{"title":"A note on a bilevel problem for parameter learning for inverse problems with the wave equation","authors":"Wiebke Günther, A. Kröner","doi":"10.4171/pm/2073","DOIUrl":"https://doi.org/10.4171/pm/2073","url":null,"abstract":"","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2022-01-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42425157","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}
We consider four long-standing Rigidity Conjectures about synchrony and phase patterns for hyperbolic periodic orbits of admissible ODEs for networks. Proofs of stronger local versions of these conjectures, published in 2010-12, are now known to have a gap, but remain valid for a broad class of networks. Using different methods we prove local versions of the conjectures under a stronger condition, `strong hyperbolicity', which is related to a network analogue of the Kupka-Smale Theorem. Under this condition we also deduce global versions of the conjectures and an analogue of the $H/K$ Theorem in equivariant dynamics. We prove the Rigidity Conjectures for all 1- and 2-colourings and all 2- and 3-node networks by proving that strong hyperbolicity is generic in these cases.
{"title":"Overdetermined ODEs and rigid periodic states in network dynamics","authors":"Ian Stewart","doi":"10.4171/pm/2080","DOIUrl":"https://doi.org/10.4171/pm/2080","url":null,"abstract":"We consider four long-standing Rigidity Conjectures about synchrony and phase patterns for hyperbolic periodic orbits of admissible ODEs for networks. Proofs of stronger local versions of these conjectures, published in 2010-12, are now known to have a gap, but remain valid for a broad class of networks. Using different methods we prove local versions of the conjectures under a stronger condition, `strong hyperbolicity', which is related to a network analogue of the Kupka-Smale Theorem. Under this condition we also deduce global versions of the conjectures and an analogue of the $H/K$ Theorem in equivariant dynamics. We prove the Rigidity Conjectures for all 1- and 2-colourings and all 2- and 3-node networks by proving that strong hyperbolicity is generic in these cases.","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48151458","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}
ev : Kg,n(X , d) → ĪX , where ĪX is the rigidified inertia stack of X . See [3, Section 3] for a detailed discussion on inertia stacks and [3, Section 4.4] for the construction of evaluation maps. (2) Forgetting stable maps to X but only retaining the domain curves yields the forgetful map π : Kg,n(X , d) → M tw g,n, where M g,n is the stack of n-pointed genus g orbifold curves, see [19, Theorem 1.9]. Assuming 2g − 2 + n > 0, then passing to coarse curves and stablizing the domains yield another forgetful map p : Kg,n(X , d) → Mg,n, where Mg,n is the stack of n-pointed genus g stable curves. There is an obvious commutative diagram Kg,n(X , d)
{"title":"On orbifold Gromov–Witten classes","authors":"Hsian-Hua Tseng","doi":"10.4171/pm/2090","DOIUrl":"https://doi.org/10.4171/pm/2090","url":null,"abstract":"ev : Kg,n(X , d) → ĪX , where ĪX is the rigidified inertia stack of X . See [3, Section 3] for a detailed discussion on inertia stacks and [3, Section 4.4] for the construction of evaluation maps. (2) Forgetting stable maps to X but only retaining the domain curves yields the forgetful map π : Kg,n(X , d) → M tw g,n, where M g,n is the stack of n-pointed genus g orbifold curves, see [19, Theorem 1.9]. Assuming 2g − 2 + n > 0, then passing to coarse curves and stablizing the domains yield another forgetful map p : Kg,n(X , d) → Mg,n, where Mg,n is the stack of n-pointed genus g stable curves. There is an obvious commutative diagram Kg,n(X , d)","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-12-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42099520","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}
We study how changing the weight datum $mathcal{A}=(a_1,...,a_n)in(mathbb{Q}cap(0,1])^n$ affects the topology of tropical moduli spaces $M_{g,mathcal{A}}^{trop}$, $overline{M}_{g,mathcal{A}}^{trop}$ and $Delta_{g,mathcal{A}}$ and the homology of the latter one. We show that for fixed $g$ and $n$, there are particular filtrations of these topological spaces which we can use to compute the reduced rational homology of $Delta_{g,mathcal{A}}$ and the top weight cohomology of the moduli space $mathcal{M}_{g,mathcal{A}}$ of smooth $(g,mathcal{A})$-stable algebraic curves.
{"title":"Filtrations of moduli spaces of tropical weighted stable curves","authors":"S. Serpente","doi":"10.4171/pm/2093","DOIUrl":"https://doi.org/10.4171/pm/2093","url":null,"abstract":"We study how changing the weight datum $mathcal{A}=(a_1,...,a_n)in(mathbb{Q}cap(0,1])^n$ affects the topology of tropical moduli spaces $M_{g,mathcal{A}}^{trop}$, $overline{M}_{g,mathcal{A}}^{trop}$ and $Delta_{g,mathcal{A}}$ and the homology of the latter one. We show that for fixed $g$ and $n$, there are particular filtrations of these topological spaces which we can use to compute the reduced rational homology of $Delta_{g,mathcal{A}}$ and the top weight cohomology of the moduli space $mathcal{M}_{g,mathcal{A}}$ of smooth $(g,mathcal{A})$-stable algebraic curves.","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":"1 1","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41422930","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}
In this article we investigate the long time behavior of solutions to a class of infinitely many master equations defined from transition rates that are suitable for the description of a quantum system approaching thermodynamical equilibrium with a heat bath at fixed temperature and a reservoir consisting of one species of particles characterized by a fixed chemical potential. We do so by proving a result which pertains to the spectral resolution of the semigroup generated by the equations, whose infinitesimal generator is realized as a trace-class self-adjoint operator defined in a suitable weighted sequence space. This allows us to prove the existence of global solutions which all stabilize toward the grand canonical equilibrium probability distribution as the time variable becomes large, some of them doing so exponentially rapidly. When we set the chemical potential equal to zero, the stability statements continue to hold in the sense that all solutions converge toward the Gibbs probability distribution of the canonical ensemble which characterizes the equilibrium of the given system with a heat bath at fixed temperature.
{"title":"On the asymptotic behavior of solutions to a class of grand canonical master equations","authors":"Sabine Bogli, P. Vuillermot","doi":"10.4171/pm/2102","DOIUrl":"https://doi.org/10.4171/pm/2102","url":null,"abstract":"In this article we investigate the long time behavior of solutions to a class of infinitely many master equations defined from transition rates that are suitable for the description of a quantum system approaching thermodynamical equilibrium with a heat bath at fixed temperature and a reservoir consisting of one species of particles characterized by a fixed chemical potential. We do so by proving a result which pertains to the spectral resolution of the semigroup generated by the equations, whose infinitesimal generator is realized as a trace-class self-adjoint operator defined in a suitable weighted sequence space. This allows us to prove the existence of global solutions which all stabilize toward the grand canonical equilibrium probability distribution as the time variable becomes large, some of them doing so exponentially rapidly. When we set the chemical potential equal to zero, the stability statements continue to hold in the sense that all solutions converge toward the Gibbs probability distribution of the canonical ensemble which characterizes the equilibrium of the given system with a heat bath at fixed temperature.","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-11-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45557302","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}
{"title":"Analysis of a bulk-surface thermistor model for large-area organic LEDs","authors":"A. Glitzky, M. Liero, Grigor Nika","doi":"10.4171/pm/2066","DOIUrl":"https://doi.org/10.4171/pm/2066","url":null,"abstract":"","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-08-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41541094","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}
In this short note we construct unbounded families of minimal surfaces of general type with canonical map of degree 4 such that the limits of the slopes assume countably many different values among 6+2/3 and 8.
{"title":"Some surfaces with canonical map of degree 4","authors":"Federico Fallucca, R. Pignatelli","doi":"10.4171/pm/2106","DOIUrl":"https://doi.org/10.4171/pm/2106","url":null,"abstract":"In this short note we construct unbounded families of minimal surfaces of general type with canonical map of degree 4 such that the limits of the slopes assume countably many different values among 6+2/3 and 8.","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-07-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45899525","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}
. We give a short proof of a theorem of G.Cotti, B.Dubrovin and D.Guzzetti ([ CDG19 ] and [ Guz21 ]) asserting the vanishing of some entries of the Stokes matrices at coalescing points of an isomonodromic deformation.
{"title":"A short proof of a theorem of Cotti, Dubrovin and Guzzetti","authors":"C. Sabbah","doi":"10.4171/PM/2077","DOIUrl":"https://doi.org/10.4171/PM/2077","url":null,"abstract":". We give a short proof of a theorem of G.Cotti, B.Dubrovin and D.Guzzetti ([ CDG19 ] and [ Guz21 ]) asserting the vanishing of some entries of the Stokes matrices at coalescing points of an isomonodromic deformation.","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47154040","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}
A stationary free boundary problem modeling a three-dimensional electrostatic MEMS device is investigated. The device is made of a rigid ground plate and an elastic top plate which is hinged at its boundary, the plates being held at different voltages. The model couples a nonlocal fourth-order equation for the deformation of the top plate to a Laplace equation for the electrostatic potential in the free domain between the two plates. The strength of the coupling is tuned by a parameter λ which is proportional to the square of the applied voltage difference. Existence of a stable stationary solution is established for small values of λ. Nonexistence of stationary solutions is obtained when λ is large enough.
{"title":"On a free boundary model for three-dimensional MEMS with a hinged top plate: Stationary case","authors":"K. Nik","doi":"10.4171/pm/2067","DOIUrl":"https://doi.org/10.4171/pm/2067","url":null,"abstract":"A stationary free boundary problem modeling a three-dimensional electrostatic MEMS device is investigated. The device is made of a rigid ground plate and an elastic top plate which is hinged at its boundary, the plates being held at different voltages. The model couples a nonlocal fourth-order equation for the deformation of the top plate to a Laplace equation for the electrostatic potential in the free domain between the two plates. The strength of the coupling is tuned by a parameter λ which is proportional to the square of the applied voltage difference. Existence of a stable stationary solution is established for small values of λ. Nonexistence of stationary solutions is obtained when λ is large enough.","PeriodicalId":51269,"journal":{"name":"Portugaliae Mathematica","volume":" ","pages":""},"PeriodicalIF":0.8,"publicationDate":"2021-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47851700","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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