Pub Date : 2025-04-25DOI: 10.1016/j.jde.2025.113348
Koichi Anada , Tetsuya Ishiwata , Takeo Ushijima
We deal with the area-preserving curvature flow in the plane, particularly the blow-up phenomena of curvatures on cusp singularities in contractions of convex immersed curves with self-crossing points. For Abresch-Langer type curves with highly symmetric properties, it has been known that the maximum of curvatures blows up at a finite time under some assumptions. In this paper, we consider the blow-up rates in this case.
{"title":"Type II singularities in area-preserving curvature flows of convex symmetric immersed closed plane curves","authors":"Koichi Anada , Tetsuya Ishiwata , Takeo Ushijima","doi":"10.1016/j.jde.2025.113348","DOIUrl":"10.1016/j.jde.2025.113348","url":null,"abstract":"<div><div>We deal with the area-preserving curvature flow in the plane, particularly the blow-up phenomena of curvatures on cusp singularities in contractions of convex immersed curves with self-crossing points. For Abresch-Langer type curves with highly symmetric properties, it has been known that the maximum of curvatures blows up at a finite time under some assumptions. In this paper, we consider the blow-up rates in this case.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"437 ","pages":"Article 113348"},"PeriodicalIF":2.4,"publicationDate":"2025-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143869741","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 : 2025-04-25DOI: 10.1016/j.jde.2025.113351
Yanqing Wang , Wei Wei , Gang Wu , Daoguo Zhou
In this paper, we are concerned with the critical mixed norm regularity of Leray-Hopf weak solutions of the Navier-Stokes equations in three dimensions and higher dimensions. It is shown that with ensure that Leray-Hopf weak solutions are regular. A new ingredient is ε-regularity criterion derived by the De Giorgi iteration technique under this critical regularity in high spatial dimension.
{"title":"On the borderline regularity criterion in anisotropic Lebesgue spaces of the Navier-Stokes equations","authors":"Yanqing Wang , Wei Wei , Gang Wu , Daoguo Zhou","doi":"10.1016/j.jde.2025.113351","DOIUrl":"10.1016/j.jde.2025.113351","url":null,"abstract":"<div><div>In this paper, we are concerned with the critical mixed norm regularity of Leray-Hopf weak solutions of the Navier-Stokes equations in three dimensions and higher dimensions. It is shown that <span><math><mi>u</mi><mo>∈</mo><msup><mrow><mi>L</mi></mrow><mrow><mo>∞</mo></mrow></msup><mo>(</mo><mn>0</mn><mo>,</mo><mi>T</mi><mo>;</mo><msup><mrow><mi>L</mi></mrow><mrow><mover><mrow><mi>q</mi></mrow><mrow><mo>→</mo></mrow></mover></mrow></msup><mo>(</mo><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>)</mo><mo>)</mo></math></span> with <span><math><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></munderover><mfrac><mrow><mn>1</mn></mrow><mrow><msub><mrow><mi>q</mi></mrow><mrow><mi>i</mi></mrow></msub></mrow></mfrac><mo>=</mo><mn>1</mn></math></span> ensure that Leray-Hopf weak solutions are regular. A new ingredient is <em>ε</em>-regularity criterion derived by the De Giorgi iteration technique under this critical regularity in high spatial dimension.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"438 ","pages":"Article 113351"},"PeriodicalIF":2.4,"publicationDate":"2025-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143870203","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 : 2025-04-25DOI: 10.1016/j.jcta.2025.106059
Ishay Haviv
The storage capacity of a graph measures the maximum amount of information that can be stored across its vertices, such that the information at any vertex can be recovered from the information stored at its neighborhood. The study of this graph quantity is motivated by applications in distributed storage and by its intimate relations to the index coding problem from the area of network information theory. In the latter, one wishes to minimize the amount of information that has to be transmitted to a collection of receivers, in a way that enables each of them to discover its required data using some prior side information.
In this paper, we initiate the study of the and problems from the perspective of parameterized complexity. We prove that the problem parameterized by the solution size admits a kernelization algorithm producing kernels of linear size. We also provide such a result for the problem, in the linear and non-linear settings, where it is parameterized by the dual value of the solution, i.e., the length of the transmission that can be saved using the side information. A key ingredient in the proofs is the crown decomposition technique due to Chor, Fellows, and Juedes [14], [11]. As an application, we significantly extend an algorithmic result of Dau, Skachek, and Chee [13].
{"title":"Kernels for storage capacity and dual index coding","authors":"Ishay Haviv","doi":"10.1016/j.jcta.2025.106059","DOIUrl":"10.1016/j.jcta.2025.106059","url":null,"abstract":"<div><div>The storage capacity of a graph measures the maximum amount of information that can be stored across its vertices, such that the information at any vertex can be recovered from the information stored at its neighborhood. The study of this graph quantity is motivated by applications in distributed storage and by its intimate relations to the index coding problem from the area of network information theory. In the latter, one wishes to minimize the amount of information that has to be transmitted to a collection of receivers, in a way that enables each of them to discover its required data using some prior side information.</div><div>In this paper, we initiate the study of the <figure><img></figure> and <figure><img></figure> problems from the perspective of parameterized complexity. We prove that the <figure><img></figure> problem parameterized by the solution size admits a kernelization algorithm producing kernels of linear size. We also provide such a result for the <figure><img></figure> problem, in the linear and non-linear settings, where it is parameterized by the dual value of the solution, i.e., the length of the transmission that can be saved using the side information. A key ingredient in the proofs is the crown decomposition technique due to Chor, Fellows, and Juedes <span><span>[14]</span></span>, <span><span>[11]</span></span>. As an application, we significantly extend an algorithmic result of Dau, Skachek, and Chee <span><span>[13]</span></span>.</div></div>","PeriodicalId":50230,"journal":{"name":"Journal of Combinatorial Theory Series A","volume":"216 ","pages":"Article 106059"},"PeriodicalIF":0.9,"publicationDate":"2025-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143868647","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 : 2025-04-25DOI: 10.1016/j.chaos.2025.116443
Tugba Palabas
Inverse Chaotic Resonance (ICR) refers to the phenomenon in which the mean firing rate is reduced with an optimal intensity of the chaotic activity. In this study, ICR is numerically investigated by modeling the scale-free network topology of Hodgkin–Huxley neurons coupled electrical, excitatory, and inhibitory chemical synapses. First, it is shown that chaotic signals play an important role in changing the average firing frequency of the network consisting of neurons connected by any synaptic coupling. Then it is expressed that the ICR phenomenon occurs depending on the synaptic strength and that even double ICR behavior can also emerge at two different optimal levels in the case of inhibitory synapse. Moreover, ICR can be modulated by a constant stimulus, and this phenomenon covers a wider range of chaotic current densities at a constant current level close to the excitation threshold. In addition, the effects of the synaptic time constant and network inputs on the appearance of the phenomenon are also examined. These extensive numerical results suggest a new perspective on ICR effect is a robust phenomenon that is observed in neuronal networks regardless of their topological structure.
{"title":"Inverse chaotic resonance in scale-free neuronal networks based on synaptic modulation","authors":"Tugba Palabas","doi":"10.1016/j.chaos.2025.116443","DOIUrl":"10.1016/j.chaos.2025.116443","url":null,"abstract":"<div><div>Inverse Chaotic Resonance (ICR) refers to the phenomenon in which the mean firing rate is reduced with an optimal intensity of the chaotic activity. In this study, ICR is numerically investigated by modeling the scale-free network topology of Hodgkin–Huxley neurons coupled electrical, excitatory, and inhibitory chemical synapses. First, it is shown that chaotic signals play an important role in changing the average firing frequency of the network consisting of neurons connected by any synaptic coupling. Then it is expressed that the ICR phenomenon occurs depending on the synaptic strength and that even double ICR behavior can also emerge at two different optimal <span><math><mi>ϵ</mi></math></span> levels in the case of inhibitory synapse. Moreover, ICR can be modulated by a constant stimulus, and this phenomenon covers a wider range of chaotic current densities at a constant current level close to the excitation threshold. In addition, the effects of the synaptic time constant and network inputs on the appearance of the phenomenon are also examined. These extensive numerical results suggest a new perspective on ICR effect is a robust phenomenon that is observed in neuronal networks regardless of their topological structure.</div></div>","PeriodicalId":9764,"journal":{"name":"Chaos Solitons & Fractals","volume":"197 ","pages":"Article 116443"},"PeriodicalIF":5.3,"publicationDate":"2025-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143869862","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 : 2025-04-25DOI: 10.1016/j.aim.2025.110293
Kevin Allen, Robert Osburn
We consider two-parameter generalizations of Hecke-Appell type expansions for the generating functions of unimodal and special unimodal sequences. We then determine their explicit representations which involve mixed false theta functions. These results complement recent striking work of Mortenson and Zwegers on the mixed mock modularity of the generalized U-function due to Hikami and Lovejoy. As an application, we demonstrate how to recover classical partial theta function identities which appear in Ramanujan's lost notebook and in work of Warnaar.
{"title":"Unimodal sequences and mixed false theta functions","authors":"Kevin Allen, Robert Osburn","doi":"10.1016/j.aim.2025.110293","DOIUrl":"10.1016/j.aim.2025.110293","url":null,"abstract":"<div><div>We consider two-parameter generalizations of Hecke-Appell type expansions for the generating functions of unimodal and special unimodal sequences. We then determine their explicit representations which involve mixed false theta functions. These results complement recent striking work of Mortenson and Zwegers on the mixed mock modularity of the generalized <em>U</em>-function due to Hikami and Lovejoy. As an application, we demonstrate how to recover classical partial theta function identities which appear in Ramanujan's lost notebook and in work of Warnaar.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"473 ","pages":"Article 110293"},"PeriodicalIF":1.5,"publicationDate":"2025-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143873941","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}
SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 976-999, April 2025. Abstract. This work is on a user-friendly reduced basis method for the solution of families of parametric partial differential equations by preconditioned Krylov subspace methods including the conjugate gradient method, generalized minimum residual method, and biconjugate gradient method. The proposed methods use a preconditioned Krylov subspace method for a high-fidelity discretization of one parameter instance to generate orthogonal basis vectors of the reduced basis subspace. Then large-scale discrete parameter-dependent problems are approximately solved in the low-dimensional Krylov subspace. We prove convergence estimates for the proposed method when the differential operator depends on two parameter coefficients and the preconditioner is the inverse of the operator at a fixed parameter. As is shown in numerical experiments, only a small number of Krylov subspace iterations are needed to simultaneously generate approximate solutions of a family of high-fidelity and large-scale parametrized systems in the reduced basis subspace. This reduces the computational cost by orders of magnitude, because (1) to construct the reduced basis vectors, we only solve one large-scale problem in the high-fidelity level; and (2) the family of problems restricted to the reduced basis subspace have much smaller sizes.
{"title":"Reduced Krylov Basis Methods for Parametric Partial Differential Equations","authors":"Yuwen Li, Ludmil T. Zikatanov, Cheng Zuo","doi":"10.1137/24m1661236","DOIUrl":"https://doi.org/10.1137/24m1661236","url":null,"abstract":"SIAM Journal on Numerical Analysis, Volume 63, Issue 2, Page 976-999, April 2025. <br/> Abstract. This work is on a user-friendly reduced basis method for the solution of families of parametric partial differential equations by preconditioned Krylov subspace methods including the conjugate gradient method, generalized minimum residual method, and biconjugate gradient method. The proposed methods use a preconditioned Krylov subspace method for a high-fidelity discretization of one parameter instance to generate orthogonal basis vectors of the reduced basis subspace. Then large-scale discrete parameter-dependent problems are approximately solved in the low-dimensional Krylov subspace. We prove convergence estimates for the proposed method when the differential operator depends on two parameter coefficients and the preconditioner is the inverse of the operator at a fixed parameter. As is shown in numerical experiments, only a small number of Krylov subspace iterations are needed to simultaneously generate approximate solutions of a family of high-fidelity and large-scale parametrized systems in the reduced basis subspace. This reduces the computational cost by orders of magnitude, because (1) to construct the reduced basis vectors, we only solve one large-scale problem in the high-fidelity level; and (2) the family of problems restricted to the reduced basis subspace have much smaller sizes.","PeriodicalId":49527,"journal":{"name":"SIAM Journal on Numerical Analysis","volume":"44 1","pages":"976-999"},"PeriodicalIF":2.9,"publicationDate":"2025-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143876070","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 : 2025-04-25DOI: 10.1016/j.jde.2025.113312
Feng Liang , Maoan Han
Under a suitable assumption we obtain some new results on expansion coefficients and their relation for the first order Melnikov functions near any m-polycycle with hyperbolic saddles, , which establish a general bifurcation theory on limit cycles near the m-polycycles. As an application we consider 2-polycyclic bifurcations for a φ-Laplacian Liénard system and gain the number of limit cycles near the polycycle with two hyperbolic saddles.
{"title":"Expansion coefficients and their relation for Melnikov functions near polycycles","authors":"Feng Liang , Maoan Han","doi":"10.1016/j.jde.2025.113312","DOIUrl":"10.1016/j.jde.2025.113312","url":null,"abstract":"<div><div>Under a suitable assumption we obtain some new results on expansion coefficients and their relation for the first order Melnikov functions near any <em>m</em>-polycycle with hyperbolic saddles, <span><math><mi>m</mi><mo>∈</mo><msup><mrow><mi>N</mi></mrow><mrow><mo>+</mo></mrow></msup></math></span>, which establish a general bifurcation theory on limit cycles near the <em>m</em>-polycycles. As an application we consider 2-polycyclic bifurcations for a <em>φ</em>-Laplacian Liénard system and gain the number of limit cycles near the polycycle with two hyperbolic saddles.</div></div>","PeriodicalId":15623,"journal":{"name":"Journal of Differential Equations","volume":"435 ","pages":"Article 113312"},"PeriodicalIF":2.4,"publicationDate":"2025-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143869155","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 : 2025-04-25DOI: 10.1016/j.aim.2025.110287
Kenneth L. Baker , Marc Kegel , Duncan McCoy
We present two examples of strongly invertible L-space knots whose surgeries are never the double branched cover of a Khovanov thin link in the 3-sphere. Consequently, these knots provide counterexamples to a conjectural characterization of strongly invertible L-space knots due to Watson. We also discuss other exceptional properties of these two knots, for example, these two L-space knots have formal semigroups that are actual semigroups.
{"title":"Two curious strongly invertible L-space knots","authors":"Kenneth L. Baker , Marc Kegel , Duncan McCoy","doi":"10.1016/j.aim.2025.110287","DOIUrl":"10.1016/j.aim.2025.110287","url":null,"abstract":"<div><div>We present two examples of strongly invertible L-space knots whose surgeries are never the double branched cover of a Khovanov thin link in the 3-sphere. Consequently, these knots provide counterexamples to a conjectural characterization of strongly invertible L-space knots due to Watson. We also discuss other exceptional properties of these two knots, for example, these two L-space knots have formal semigroups that are actual semigroups.</div></div>","PeriodicalId":50860,"journal":{"name":"Advances in Mathematics","volume":"473 ","pages":"Article 110287"},"PeriodicalIF":1.5,"publicationDate":"2025-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143869129","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 : 2025-04-25DOI: 10.1016/j.camwa.2025.04.017
L. Liu, L.L. Zhang, M. Lei, R.P. Niu
An improved boundary knot method (IBKM) is proposed to enhance the performance of BKM in solving homogeneous high-order Helmholtz-type partial differential equations. Compared with the classical BKM where the sources are always placed on the physical boundary as collocation points, the new sources named fictitious points are now placed on multi-layer extended pseudo boundaries. This modification leads to higher accuracy without the loss of efficiency. Numerical examples are presented to demonstrate the superiority of the IBKM.
{"title":"The improved boundary knot method with fictitious points for solving high-order Helmholtz-type PDEs","authors":"L. Liu, L.L. Zhang, M. Lei, R.P. Niu","doi":"10.1016/j.camwa.2025.04.017","DOIUrl":"10.1016/j.camwa.2025.04.017","url":null,"abstract":"<div><div>An improved boundary knot method (IBKM) is proposed to enhance the performance of BKM in solving homogeneous high-order Helmholtz-type partial differential equations. Compared with the classical BKM where the sources are always placed on the physical boundary as collocation points, the new sources named fictitious points are now placed on multi-layer extended pseudo boundaries. This modification leads to higher accuracy without the loss of efficiency. Numerical examples are presented to demonstrate the superiority of the IBKM.</div></div>","PeriodicalId":55218,"journal":{"name":"Computers & Mathematics with Applications","volume":"191 ","pages":"Pages 36-47"},"PeriodicalIF":2.9,"publicationDate":"2025-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143869978","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 : 2025-04-25DOI: 10.1007/s00006-025-01376-9
Giulio Binosi
Holomorphic Cliffordian functions of order k are functions in the kernel of the differential operator (overline{partial }Delta ^k). When (overline{partial }Delta ^k) is applied to functions defined in the paravector space of some Clifford Algebra (mathbb {R}_m) with an odd number of imaginary units, the Fueter–Sce construction establishes a critical index (k=frac{m-1}{2}) (sometimes called Sce exponent) for which the class of slice regular functions is contained in the one of holomorphic Cliffordian functions of order (frac{m-1}{2}). In this paper, we analyze the case (k<frac{m-1}{2}) and find that the polynomials of degree at most 2k are the only slice regular holomorphic Cliffordian functions of order k.
{"title":"Slice Regular Holomorphic Cliffordian Functions of Order k","authors":"Giulio Binosi","doi":"10.1007/s00006-025-01376-9","DOIUrl":"10.1007/s00006-025-01376-9","url":null,"abstract":"<div><p>Holomorphic Cliffordian functions of order <i>k</i> are functions in the kernel of the differential operator <span>(overline{partial }Delta ^k)</span>. When <span>(overline{partial }Delta ^k)</span> is applied to functions defined in the paravector space of some Clifford Algebra <span>(mathbb {R}_m)</span> with an odd number of imaginary units, the Fueter–Sce construction establishes a critical index <span>(k=frac{m-1}{2})</span> (sometimes called Sce exponent) for which the class of slice regular functions is contained in the one of holomorphic Cliffordian functions of order <span>(frac{m-1}{2})</span>. In this paper, we analyze the case <span>(k<frac{m-1}{2})</span> and find that the polynomials of degree at most 2<i>k</i> are the only slice regular holomorphic Cliffordian functions of order <i>k</i>.</p></div>","PeriodicalId":7330,"journal":{"name":"Advances in Applied Clifford Algebras","volume":"35 3","pages":""},"PeriodicalIF":1.1,"publicationDate":"2025-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00006-025-01376-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143871297","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}
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