Pub Date : 2017-01-01DOI: 10.4310/MAA.2017.V24.N1.A6
K. Konno
Koyama’s inequality for normal surface singularities gives the upper bound on the self-intersection number of the canonical cycle in terms of the arithmetic genus. For those singularities of fundamental genus two attaining the bound, a formula for computing the geometric genus is shown and the resolution dual graphs are roughly classified. In Gorenstein case, the multiplicity and the embedding dimension are also computed.
{"title":"Certain normal surface singularities of general type","authors":"K. Konno","doi":"10.4310/MAA.2017.V24.N1.A6","DOIUrl":"https://doi.org/10.4310/MAA.2017.V24.N1.A6","url":null,"abstract":"Koyama’s inequality for normal surface singularities gives the upper bound on the self-intersection number of the canonical cycle in terms of the arithmetic genus. For those singularities of fundamental genus two attaining the bound, a formula for computing the geometric genus is shown and the resolution dual graphs are roughly classified. In Gorenstein case, the multiplicity and the embedding dimension are also computed.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70488240","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-01-01DOI: 10.4310/MAA.2017.v24.n2.a4
A. Harris
We introduce the notion of involutive Kodaira-Spencer deformations of the regular part X0 of a normal surface singularity, which form a subspace of the analytic cohomology H(X0, T X0). Examples of involutive deformations for which the Stein completion does not embed in a complex Euclidean space of stable dimension are in fact well-known. Under the assumption that X0 admits a Kähler metric with L-curvature, we show that unstable deformations are avoided if the holomorphic functions which determine an embedding of the central fibre are correspondingly deformed into functions which can be uniformly bounded on compact subsets.
{"title":"An intrinsic approach to stable embedding of normal surface deformations","authors":"A. Harris","doi":"10.4310/MAA.2017.v24.n2.a4","DOIUrl":"https://doi.org/10.4310/MAA.2017.v24.n2.a4","url":null,"abstract":"We introduce the notion of involutive Kodaira-Spencer deformations of the regular part X0 of a normal surface singularity, which form a subspace of the analytic cohomology H(X0, T X0). Examples of involutive deformations for which the Stein completion does not embed in a complex Euclidean space of stable dimension are in fact well-known. Under the assumption that X0 admits a Kähler metric with L-curvature, we show that unstable deformations are avoided if the holomorphic functions which determine an embedding of the central fibre are correspondingly deformed into functions which can be uniformly bounded on compact subsets.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70488355","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-01-01DOI: 10.4310/MAA.2017.V24.N1.A4
G. Ishikawa, S. Janeczko
We study germs of differential forms over singular varieties. The geometric restriction of differential forms to singular varieties is introduced and algebraic restrictions of differential forms with vanishing geometric restrictions, called residual algebraic restrictions, are investigated. Residues of plane curves-germs, hypersurfaces, Lagrangian varieties as well as the geometric and algebraic restriction via a mapping were calculated.
{"title":"Residual algebraic restrictions of differential forms","authors":"G. Ishikawa, S. Janeczko","doi":"10.4310/MAA.2017.V24.N1.A4","DOIUrl":"https://doi.org/10.4310/MAA.2017.V24.N1.A4","url":null,"abstract":"We study germs of differential forms over singular varieties. The geometric restriction of differential forms to singular varieties is introduced and algebraic restrictions of differential forms with vanishing geometric restrictions, called residual algebraic restrictions, are investigated. Residues of plane curves-germs, hypersurfaces, Lagrangian varieties as well as the geometric and algebraic restriction via a mapping were calculated.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70488230","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-01-01DOI: 10.4310/MAA.2017.V24.N2.A6
K. Konno, Ezio Stagnaro
Let C6 be a plane sextic curve with 6 double points that are not nodes. It is shown that if they are on a conic C2, then the unique possible case is that all of them are ordinary cusps. From this it follows that C6 is irreducible. Moreover, there is a plane cubic curve C3 such that C6 = C 3 2 + C 3 . Such curves are closely related to both the branch curve of the projection to a plane of the general cubic surface from a point outside it and canonical surfaces in P or P whose desingularizations have birational invariants q > 0, pg = 4 or pg = 5, P2 ≤ 23.
{"title":"Sextic curves with six double points on a conic","authors":"K. Konno, Ezio Stagnaro","doi":"10.4310/MAA.2017.V24.N2.A6","DOIUrl":"https://doi.org/10.4310/MAA.2017.V24.N2.A6","url":null,"abstract":"Let C6 be a plane sextic curve with 6 double points that are not nodes. It is shown that if they are on a conic C2, then the unique possible case is that all of them are ordinary cusps. From this it follows that C6 is irreducible. Moreover, there is a plane cubic curve C3 such that C6 = C 3 2 + C 3 . Such curves are closely related to both the branch curve of the projection to a plane of the general cubic surface from a point outside it and canonical surfaces in P or P whose desingularizations have birational invariants q > 0, pg = 4 or pg = 5, P2 ≤ 23.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70488391","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2017-01-01DOI: 10.4310/MAA.2017.V24.N2.A1
A. Aleksandrov, Hui-Qin Zuo, Henry Laufer
. We discuss an approach to the problem of classifying zero-dimensional gradient quasihomogeneous singularities using simple properties of deformation theory. As an example, we enumerate all such singularities with modularity ℘ = 0 and with Milnor number not greater than 12. We also compute normal forms and monomial vector-bases of the first cotangent homology and cohomology modules, the corresponding Poincar´e polynomials, inner modality, inner modularity, primitive ideals, etc.
{"title":"Zero-dimensional gradient singularities","authors":"A. Aleksandrov, Hui-Qin Zuo, Henry Laufer","doi":"10.4310/MAA.2017.V24.N2.A1","DOIUrl":"https://doi.org/10.4310/MAA.2017.V24.N2.A1","url":null,"abstract":". We discuss an approach to the problem of classifying zero-dimensional gradient quasihomogeneous singularities using simple properties of deformation theory. As an example, we enumerate all such singularities with modularity ℘ = 0 and with Milnor number not greater than 12. We also compute normal forms and monomial vector-bases of the first cotangent homology and cohomology modules, the corresponding Poincar´e polynomials, inner modality, inner modularity, primitive ideals, etc.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2017-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70488328","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-11-29DOI: 10.4310/MAA.2016.V23.N3.A4
C. Argáez, M. Melgaard
We study the open-shell, spin-polarized Kohn-Sham models for non-relativistic and quasi-relativistic N-electron Coulomb systems, that is, systems where the kinetic energy of the electrons is given by either the non-relativistic operator −Δxn or the quasi-relativistic operator √−α−²Δxn + α−4 − α−². For standard and extended Kohn-Sham models in the local density approximation, we prove existence of a ground state (or minimizer) provided that the total charge Ztot of K nuclei is greater than N − 1. For the quasi-relativistic setting we also need that Ztot is smaller than a critical charge Zc = 2α−¹π−¹.
{"title":"Minimizers for open-shell, spin-polarised Kohn–Sham equations for non-relativistic and quasi-relativistic molecular systems","authors":"C. Argáez, M. Melgaard","doi":"10.4310/MAA.2016.V23.N3.A4","DOIUrl":"https://doi.org/10.4310/MAA.2016.V23.N3.A4","url":null,"abstract":"We study the open-shell, spin-polarized Kohn-Sham models for non-relativistic and quasi-relativistic N-electron Coulomb systems, that is, systems where the kinetic energy of the electrons is given by either the non-relativistic operator −Δxn or the quasi-relativistic operator √−α−²Δxn + α−4 − α−². For standard and extended Kohn-Sham models in the local density approximation, we prove existence of a ground state (or minimizer) provided that the total charge Ztot of K nuclei is greater than N − 1. For the quasi-relativistic setting we also need that Ztot is smaller than a critical charge Zc = 2α−¹π−¹.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2016-11-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70488616","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-11-03DOI: 10.4310/MAA.2017.V24.N2.A2
Enrique Artal Bartolo, P. Cassou-Noguès, I. Luengo, A. Melle-Hern'andez
In 1982, Tamaki Yano proposed a conjecture predicting the set of b-exponents of an irreducible plane curve singularity germ which is generic in its equisingularity class. In cite{ACLM-Yano2} we proved the conjecture for the case in which the germ has two Puiseux pairs and its algebraic monodromy has distinct eigenvalues. In this article we aim to study the Bernstein polynomial for any function with the hypotheses above. In particular the set of all common roots of those Bernstein polynomials is given. We provide also bounds for some analytic invariants of singularities and illustrate the computations in suitable examples.
{"title":"Bernstein polynomial of $2$-Puiseux pairs irreducible plane curve singularities","authors":"Enrique Artal Bartolo, P. Cassou-Noguès, I. Luengo, A. Melle-Hern'andez","doi":"10.4310/MAA.2017.V24.N2.A2","DOIUrl":"https://doi.org/10.4310/MAA.2017.V24.N2.A2","url":null,"abstract":"In 1982, Tamaki Yano proposed a conjecture predicting the set of b-exponents of an irreducible plane curve singularity germ which is generic in its equisingularity class. In cite{ACLM-Yano2} we proved the conjecture for the case in which the germ has two Puiseux pairs and its algebraic monodromy has distinct eigenvalues. In this article we aim to study the Bernstein polynomial for any function with the hypotheses above. In particular the set of all common roots of those Bernstein polynomials is given. We provide also bounds for some analytic invariants of singularities and illustrate the computations in suitable examples.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2016-11-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70488343","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-03-10DOI: 10.4310/maa.2018.v25.n3.a3
A. Fish, L. Paunescu
. Inspired by the previous works by Freedman and He [FH], and Katznelson, Subhashis Nag, and Sullivan [KNS], we study the spiraling behaviour around a singularity of bi-Lipschitz homeomorphisms in R 2 . In particular, we show that there is no bi-Lipschitz homeomorphism of R 2 that maps a spiral with a sub-exponential decay of winding radii to an unwound arc. This result is sharp as shows an example of a logarithmic spiral.
． 受Freedman和He [FH]以及Katznelson, Subhashis Nag, and Sullivan [KNS]先前工作的启发，我们研究了r2中双lipschitz同纯态在奇点周围的螺旋行为。特别地，我们证明了r2中不存在双lipschitz同纯映射，它将一个绕圈半径呈次指数衰减的螺旋映射为一个未绕圈的弧。作为对数螺旋的一个例子，这个结果是清晰的。
{"title":"Unwinding spirals I","authors":"A. Fish, L. Paunescu","doi":"10.4310/maa.2018.v25.n3.a3","DOIUrl":"https://doi.org/10.4310/maa.2018.v25.n3.a3","url":null,"abstract":". Inspired by the previous works by Freedman and He [FH], and Katznelson, Subhashis Nag, and Sullivan [KNS], we study the spiraling behaviour around a singularity of bi-Lipschitz homeomorphisms in R 2 . In particular, we show that there is no bi-Lipschitz homeomorphism of R 2 that maps a spiral with a sub-exponential decay of winding radii to an unwound arc. This result is sharp as shows an example of a logarithmic spiral.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2016-03-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70488902","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-01-01DOI: 10.4310/MAA.2016.V23.N2.A2
Xiang-Sheng Wang
In this paper, we study asymptotic solutions of second-order difference equations with quadratic coefficients. According to the parameter values, we classify the difference equations into three cases and derive Plancherel-Rotach type asymptotic formulas of the solutions respectively. As direct applications of our main results, we also provide asymptotic formulas of associated Meixner-Pollaczek polynomials, associated Meixner polynomials, and associated Laguerre polynomials, respectively.
{"title":"Asymptotic analysis of difference equations with quadratic coefficients","authors":"Xiang-Sheng Wang","doi":"10.4310/MAA.2016.V23.N2.A2","DOIUrl":"https://doi.org/10.4310/MAA.2016.V23.N2.A2","url":null,"abstract":"In this paper, we study asymptotic solutions of second-order difference equations with quadratic coefficients. According to the parameter values, we classify the difference equations into three cases and derive Plancherel-Rotach type asymptotic formulas of the solutions respectively. As direct applications of our main results, we also provide asymptotic formulas of associated Meixner-Pollaczek polynomials, associated Meixner polynomials, and associated Laguerre polynomials, respectively.","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70488380","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2016-01-01DOI: 10.4310/MAA.2016.V23.N2.A3
Xiangsheng Xu
In this article we study the initial-boundary value problem for a family of nonlinear fourth order parabolic equations. The classical quantum drift-diffusion model is a member of the family. Two new existence theorems are established. Our approach is based upon a semi-discretization scheme, which generates a sequence of positive approximate solutions, and a functional inequality of the type
{"title":"A functional inequality and its applications to a class of nonlinear fourth-order parabolic equations","authors":"Xiangsheng Xu","doi":"10.4310/MAA.2016.V23.N2.A3","DOIUrl":"https://doi.org/10.4310/MAA.2016.V23.N2.A3","url":null,"abstract":"In this article we study the initial-boundary value problem for a family of nonlinear fourth order parabolic equations. The classical quantum drift-diffusion model is a member of the family. Two new existence theorems are established. Our approach is based upon a semi-discretization scheme, which generates a sequence of positive approximate solutions, and a functional inequality of the type","PeriodicalId":18467,"journal":{"name":"Methods and applications of analysis","volume":null,"pages":null},"PeriodicalIF":0.3,"publicationDate":"2016-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70488417","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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