Kirchhoff–Schrödinger type, incorporating a convection term. By imposing certain assumptions and utilizing the topological degree for a class of <mml:math altimg=\"si1.svg\" display=\"inline\"><mml:mrow><mml:mo>(</mml:mo><mml:msub><mml:mrow><mml:mi>S</mml:mi></mml:mrow><mml:mrow><mml:mo>+</mml:mo></mml:mrow></mml:msub><mml:mo>)</mml:mo></mml:mrow></mml:math>-demicontinuous operators, along with the Galerkin method within the framework of Musielak–Orlicz–Sobolev spaces, we establish the existence of strong generalized solutions and weak solutions for the problems under consideration.","PeriodicalId":50658,"journal":{"name":"Communications in Nonlinear Science and Numerical Simulation","volume":"18 1","pages":""},"PeriodicalIF":3.9,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142679155","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}
This paper is concerned with the billiard version of Jacobi's last geometric statement and its generalizations. Given a non-focal point inside an elliptic billiard table, one considers the family of rays emanating from and the caustic of the reflected family after reflections off the ellipse, for each positive integer . It is known that has at least four cusps and it has been conjectured that it has exactly four (ordinary) cusps. The present paper presents a proof of this conjecture in the special case when the ellipse is a circle. In the case of an arbitrary ellipse, we give an explicit description of the location of four of the cusps of , though we do not prove that these are the only cusps.
本文关注的是雅可比最后一个几何陈述的台球桌版本及其一般化。给定一个椭圆台球桌内的非焦点 O $O$，考虑从 O $O$ 射出的射线族，以及在每个正整数 n $n$ 反射出椭圆 n $n$ 后反射族的苛值 Γ n $ Gamma _n$ 。众所周知，Γ n $Gamma _n$至少有四个尖顶，而且有人猜想它正好有四个（普通）尖顶。本文在椭圆是圆的特殊情况下证明了这一猜想。在任意椭圆的情况下，我们明确描述了 Γ n $Gamma _n$ 的四个尖顶的位置，尽管我们并没有证明这些尖顶是唯一的尖顶。
{"title":"Cusps of caustics by reflection in ellipses","authors":"Gil Bor, Mark Spivakovsky, Serge Tabachnikov","doi":"10.1112/jlms.70033","DOIUrl":"https://doi.org/10.1112/jlms.70033","url":null,"abstract":"<p>This paper is concerned with the billiard version of Jacobi's last geometric statement and its generalizations. Given a non-focal point <span></span><math>\u0000 <semantics>\u0000 <mi>O</mi>\u0000 <annotation>$O$</annotation>\u0000 </semantics></math> inside an elliptic billiard table, one considers the family of rays emanating from <span></span><math>\u0000 <semantics>\u0000 <mi>O</mi>\u0000 <annotation>$O$</annotation>\u0000 </semantics></math> and the caustic <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>Γ</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <annotation>$ Gamma _n$</annotation>\u0000 </semantics></math> of the reflected family after <span></span><math>\u0000 <semantics>\u0000 <mi>n</mi>\u0000 <annotation>$n$</annotation>\u0000 </semantics></math> reflections off the ellipse, for each positive integer <span></span><math>\u0000 <semantics>\u0000 <mi>n</mi>\u0000 <annotation>$n$</annotation>\u0000 </semantics></math>. It is known that <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>Γ</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <annotation>$Gamma _n$</annotation>\u0000 </semantics></math> has at least four cusps and it has been conjectured that it has exactly four (ordinary) cusps. The present paper presents a proof of this conjecture in the special case when the ellipse is a circle. In the case of an arbitrary ellipse, we give an explicit description of the location of four of the cusps of <span></span><math>\u0000 <semantics>\u0000 <msub>\u0000 <mi>Γ</mi>\u0000 <mi>n</mi>\u0000 </msub>\u0000 <annotation>$Gamma _n$</annotation>\u0000 </semantics></math>, though we do not prove that these are the only cusps.</p>","PeriodicalId":49989,"journal":{"name":"Journal of the London Mathematical Society-Second Series","volume":"110 6","pages":""},"PeriodicalIF":1.0,"publicationDate":"2024-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142674270","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}
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