• Volume 115, Issue 1

February 2005,   pages  1-116

• Zeta function of the projective curveaY21 =bX21 +cZ21 over a class of finite fields, for odd primesl

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 curvesaY21 =bX21 +cZ21 defined over finite fields Fq such thatq = pα? 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 Fq. 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 Fq, as rational functions in the variablet, for distinct cases ofa, b, andc, in Fq*. 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 Fq. 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).

• 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.

• Localization of tight closure in two-dimensional rings

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.

• 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.

• On the structure ofp-zero-sum free sequences and its application to a variant of Erdös-Ginzburg-Ziv theorem

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 = (a1,a2,...,a2p−k) be any sequence in ℤp such that there is no subsequence of lengthp of S whose sum is zero in ℤp. Then we prove that we can arrange the sequence S as follows: $$S = (\underbrace {a,a,...,a,}_{u times}\underbrace {b,b,...,b,}_{v times}a'_1 ,a'_2 ,...,a'_{2p - k - u - v} )$$whereuv,u +v ≥ 2p - 2k + 2 anda -b generates ℤp. 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 ℤp 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 ℤp.

• Inequalities for dual quermassintegrals of mixed intersection bodies

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.

• On two functionals connected to the Laplacian in a class of doubly connected domains in space-forms

LetB1 be a ball of radiusr1 inSn (ℍn), and letB0 be a smaller ball of radiusr0 such thatB0B1. ForSn we considerr1π. Let u be a solution of the problem- δm = 1 in Ω :=B1 /B0 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.

• Isometric multipliers ofLp(G, X)

Let G be a locally compact group with a fixed right Haar measure andX a separable Banach space. LetLp(G, X) be the space of X-valued measurable functions whose norm-functions are in the usualLp. A left multiplier ofLp(G, X) is a bounded linear operator onBp(G, X) which commutes with all left translations. We use the characterization of isometries ofLp(G, X) onto itself to characterize the isometric, invertible, left multipliers ofLp(G, X) for 1 ≤p ∞,p ≠ 2, under the assumption thatX is not thelp-direct sum of two non-zero subspaces. In fact we prove that ifT is an isometric left multiplier ofLp(G, X) onto itself then there existsa y ∃ G and an isometryU ofX onto itself such thatTf(x) = U(Ryf)(x). As an application, we determine the isometric left multipliers of L1Lp(G, X) and L1C0(G, X) whereG is non-compact andX is not the lp-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 ofAp(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.

• Measure free martingales

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.

• # Proceedings – Mathematical Sciences

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