Showing papers in "Forum Mathematicum in 2007"
TL;DR: In this paper, it was proved that a cuspidal newform (holomorphic or Maass) of arbitrary level and nebentypus, a primitive character of conductor q, and s a point on the critical line
Abstract: Let g be a cuspidal newform (holomorphic or Maass) of arbitrary level and nebentypus, ´ a primitive character of conductor q, and s a point on the critical line
95 citations
TL;DR: In this article, Kramar et al. extend the results of [Kramar M. et al., 2005] to more general transport processes in networks allowing space dependent velocities and absorption.
Abstract: Abstract Using functional analytical and graph theoretical methods, we extend the results of [Kramar M. und Sikolya E.: Spectral properties and asymptotic periodicity of flows in networks. Math. Z. 249 (2005), 139–162] to more general transport processes in networks allowing space dependent velocities and absorption. We characterize asymptotic periodicity and convergence to an equilibrium by conditions on the underlying directed graph and the (average) velocities.
48 citations
TL;DR: In this article, a general class of symmetric spaces, called lineated symmetric space (LSS) are defined, which satisfy the axioms of Loos together with an additional axiom that guarantees unique midpoints of symmetry.
Abstract: Abstract We develop the basic theory of a general class of symmetric spaces, called lineated symmetric spaces, that satisfy the axioms of Loos together with an additional axiom that guarantees unique midpoints of symmetry. Our primary interest is the case that these symmetric spaces are Banach manifolds, in which case they exhibit an interesting geometric structure, and particularly in the metric case, where it is assumed the symmetric space carries a convex metric, an invariant complete metric contracting the square root function. One major result is that the distance function between points evolving over time on two geodesics is a convex function. Primary examples arise from involutive Banach-Lie groups (G,σ) admitting a polar decomposition G = P · K, where K is the subgroup fixed by σ and P is the associated symmetric set. We consider an appropriate notion of seminegative curvature for such symmetric spaces endowed with an invariant Finsler metric and prove that the corresponding length metric must be a convex metric. The preceding results provide a general framework for the interesting Finsler geometry of the space of positive Hermitian elements of a C*-algebra that has emerged in recent years.
44 citations
TL;DR: In this paper, a parabolic Harnack inequality for the heat kernel was proved, which implies a sharp two-sided estimate for the associated heat kernel, based on the unitary equivalence between the Schrödinger operator and the weighted Laplacian.
Abstract: Abstract A parabolic Harnack inequality for the equation is proved; in particular, this implies a sharp two-sided estimate for the associated heat kernel. Our approach relies on the unitary equivalence between the Schrödinger operator and the weighted Laplacian when .
44 citations
TL;DR: In this paper, the authors studied the chain transitive sets and Morse decompositions of flows on fiber bundles whose fibers are compact homogeneous spaces of Lie groups, and the emphasis was put on generalized flag manifolds of semi-simple (and reductive) Lie groups.
Abstract: We study the chain transitive sets and Morse decompositions of flows on fiber bundles whose fibers are compact homogeneous spaces of Lie groups. The emphasis is put on generalized flag manifolds of semi-simple (and reductive) Lie groups. In this case an algebraic description of the chain transitive sets is given. Our approach consists in shadowing the flow by semigroups of homeomorphisms to take advantage of the good properties of the semigroup actions on flag manifolds. The description of the chain components in the flag bundles generalizes a theorem of Selgrade for projective bundles with an independent proof.
44 citations
TL;DR: In this paper, the cost of the classical Kummer descent has been removed from the previous best upper bounds for the ring of integers in algebraic numbers, where b 1, b n being rational integers and Ξ ≠ 1.
Abstract: Abstract Let α1, … , α n be non-zero algebraic numbers and K be a number field containing α1, … , α n . Denote by 𝔭 a prime ideal of the ring of integers in K. We present completely explicit upper bounds for , where with b 1, … , b n being rational integers and Ξ ≠ 1. The cost n!/2 n−1 of the classical Kummer descent has been removed from the previous best upper bounds.
43 citations
TL;DR: In this article, the authors define recurrence at a point and then show that the conditions (i) pointwise recurrence, (ii) X is a union of minimal sets and (iii) the orbit closure relation is closed in X × X, are equivalent.
Abstract: Abstract For an action of a finitely generated group G on a compact space X we define recurrence at a point and then show that, when X is zero dimensional, the conditions (i) pointwise recurrence, (ii) X is a union of minimal sets and (iii) the orbit closure relation is closed in X × X, are equivalent. As a corollary we get that for such flows distality is the same as equicontinuity. In the last part of the paper we describe an example of a ℤ-flow where all points are positively recurrent, but there are points which are not negatively recurrent.
37 citations
TL;DR: De Gruyter et al. as discussed by the authors proved a rank 3 criterion for the simple connectedness of certain subsets of buildings and gave two applications of this criterion, one generalizing a result of Tits for Chevalley groups to 3-spherical Kac-Moody groups.
Abstract: We prove a rank 3 criterion for the simple connectedness of certain subsets of buildings and we give two applications of this criterion. The first generalizes a result of Tits for Chevalley groups to 3-spherical Kac-Moody groups. The second is the proof of the simple connectedness of certain flipflop geometries introduced in [BGHS]. © Walter de Gruyter 2007.
36 citations
TL;DR: For the case of 0 < r < 1, this paper showed that the families of multiparameter paraproducts are much richer than in the one-parameter case.
Abstract: For multiparameter bilinear paraproduct operators B we prove the estimate B : L pL q 7 L r ; 1 < p; qay: Here, 1=p þ 1=q ¼ 1=r and special attention is paid to the case of 0 < r < 1. (Note that the families of multiparameter paraproducts are much richer than in the one parameter case.) These estimates are the essential step in the version of the multiparameter Coifman-Meyer theorem proved by C. Muscalu, J. Pipher, T. Tao, and C. Thiele (10, 11). We oer a dierent proof of these inequalities.
34 citations
TL;DR: In this article, the authors extend the Farrell-Jones and Baum-Connes Conjecture for finite groups to infinite groups by reducing these families of subgroups to a smaller family.
Abstract: The Farrell-Jones and the Baum-Connes Conjecture say that one can compute the algebraic K- and L-theory of the group ring and the topological K-theory of the reduced group C -algebra of a group G in terms of these functors for the virtually cyclic subgroups or the finite subgroups of G. By induction theory we want to reduce these families of subgroups to a smaller family, for instance to the family of subgroups which are either finite hyperelementary or extensions of finite hyperelementary groups with Z as kernel or to the family of finite cyclic subgroups. Roughly speaking, we extend the induction theorems of Dress for finite groups to infinite groups.
30 citations
TL;DR: In this paper, a new presentation for simply connected Kac-Moody groups of 2-spherical type and their universal central extensions is presented. But their results extend to the more general class of groups endowed with a root datum.
Abstract: We provide a new presentation for simply connected Kac-Moody groups of 2- spherical type and for their universal central extensions. Under mild local restrictions, these results extend to the more general class of groups of Kac-Moody type (i.e. groups endowed with a root datum).
TL;DR: In this article, the authors extend the Lewis Correspondence between Maas wave forms and period functions to subgroups of the modular group of finite index, and the period functions then become vector valued functions.
Abstract: We extend the Lewis Correspondence between Maas wave forms and period functions to subgroups of the modular group of finite index. The period functions then become vector valued functions.
Abstract: (Communicated by Ru¨diger Go¨bel)Abstract. Complete cotorsion pairs are among the main sources of module approximations.Given a ring R and a cotorsion pair C ¼ðA;BÞ, we consider closure properties of the classesA and B that imply completeness of C.Assuming Go¨del’s Axiom of Constructibility (V ¼ L) we prove that C is complete providedC is generated by a set, and either (i) A is closed under pure submodules, or (ii) C is hereditaryand B consists of modules of finite injective dimension. These two results are independent ofZFC þGCH. However, (i) or (ii) implies completeness of C in ZFC provided B is closed underarbitrary direct sums.In ZFC, we also show that C is complete whenever C is hereditary, A closed under arbitrarydirect products, and B consists of modules of finite injective dimension. This yields a charac-terization of n-cotilting cotorsion pairs as the hereditary cotorsion pairs ðC;DÞ such that C isclosed under arbitrary direct products and D consists of modules of injective dimensionan.1991 Mathematics Subject Classification: 16D90, 16E30, 18E40, 03E35, 03E45.
TL;DR: In this article, the boundaries of analyticity and meromorphy for a multivariable Euler product determined by any toric variety (split over Q) were described.
Abstract: This article extends classical one variable results about Euler products, defined by integral valued polynomial or analytic functions, to several variables. We show there exists a meromorphic continuation up to a presumed natural boundary, and give a criterion, ala Estermann-Dahlquist, for the existence of a meromorphic extension to C n . In addition, we precisely describe the boundaries of analyticity and meromorphy for a multivariable Euler product determined by any toric variety (split over Q). Using our method, we are also able to calculate a precise asymptotic for the number of n-fold products of integers that equal the n th
TL;DR: In this article, a class of uniformly elliptic operators with unbounded coefficients in unbounded domains is considered and it is shown that the Cauchy-Neumann problem associated with the operator admits a unique bounded classical solution u for any initial datum f which is bounded and continuous in.
Abstract: Abstract We consider a class of uniformly elliptic operators 𝒜 with unbounded coefficients in unbounded domains . Under suitable assumptions on the geometry of and on the coefficients, we prove that the Cauchy-Neumann problem associated with the operator 𝒜 admits a unique bounded classical solution u for any initial datum f which is bounded and continuous in . Moreover, we prove uniform and pointwise gradient estimates for u. Finally, we give some applications of the so obtained estimates.
TL;DR: In this article, the authors give a characterization of cotilting modules over Prüfer domains, up to equivalence, and show that tilting modules are of projective dimension at most one.
Abstract: Abstract We give a characterization of cotilting modules over Prüfer domains, up to equivalence; moreover we show that tilting modules over Prüfer domains are of projective dimension at most one.
TL;DR: In this article, two unified approaches to the Nagata rings and the Kronecker function rings are studied, which yield these rings and their classical generalizations as special cases of integral domains of rational functions.
Abstract: Let D be an integral domain with quotient field K. The Nagata ring D(X) and the Kronecker function ring Kr(D) are both subrings of the field of rational functions K(X) containing as a subring the ring D(X) of poly- nomials in the variable X. Both of these function rings have been extensively studied and generalized. The principal interest in these two extensions of D lies in the reflection of various algebraic and spectral properties of D and Spec(D) in algebraic and spectral properties of the function rings. Despite the obvious similarities in definitions and properties, these two kinds of domains of rational functions have been classically treated independently, when D is not a Prufer domain. The purpose of this note is to study two different unified approaches to the Nagata rings and the Kronecker function rings, which yield these rings and their classical generalizations as special cases.
TL;DR: In this article, weakly half-factorial sets in finite abelian groups were studied and the maximum cardinality of such a set was determined in terms of factorization lengths in block monoids.
Abstract: We investigate weakly half-factorial sets in finite abelian groups, a concept introduced by J. Śliwa to study half-factorial sets. We fully characterize weakly half-factorial sets in a given group, and determine the maximum cardinality of such a set. This leads to several new results on half-factorial sets; in particular we solve a problem of W. Narkiewicz in some special cases. We also study the arithmetical consequences of weakly-half-factoriality in terms of factorization lengths in block monoids.
TL;DR: In this paper, the authors characterized homology classes z in a regular covering over a finite polyhedron which can be realized in arbitrary small neighborhoods of infinity in X. This problem was motivated by applications in the theory of critical points of closed 1-forms initiated in [2], [3].
Abstract: Let q : X → X be a regular covering over a finite polyhedron with free abelian group of covering translations. Each nonzero cohomology class ξ ∈ H(X;R) with q∗ξ = 0 determines a notion of “infinity” of the noncompact space X. In this paper we characterize homology classes z in X which can be realized in arbitrary small neighborhoods of infinity in X. This problem was motivated by applications in the theory of critical points of closed 1-forms initiated in [2], [3].
TL;DR: In this article, it was shown that if R is a commutative Bezout domain, then the division closure of the image of RF in RhhX@ii is a universal localization of RF at §.
Abstract: Let R be a ring, let F be a free group, and let X be a basis of F. Let †: RF ! R denote the usual augmentation map for the group ring RF, let X@ := fx i 1 j x 2 Xg µ RF, let § denote the set of matrices over RF that are sent to invertible matrices by †, and let (RF)§ i1 denote the universal localization of RF at §. A classic result of Magnus and Fox gives an embedding of RF in the power-series ring RhhX@ii. We show that if R is a commutative Bezout domain, then the division closure of the image of RF in RhhX@ii is a universal localization of RF at §. We also show that if R is a von Neumann regular ring or a commutative Bezout domain, then (RF)§ i1 is stably flat as an RF-ring, in the sense of
TL;DR: In this article, Bendersky and Davis used the BP-based unstable Novikov spectral sequence to study the 2-primary v 1-periodic homotopy groups of SU(n).
Abstract: Abstract In 1991, Bendersky and Davis used the BP-based unstable Novikov spectral sequence to study the 2-primary v 1-periodic homotopy groups of SU(n). Here we use a K-theoretic approach to add more detail to those results. In particular, whereas only the order of the groups was determined in the 1991 paper, here we determine the number of summands in these groups and much information about the orders of those summands. In addition, we give explicit conditions for certain differentials and extensions in a spectral sequence, which affect the homotopy groups. Finally, we give complete results for for n ≤ 13.
TL;DR: In this article, the derivation algebra of multi-loop algebras is studied in the context of extended affine Lie algesia, which is a generalization of a process known as twisting by automorphisms in the theory of Kac-Moody algesias.
Abstract: Abstract This article is about the derivation algebra of multi-loop algebras. Multi-loop algebras are algebras obtained by a generalization of a process known as twisting by automorphisms in the theory of Kac–Moody algebras. Multi-loop algebras are used in the realization of extended affine Lie algebras. Under certain conditions on an algebra 𝒜, we determine the derivation algebra of an n-step multi-loop algebra based on 𝒜 as the semidirect product of a multi-loop algebra based on the derivation algebra of 𝒜 and the derivation algebra of the Laurent polynomials in n-variables. This in particular determines the derivation algebras of the core modulo center of (almost all) extended affine Lie algebras.
TL;DR: In this article, it was shown that the spinor L-function of the automorphic cuspidal representations of the similitude symplectic group of order four over the rational numbers is Eulerian.
Abstract: In this paper we prove two seemingly unrelated theorems. First we establish the entireness of the spinor L-functions of certain automorphic cuspidal representations of the similitude symplectic group of order four over the rational numbers. We also prove a theorem related to the existence of Bessel models for generic discrete series representations of the same group over the real numbers. The two results are linked by the method of proof; in both cases it is based on the pull-back of an appropriately chosen global Bessel functional via the theta correspondence for the dual pair (GO(2, 2),GSp(4)). The first main theorem is related to analytic properties of spinor L-functions. We prove the entireness of the spinor L-function for those generic automorphic cuspidal representation which satisfy a condition at the archimedean place (see below). Our study of the spinor L-function is based on an integral representation which works for generic representations. These integrals which were introduced by M. Novodvorsky in the Corvallis conference [26] serve as one of the few available integral representations for the Spinor L-function of GSp(4). Some of the details missing in Novodvorsky’s original paper have been reproduced in Daniel Bump’s survey article [4]. Further details have been supplied by [40]. Novodvorsky’s integral was first generalized by Ginzburg [10], and further generalized by Soudry [39], to orthogonal groups of arbitrary odd degree. In light of the results of [40], it is sufficient to study the integral of Novodvorsky at the archimedean place. Archimedean computations are often forbidding, and unless one expects major simplifications due to the nature of the parameters, the resulting integrals are often quite hard to manage. In our case of interest, the work of Moriyama [25] benefits from exactly such simplifications when he treats the case of cuspidal representations with archimedean components in the generic (limit of) discrete series. In this work, we concentrate on those archimedean representations for which direct computations have yielded very little. For this reason, our methods are a bit indirect, in fact somewhat more indirect than what at first seems necessary. Our method is based on the theta correspondence. First we observe in Lemma 2.2 that Novodvorsky’s integral is in fact a split Bessel functional. Then in 2.1 we pull the Bessel functional back via the theta correspondence for the dual reductive pair (GO(2, 2),GSp(4)), and prove that the resulting functional on GO(2, 2) is Eulerian. On the other hand,
TL;DR: For a fixed hyperelliptic curve C given by the equation y 1/4 f ðxÞ with f A Z 1/2x having distinct roots and degree at least 5, the Dirichlet series Df ðsÞ 1 Ω4 P 0 m00aCmðQÞjmj s where the summation is over all non-zero squarefree integers was studied in this paper.
Abstract: For a fixed hyperelliptic curve C given by the equation y 1⁄4 f ðxÞ with f A Z1⁄2x having distinct roots and degree at least 5, we study the variation of rational points on the quadratic twists Cm whose equation is given by my 2 1⁄4 f ðxÞ. More precisely, we study the Dirichlet series Df ðsÞ 1⁄4 P 0 m00aCmðQÞjmj s where the summation is over all non-zero squarefree integers. We show that Df ðsÞ converges for 1. We extend its range of convergence assuming the ABC conjecture. This leads us to study related Dirichlet series attached to binary forms. We are then led to investigate the variation of rational points on twists of superelliptic curves. We apply this study to certain classical problems of analytic number theory such as the number of powerfree values of a fixed polynomial in Z1⁄2x . 2000 Mathematics Subject Classification: 11G30; 11M41.
TL;DR: In this article, the authors show that the Cauchy problem of the 3D nonlinear Schrodinger equation with repulsive potential is globally wellposed if the initial data u0 is spherically sym- metric and u0 A S ¼f f ; f A H 1 ; xf A L 2 g.
Abstract: In this paper, we show that the Cauchy problem of the 3D nonlinear Schrodinger equation with repulsive potential is globally wellposed if the initial data u0 is spherically sym- metric and u0 A S ¼f f ; f A H 1 ; xf A L 2 g. We also prove that the scattering operator is holo- morphic from the radial functions in S to themselves. In order to preclude the possible energy concentration, we first show the energy concentration may occur only at finite time by using the decay estimate of potential energy kuðtÞk 6 , then we preclude the possible finite time energy concentration by inductive arguments.
TL;DR: In this paper, the authors characterize H-spaces which are p-torsion Postnikov pieces of finite type by a cohomological property together with a necessary acyclicity condition.
Abstract: We characterize H-spaces which are p-torsion Postnikov pieces of finite type by a cohomological property together with a necessary acyclicity condition. When the mod p co- homology of an H-space is finitely generated as an algebra over the Steenrod algebra we prove that its homotopy groups behave like those of a finite complex. In particular, a p-complete infinite loop space has a finite number of non-trivial homotopy groups if and only if its mod p cohomology satisfies this finiteness condition.
TL;DR: In this paper, the authors studied T-groups with min-p for all primes p in which every descendant subgroup is normal and showed that these groups are precisely T groups, that is, groups whose subnormal subgroups are normal.
Abstract: Abstract Radical locally finite groups with min-p for all primes p in which every descendant subgroup is normal are studied in the paper. It turns out that these groups are precisely T-groups, that is, groups whose subnormal subgroups are normal.
TL;DR: In this paper, it was shown that if an abelian partial monoid M is embedded in a topological group, then the natural map C M (R 1, X) induced by the inclusion M ± M ={a b| a, b 2 M}.
Abstract: Given a pair of an abelian partial monoid M and a pointed space X, let C M (R 1 ,X) denote the configuration space of finite distinct points in R 1 parametrized by the partial monoid X^ M. In this note we will show that if M is embedded in a topological abelian group then the natural map C M (R 1 ,X)! C ±M (R 1 ,X), induced by the inclusion M ± M ={a b| a, b 2 M}, is a group completion. This generalizes the result of Caruso (1) that the space of "positive and negative particles" in R 1 parametrized by X is weakly equivalent to 1 1 X.
TL;DR: In this article, the authors studied the quadratic extensions of an algebraic number field K which can be embedded in a cyclic extension of K of degree 2 n for all natural numbers n, and also in an infinite normal extension with the additive group of 2-adic integers as Galois group.
Abstract: Abstract For an algebraic number field K we study the quadratic extensions of K which can be embedded in a cyclic extension of K of degree 2 n for all natural numbers n, as well as the quadratic extensions which can be embedded in an infinite normal extension with the additive group of 2-adic integers as Galois group. For shortness we call a normal extension of K whose Galois group is the cyclic group ℤ/2 n ℤ of order 2 n with n ∈ ℕ, resp. , a (ℤ/2 n ℤ)-extension resp. a -extension of K. A quadratic extension L|K is called (ℤ/2 n ℤ)-embeddable, resp. -embeddable, if there exists a (ℤ/2 n ℤ)-extension, resp. a -extension, of K containing L. One main result of this paper is the following observation, the exact formulation of which is given in theorems 6 to 8 in §3: Theorem 0. Let K be an imaginary quadratic number field whose discriminant has m prime divisors. Then the number of quadratic extensions L|K which are (ℤ/2 n ℤ)-embeddable for all n is 2 m−1 − 1, 2 m − 1 or 2 m+1 − 1, depending on certain congruences for the discriminant and its prime divisors. But the number of quadratic extensions L|K which are -embeddable is only 3.