We compare on the one hand the combinatorial procedure described in [1] which gives a lower bound for the Newton polygon of the $L$-function attached to a commode, non-degenerate polynomial with coefficients in a finite field and on the other hand the procedure which gives the Hodge theoretical spectrum at infinity of a polynomial with complex coefficients and with the same Newton polyhedron. The outcome is that they are essentially the same, thus providing a Hodge theoretical interpretation of the Adolphson-Sperber lower bound which was conjectured in [1].
{"title":"A note on L-series and Hodge spectrum of polynomials","authors":"R. G. López","doi":"10.3934/ERA.2009.16.56","DOIUrl":"https://doi.org/10.3934/ERA.2009.16.56","url":null,"abstract":"We compare on the one hand the combinatorial procedure described in [1] which gives a lower bound for the Newton polygon of the $L$-function attached to a commode, non-degenerate polynomial with coefficients in a finite field \u0000and on the other hand the procedure which gives the Hodge theoretical spectrum at infinity of a polynomial with complex coefficients and with the same Newton polyhedron. The outcome is that they are essentially the same, thus providing a Hodge theoretical interpretation of the Adolphson-Sperber lower bound which was conjectured in [1].","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"16 1","pages":"56-62"},"PeriodicalIF":0.0,"publicationDate":"2009-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232252","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}
Let $G$ be a word-hyperbolic group with a quasiconvex hierarchy. We show that $G$ has a finite index subgroup $G'$ that embeds as a quasiconvex subgroup of a right-angled Artin group. It follows that every quasiconvex subgroup of $G$ is a virtual retract, and is hence separable. The results are applied to certain 3-manifold and one-relator groups.
{"title":"RESEARCH ANNOUNCEMENT: THE STRUCTURE OF GROUPS WITH A QUASICONVEX HIERARCHY","authors":"D. Wise","doi":"10.3934/ERA.2009.16.44","DOIUrl":"https://doi.org/10.3934/ERA.2009.16.44","url":null,"abstract":"Let $G$ be a word-hyperbolic group with a quasiconvex hierarchy. \u0000We show that $G$ has a finite index subgroup $G'$ that embeds as a \u0000 quasiconvex subgroup of a right-angled Artin group. \u0000It follows that every quasiconvex subgroup of $G$ is a virtual retract, \u0000and is hence separable. \u0000The results are applied to certain 3-manifold and one-relator groups.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"16 1","pages":"44-55"},"PeriodicalIF":0.0,"publicationDate":"2009-10-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232211","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}
The goal of this note is to outline a proof that, for any l $geq 0$, the JLO bivariant cocycle associated with a family of Dirac type operators along a smooth fibration $Mto B$ over the pair of algebras $(C^infty (M), C^infty(B))$, is entire when we endow $C^infty(M)$ with the $C^{l+1}$ topology and $C^infty(B)$ with the $C^{l}$ topology. As a corollary, we deduce that this cocycle is analytic when we consider the Frechet smooth topologies on $C^infty(M)$ and $C^infty(B)$.
{"title":"On the analyticity of the bivariant JLO cocycle","authors":"M. Benameur, A. L. Carey","doi":"10.3934/ERA.2009.16.37","DOIUrl":"https://doi.org/10.3934/ERA.2009.16.37","url":null,"abstract":"The goal of this note is to outline a proof that, for any l $geq 0$, the JLO bivariant cocycle associated with a family of Dirac type operators along a smooth fibration $Mto B$ over the pair of algebras $(C^infty (M), C^infty(B))$, is entire when we endow $C^infty(M)$ with the $C^{l+1}$ topology and $C^infty(B)$ with the $C^{l}$ topology. As a corollary, we deduce that this cocycle is analytic when we consider the Frechet smooth topologies on $C^infty(M)$ and $C^infty(B)$.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"16 1","pages":"37-43"},"PeriodicalIF":0.0,"publicationDate":"2009-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232204","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 describe a method to study the existence of whiskered quasi-periodic solutions in Hamiltonian systems. The method applies to finite dimensional systems and also to lattice systems and to PDE's including some ill posed ones. In coupled map lattices, we can also construct solutions of infinitely many frequencies which do not vanish asymptotically.
{"title":"A METHOD FOR THE STUDY OF WHISKERED QUASI-PERIODIC AND ALMOST-PERIODIC SOLUTIONS IN FINITE AND INFINITE DIMENSIONAL HAMILTONIAN SYSTEMS","authors":"E. Fontich, R. Llave, Y. Sire","doi":"10.3934/ERA.2009.16.9","DOIUrl":"https://doi.org/10.3934/ERA.2009.16.9","url":null,"abstract":"We describe a method to study the existence of \u0000whiskered quasi-periodic solutions in Hamiltonian \u0000systems. \u0000The method applies to finite dimensional systems \u0000and also to lattice systems and to PDE's including \u0000some ill posed ones. \u0000In coupled map lattices, we can also \u0000construct solutions of infinitely many frequencies \u0000which do not vanish asymptotically.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"16 1","pages":"9-22"},"PeriodicalIF":0.0,"publicationDate":"2009-05-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232311","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 consider the spectrum of the Fibonacci Hamiltonian for small values of the coupling constant. It is known that this set is a Cantor set of zero Lebesgue measure. Here we study the limit, as the value of the coupling constant approaches zero, of its thickness and its Hausdorff dimension. We announce the following results and explain some key ideas that go into their proofs. The thickness tends to infinity and, consequently, the Hausdorff di- mension of the spectrum tends to one. Moreover, the length of every gap tends to zero linearly. Finally, for sufficiently small coupling, t sum of the spec- trum with itself is an interval. This last result provides a rigorous explanation of a phenomenon for the Fibonacci square lattice discovered numerically by Even-Dar Mandel and Lifshitz.
{"title":"THE SPECTRUM OF THE WEAKLY COUPLED FIBONACCI HAMILTONIAN","authors":"D. Damanik, A. Gorodetski","doi":"10.3934/ERA.2009.16.23","DOIUrl":"https://doi.org/10.3934/ERA.2009.16.23","url":null,"abstract":"We consider the spectrum of the Fibonacci Hamiltonian for small values of the coupling constant. It is known that this set is a Cantor set of zero Lebesgue measure. Here we study the limit, as the value of the coupling constant approaches zero, of its thickness and its Hausdorff dimension. We announce the following results and explain some key ideas that go into their proofs. The thickness tends to infinity and, consequently, the Hausdorff di- mension of the spectrum tends to one. Moreover, the length of every gap tends to zero linearly. Finally, for sufficiently small coupling, t sum of the spec- trum with itself is an interval. This last result provides a rigorous explanation of a phenomenon for the Fibonacci square lattice discovered numerically by Even-Dar Mandel and Lifshitz.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"16 1","pages":"23-29"},"PeriodicalIF":0.0,"publicationDate":"2009-01-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232155","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}
The purpose of this note is to announce an extension of the descent method of Ginzburg, Rallis, and Soudry to the setting of essentially self dual representations. This extension of the descent construction provides a complement to recent work of Asgari and Shahidi [2] on the generic transfer for general Spin groups as well as to the work of Asgari and Raghuram [1] on cuspidality of the exterior square lift for representations of $GL_4$. Complete proofs of the results announced in the present note will appear in our forthcoming article(s).
{"title":"DESCENT CONSTRUCTION FOR GSPIN GROUPS: MAIN RESULTS AND APPLICATIONS","authors":"Joseph Hundley, E. Sayag","doi":"10.3934/ERA.2009.16.30","DOIUrl":"https://doi.org/10.3934/ERA.2009.16.30","url":null,"abstract":"The purpose of this note is to announce an extension of the \u0000descent method of Ginzburg, Rallis, and Soudry to the setting of \u0000 essentially self dual representations. This extension of the \u0000descent construction provides a complement to recent work of \u0000Asgari and Shahidi [2] \u0000 \u0000on the generic transfer for general Spin groups as well as to the \u0000work of Asgari and Raghuram [1] on cuspidality \u0000of the exterior square lift for representations of $GL_4$. \u0000Complete proofs of the results announced in the present note will \u0000appear in our forthcoming article(s).","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"16 1","pages":"30-36"},"PeriodicalIF":0.0,"publicationDate":"2008-08-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232194","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 announce the development of a functional calculus for operators defined on quaternionic Banach spaces. The definition is based on a new notion of slice regularity, see [6], and the key tools are a new resolvent operator and a new eigenvalue problem. This approach allows us to deal both with bounded and unbounded operators.
{"title":"A Functional Calculus in a Non Commutative Setting","authors":"F. Colombo, G. Gentili, I. Sabadini, D. Struppa","doi":"10.3934/ERA.2007.14.60","DOIUrl":"https://doi.org/10.3934/ERA.2007.14.60","url":null,"abstract":"In this paper we announce the development of a functional calculus for operators defined on quaternionic Banach spaces. The definition is based on a new notion of slice regularity, see [6], and the key tools are a new resolvent operator and a new eigenvalue problem. This approach allows us to deal both with bounded and unbounded operators.","PeriodicalId":53151,"journal":{"name":"Electronic Research Announcements in Mathematical Sciences","volume":"14 1","pages":"60-68"},"PeriodicalIF":0.0,"publicationDate":"2007-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"70232603","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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