Volume 115, Issue 1
February 2005, pages 1-116
pp 1-14 February 2005
Letp andl be rational primes such thatl is odd and the order ofp modulol is even. For such primesp andl, and fore = l, 2l, we consider the non-singular projective curvesaY^{21} =bX^{21} +cZ^{21} defined over finite fields F_{q} such thatq = p^{α}? l(mode).We see that the Fermat curves correspond precisely to those curves among each class (fore = l, 2l), that are maximal or minimal over F_{q}. We observe that each Fermat prime gives rise to explicit maximal and minimal curves over finite fields of characteristic 2. Fore = 2l, we explicitly determine the ζ -function(s) for this class of curves, over F_{q}, as rational functions in the variablet, for distinct cases ofa, b, andc, in F_{q}^{*}. Theζ-function in each case is seen to satisfy the Weil conjectures (now theorems) for this concrete class of curves.
Fore = l, 2l, we determine the class numbers for the function fields associated to each class of curves over F_{q}. As a consequence, when the field of definition of the curve(s) is fixed, this provides concrete information on the growth of class numbers for constant field extensions of the function field(s) of the curve(s).
pp 15-49 February 2005
Moduli for decorated tuples of sheaves and representation spaces for quivers
We extend the scope of a former paper to vector bundle problems involving more than one vector bundle. As the main application, we obtain the solution of the well-known moduli problems of vector bundles associated with general quivers.
pp 51-56 February 2005
Localization of tight closure in two-dimensional rings
Kamran Divaani-Aazar Massoud Tousi
It is shown that tight closure commutes with localization in any two-dimensional ringR of prime characteristic if eitherR is a Nagata ring orR possesses a weak test element. Moreover, it is proved that tight closure commutes with localization at height one prime ideals in any ring of prime characteristic.
pp 57-65 February 2005
Fields and forms on ρ-algebras
In this paper we introduce non-commutative fields and forms on a new kind of non-commutative algebras: ρ-algebras. We also define the Frölicher-Nijenhuis bracket in the non-commutative geometry on ρ-algebras.
pp 67-77 February 2005
W D Gao A Panigrahi R Thangadurai
Letp be any odd prime number. Letk be any positive integer such that $$2 \leqslant k \leqslant \left[ {\frac{{p + 1}}{3}} \right] + 1$$. LetS = (a_{1},a_{2},...,a_{2p−k}) be any sequence in ℤ_{p} such that there is no subsequence of lengthp of S whose sum is zero in ℤ_{p}. Then we prove that we can arrange the sequence S as follows: $$S = (\underbrace {a,a,...,a,}_{u times}\underbrace {b,b,...,b,}_{v times}a'_1 ,a'_2 ,...,a'_{2p - k - u - v} )$$whereu ≥v,u +v ≥ 2p - 2k + 2 anda -b generates ℤ_{p}. This extends a result in [13] to all primesp andk satisfying (p + 1)/4 + 3 ≤k ≤ (p + 1)/3 + 1. Also, we prove that ifg denotes the number of distinct residue classes modulop appearing in the sequenceS in ℤ_{p} of length 2p -k (2≤k ≤ [(p + 1)/4]+1), and $$g \geqslant 2\sqrt 2 \sqrt {k - 2} $$, then there exists a subsequence of S of lengthp whose sum is zero in ℤ_{p}.
pp 79-91 February 2005
Inequalities for dual quermassintegrals of mixed intersection bodies
Zhao Chang-Jian Leng Gang-Song
In this paper, we first introduce a new concept ofdual quermassintegral sum function of two star bodies and establish Minkowski’s type inequality for dual quermassintegral sum of mixed intersection bodies, which is a general form of the Minkowski inequality for mixed intersection bodies. Then, we give the Aleksandrov-Fenchel inequality and the Brunn-Minkowski inequality for mixed intersection bodies and some related results. Our results present, for intersection bodies, all dual inequalities for Lutwak’s mixed prosection bodies inequalities.
pp 93-102 February 2005
On two functionals connected to the Laplacian in a class of doubly connected domains in space-forms
LetB_{1} be a ball of radiusr_{1} inS^{n} (ℍ^{n}), and letB_{0} be a smaller ball of radiusr_{0} such thatB_{0} ⊂B_{1}. ForS^{n} we considerr_{1}π. Let u be a solution of the problem- δm = 1 in Ω :=B_{1} /B_{0} vanishing on the boundary. It is shown that the associated functionalJ (Ω) is minimal if and only if the balls are concentric. It is also shown that the first Dirichlet eigenvalue of the Laplacian on Ω is maximal if and only if the balls are concentric.
pp 103-109 February 2005
Isometric multipliers ofL^{p}(G, X)
Let G be a locally compact group with a fixed right Haar measure andX a separable Banach space. LetL^{p}(G, X) be the space of X-valued measurable functions whose norm-functions are in the usualL^{p}. A left multiplier ofL^{p}(G, X) is a bounded linear operator onB^{p}(G, X) which commutes with all left translations. We use the characterization of isometries ofL^{p}(G, X) onto itself to characterize the isometric, invertible, left multipliers ofL^{p}(G, X) for 1 ≤p ∞,p ≠ 2, under the assumption thatX is not thel^{p}-direct sum of two non-zero subspaces. In fact we prove that ifT is an isometric left multiplier ofL^{p}(G, X) onto itself then there existsa y ∃ G and an isometryU ofX onto itself such thatTf(x) = U(R_{y}f)(x). As an application, we determine the isometric left multipliers of L^{1} ∩L^{p}(G, X) and L^{1} ∩C_{0}(G, X) whereG is non-compact andX is not the l^{p}-direct sum of two non-zero subspaces. If G is a locally compact abelian group andH is a separable Hubert space, we define $$A^p (G,H) = \{ f \in L^1 (G,H):\hat f \in L^p (\Gamma ,H)\} $$ where г is the dual group of G. We characterize the isometric, invertible, left multipliers ofA^{p}(G, H), provided G is non-compact. Finally, we use the characterization of isometries ofC(G, X) for G compact to determine the isometric left multipliers ofC(G, X) providedX^{*} is strictly convex.
pp 111-116 February 2005
Rajeeva L Karandikar M G Nadkarni
We give a necessary and sufficient condition on a sequence of functions on a set Ω under which there is a measure on Ω which renders the given sequence of functions a martingale. Further such a measure is unique if we impose a natural maximum entropy condition on the conditional probabilities.
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