It is long known that with respect to the property of having a finitely axiomatizable equational theory, there is no relationship between a general involution semigroup and its semigroup reduct. The present article establishes such a relationship within the class of involution semigroups that are unstable in the sense that the varieties they generate contain semilattices with nontrivial involution. Specifically, it is shown that the equational theory of an unstable involution semigroup is not finitely axiomatizable whenever the equational theory of its semigroup reduct satisfies the same property. Consequently, many results on equational properties of semigroups can be converted into results applicable to involution semigroups.
{"title":"Equational theories of unstable involution semigroups","authors":"Edmond W. H. Lee","doi":"10.3934/ERA.2017.24.002","DOIUrl":"https://doi.org/10.3934/ERA.2017.24.002","url":null,"abstract":"It is long known that with respect to the property of having a finitely axiomatizable equational theory, there is no relationship between a general involution semigroup and its semigroup reduct. The present article establishes such a relationship within the class of involution semigroups that are unstable in the sense that the varieties they generate contain semilattices with nontrivial involution. Specifically, it is shown that the equational theory of an unstable involution semigroup is not finitely axiomatizable whenever the equational theory of its semigroup reduct satisfies the same property. Consequently, many results on equational properties of semigroups can be converted into results applicable to involution semigroups.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"24 1","pages":"10-20"},"PeriodicalIF":0.0,"publicationDate":"2017-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44267163","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}
We prove a result for maps of surfaces that illustrates how singularhyperbolic flows can be desingularized if a global section can be collapsed to a surface along stable leaves.
{"title":"Desingularization of surface maps","authors":"Erica Clay, B. Hasselblatt, E. Pujals","doi":"10.3934/ERA.2017.24.001","DOIUrl":"https://doi.org/10.3934/ERA.2017.24.001","url":null,"abstract":"We prove a result for maps of surfaces that illustrates how singularhyperbolic flows can be desingularized if a global section can be collapsed to a surface along stable leaves.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"24 1","pages":"1-9"},"PeriodicalIF":0.0,"publicationDate":"2017-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47975246","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}
Using Langer's variation on the Bogomolov-Miyaoka-Yau inequality, we provide some Hirzebruch-type inequalities for curve arrangements in the complex projective plane.
{"title":"The orbifold Langer-Miyaoka-Yau Inequality and Hirzebruch-type inequalities","authors":"Piotr Pokora","doi":"10.3934/era.2017.24.003","DOIUrl":"https://doi.org/10.3934/era.2017.24.003","url":null,"abstract":"Using Langer's variation on the Bogomolov-Miyaoka-Yau inequality, we provide some Hirzebruch-type inequalities for curve arrangements in the complex projective plane.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"24 1","pages":"21-27"},"PeriodicalIF":0.0,"publicationDate":"2016-12-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70233690","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}
In this note we construct Banach limits on the class of sequences of convex bodies. Surprisingly, the construction uses the recently introduced geometric mean of convex bodies. In the opposite direction, we explain how Banach limits can be used to construct a new variant of the geometric mean that has some desirable properties.
{"title":"Banach limit in convexity and geometric means for convex bodies","authors":"Liran Rotem","doi":"10.3934/ERA.2016.23.005","DOIUrl":"https://doi.org/10.3934/ERA.2016.23.005","url":null,"abstract":"In this note we construct Banach limits on the class of sequences \u0000of convex bodies. Surprisingly, the construction uses the recently \u0000introduced geometric mean of convex bodies. In the opposite direction, \u0000we explain how Banach limits can be used to construct a new variant \u0000of the geometric mean that has some desirable properties.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"23 1","pages":"41-51"},"PeriodicalIF":0.0,"publicationDate":"2016-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70233679","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}
In this paper, a Liouville type theorem is proved for some global fully nonlinear evolution inequality via suitable choices of test functions and the argument of integration by parts.
{"title":"NONEXISTENCE RESULTS FOR A FULLY NONLINEAR EVOLUTION INEQUALITY","authors":"Qianzhong Ou","doi":"10.3934/ERA.2016.23.003","DOIUrl":"https://doi.org/10.3934/ERA.2016.23.003","url":null,"abstract":"In this paper, a Liouville type theorem is proved for some global fully nonlinear evolution inequality via suitable choices of test functions and the argument of integration by parts.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"27 1","pages":"19-24"},"PeriodicalIF":0.0,"publicationDate":"2016-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70233645","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}
In this paper, we study a combined incompressible and vanishing capillarity limit in the barotropic compressible Navier-Stokes-Korteweg equations for weak solutions. For well prepared initial data, the convergence of solutions of the compressible Navier-Stokes-Korteweg equations to the solutions of the incompressible Navier-Stokes equation are justified rigorously by adapting the modulated energy method. Furthermore, the corresponding convergence rates are also obtained.
{"title":"Asymptotic limit of a Navier-Stokes-Korteweg system with density-dependent viscosity","authors":"Jianwei Yang, Peng Cheng, Yudong Wang","doi":"10.3934/ERA.2015.22.20","DOIUrl":"https://doi.org/10.3934/ERA.2015.22.20","url":null,"abstract":"In this paper, we study a combined incompressible and vanishing \u0000capillarity limit in the barotropic compressible \u0000Navier-Stokes-Korteweg equations for weak solutions. For well \u0000prepared initial data, the convergence of solutions of the \u0000compressible Navier-Stokes-Korteweg equations to the \u0000solutions of the incompressible Navier-Stokes equation are justified \u0000rigorously by adapting the modulated energy method. Furthermore, the \u0000corresponding convergence rates are also obtained.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"22 1","pages":"20-31"},"PeriodicalIF":0.0,"publicationDate":"2015-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70233967","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}
We announce an analogue of the celebrated theorem by Campbell, Baker, Hausdorff, and Dynkin for the $q$-exponential $exp_q(x)=sum_{n=0}^{infty} frac{x^n}{[n]_q!}$, with the usual notation for $q$-factorials: $[n]_q!:=[n-1]_q!cdot(q^n-1)/(q-1)$ and $[0]_q!:=1$. Our result states that if $x$ and $y$ are non-commuting indeterminates and $[y,x]_q$ is the $q$-commutator $yx-q,xy$, then there exist linear combinations $Q_{i,j}(x,y)$ of iterated $q$-commutators with exactly $i$ $x$'s, $j$ $y$'s and $[y,x]_q$ in their central position, such that $exp_q(x)exp_q(y)=exp_q!big(x+y+sum_{i,jgeq 1}Q_{i,j}(x,y)big)$. Our expansion is consistent with the well-known result by Schutzenberger ensuring that one has $exp_q(x)exp_q(y)=exp_q(x+y)$ if and only if $[y,x]_q=0$, and it improves former partial results on $q$-deformed exponentiation. Furthermore, we give an algorithm which produces conjecturally a minimal generating set for the relations between $[y,x]_q$-centered $q$-commutators of any bidegree $(i,j)$, and it allows us to compute all possible $Q_{i,j}$.
{"title":"The $boldsymbol{q}$-deformed Campbell-Baker-Hausdorff-Dynkin theorem","authors":"Rüdiger Achilles, A. Bonfiglioli, J. Katriel","doi":"10.3934/ERA.2015.22.32","DOIUrl":"https://doi.org/10.3934/ERA.2015.22.32","url":null,"abstract":"We announce an analogue of the celebrated theorem by Campbell, Baker, Hausdorff, and Dynkin for the $q$-exponential $exp_q(x)=sum_{n=0}^{infty} frac{x^n}{[n]_q!}$, with the usual notation for $q$-factorials: $[n]_q!:=[n-1]_q!cdot(q^n-1)/(q-1)$ and $[0]_q!:=1$. Our result states that if $x$ and $y$ are non-commuting indeterminates and $[y,x]_q$ is the $q$-commutator $yx-q,xy$, then there exist linear combinations $Q_{i,j}(x,y)$ of iterated $q$-commutators with exactly $i$ $x$'s, $j$ $y$'s and $[y,x]_q$ in their central position, such that $exp_q(x)exp_q(y)=exp_q!big(x+y+sum_{i,jgeq 1}Q_{i,j}(x,y)big)$. Our expansion is consistent with the well-known result by Schutzenberger ensuring that one has $exp_q(x)exp_q(y)=exp_q(x+y)$ if and only if $[y,x]_q=0$, and it improves former partial results on $q$-deformed exponentiation. Furthermore, we give an algorithm which produces conjecturally a minimal generating set for the relations between $[y,x]_q$-centered $q$-commutators of any bidegree $(i,j)$, and it allows us to compute all possible $Q_{i,j}$.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"22 1","pages":"32-45"},"PeriodicalIF":0.0,"publicationDate":"2015-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70234032","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}
We extend the results of B. Minemyer by showing that any indefinite metric polyhedron (either compact or not) with the vertex degree bounded from above admits an isometric simplicial embedding into a Minkowski space of the lowest possible dimension. We provide a simple algorithm for constructing such embeddings. We also show that every partial simplicial isometric embedding of such space in general position extends to a simplicial isometric embedding of the whole space.
{"title":"Extensions of isometric embeddings of pseudo-Euclidean metric polyhedra","authors":"Pavel Galashin, V. Zolotov","doi":"10.3934/era.2016.23.001","DOIUrl":"https://doi.org/10.3934/era.2016.23.001","url":null,"abstract":"We extend the results of B. Minemyer by showing that any indefinite metric polyhedron (either compact or not) with the vertex degree bounded from above admits an isometric simplicial embedding into a Minkowski space of the lowest possible dimension. We provide a simple algorithm for constructing such embeddings. We also show that every partial simplicial isometric embedding of such space in general position extends to a simplicial isometric embedding of the whole space.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"23 1","pages":"1-7"},"PeriodicalIF":0.0,"publicationDate":"2015-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70233628","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}
We use recent work on integral forms in vertex operator algebras to construct vertex algebras over general commutative rings and Chevalley groups acting on them as vertex algebra automorphisms. In this way, we get series of vertex algebras over fields whose automorphism groups are essentially those Chevalley groups (actually, an exact statement depends on the field and involves upwards extensions of these groups by outer diagonal and graph automorphisms). In particular, given a prime power $q$, we realize each finite simple group which is a Chevalley or Steinberg variations over $mathbb{F}_q$ as "most of'' the full automorphism group of a vertex algebra over $mathbb{F}_q$. These finite simple groups are [ A_n(q), B_n(q), C_n(q), D_n(q), E_6(q), E_7(q), E_8(q), F_4(q), G_2(q) ] [ text{and } ^{2}A_n(q), ^{2}D_n(q), ^{3}D_4(q), ^{2}E_6(q), ] where $q$ is a prime power. Also, we define certain reduced VAs. In characteristics 2 and 3, there are exceptionally large automorphism groups. A covering algebra idea of Frohardt and Griess for Lie algebras is applied to the vertex algebra situation. We use integral form and covering procedures for vertex algebras to complete the modular moonshine program of Borcherds and Ryba for proving an embedding of the sporadic group $F_3$ of order $2^{15}3^{10}5^3 7^2 13{cdot }19{cdot} 31$ in $E_8(3)$.
{"title":"Groups of Lie type, vertex algebras, and modular moonshine","authors":"R. Griess, C. Lam","doi":"10.3934/ERA.2014.21.167","DOIUrl":"https://doi.org/10.3934/ERA.2014.21.167","url":null,"abstract":"We use recent work on integral forms in vertex operator algebras to construct vertex algebras over general commutative rings and Chevalley groups acting on them as vertex algebra automorphisms. \u0000In this way, we get series of vertex algebras over fields whose automorphism groups are essentially those Chevalley groups (actually, an exact statement depends on the field and involves upwards extensions of these groups by outer diagonal and graph automorphisms). In particular, given a prime power $q$, we realize each finite simple group which is a Chevalley or Steinberg variations over $mathbb{F}_q$ as \"most of'' the full automorphism group of a vertex algebra over $mathbb{F}_q$. \u0000These finite simple groups are \u0000[ A_n(q), B_n(q), C_n(q), D_n(q), E_6(q), E_7(q), E_8(q), F_4(q), G_2(q) ] \u0000[ text{and } ^{2}A_n(q), ^{2}D_n(q), ^{3}D_4(q), ^{2}E_6(q), ] \u0000where $q$ is a prime power. \u0000 \u0000Also, we define certain reduced VAs. In characteristics 2 and 3, \u0000there are exceptionally large automorphism groups. A covering algebra idea of Frohardt and Griess for Lie algebras is applied to \u0000the vertex algebra situation. \u0000 \u0000We use integral form and covering procedures for vertex algebras to complete the modular moonshine program of Borcherds and \u0000Ryba for proving an embedding of the sporadic group $F_3$ of order $2^{15}3^{10}5^3 7^2 13{cdot }19{cdot} 31$ in $E_8(3)$.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"21 1","pages":"167-176"},"PeriodicalIF":0.0,"publicationDate":"2014-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232671","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}
On a real analytic $5$-dimensional CR-generic submanifold $M^5 subset mathbb{C}^4$ of codimension $3$ hence of CR dimension $1$, which enjoys the generically satisfied nondegeneracy condition begin{align*} {bf 5} &= text{rank}_mathbb{C} big( T^{1,0}M+T^{0,1}M + big[T^{1,0}M,,T^{0,1}Mbig] ,+ &qquad + big[T^{1,0}M,,[T^{1,0}M,T^{0,1}M]big] + big[T^{0,1}M,,[T^{1,0}M,T^{0,1}M]big] big), end{align*} a canonical Cartan connection is constructed after reduction to a certain partially explicit ${ e}$-structure of the concerned local biholomorphic equivalence problem.
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