• Volume 115, Issue 1

February 2005,   pages  1-116

• Zeta Function of the Projective Curve $aY^{2l} = bX^{2l} + cZ^{2l}$ over a Class of Finite Fields, for Odd Primes 𝑙

Let 𝑝 and 𝑙 be rational primes such that 𝑙 is odd and the order of 𝑝 modulo 𝑙 is even. For such primes 𝑝 and 𝑙, and for 𝑒 = 𝑙, 2𝑙, we consider the non-singular projective curves $aY^e = bX^e + cZ^e (abc≠ 0)$ defined over finite fields $F_q$ such that $q = p^𝛼 ≡ 1 (\mathrm{mod} e)$. We see that the Fermat curves correspond precisely to those curves among each class (for 𝑒 = 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. For 𝑒 = 2𝑙, we explicitly determine the 𝜁-function(s) for this class of curves, over $F_q$, as rational functions in the variable 𝑡, for distinct cases of 𝑎, 𝑏, and 𝑐, in $F^*_q$. The 𝜁-function in each case is seen to satisfy the Weil conjectures (now theorems) for this concrete class of curves.

For 𝑒 = 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).

• 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 ring 𝑅 of prime characteristic if either 𝑅 is a Nagata ring or 𝑅 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 of 𝑝-Zero-Sum Free Sequences and its Application to a Variant of Erdös–Ginzburg–Ziv Theorem

Let 𝑝 be any odd prime number. Let 𝑘 be any positive integer such that $2 ≤ k ≤ \left[\frac{p+1}{3}\right]+1$. Let $S =(a_1,a_2,\ldots,a_{2p−k})$ be any sequence in $\mathbb{Z}_p$ such that there is no subsequence of length 𝑝 of 𝑆 whose sum is zero in $\mathbb{Z}_p$. Then we prove that we can arrange the sequence 𝑆 as follows:

$$S=(\underset{u\text{times}}{\underbrace{a,a,\ldots,a}},\underset{v\text{times}}{\underbrace{b,b,\ldots,b}},{a'}_1,{a'}_2,\ldots,{a'}_{2p-k-u-v})$$

where $u≥ v, u + v ≥ 2p − 2k + 2$ and 𝑎 − 𝑏 generates $\mathbb{Z}_p$. This extends a result in [13] to all primes 𝑝 and 𝑘 satisfying $(p + 1)/4 + 3≤ k≤ (p + 1)/3 + 1$. Also, we prove that if 𝑔 denotes the number of distinct residue classes modulo 𝑝 appearing in the sequence 𝑆 in $\mathbb{Z}_p$ of length $2p − k(2≤ k≤ [(p + 1)/4]+1)$, and $g≥ 2\sqrt{2}\sqrt{k - 2}$, then there exists a subsequence of 𝑆 of length 𝑝 whose sum is zero in $\mathbb{Z}_p$.

• Inequalities for Dual Quermassintegrals of Mixed Intersection Bodies

In this paper, we first introduce a new concept of dual 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

Let $B_1$ be a ball of radius $r_1$ in $S^n(\mathbb{H}^n)$, and let $B_0$ be a smaller ball of radius $r_0$ such that $\overline{B_0} \subset B_1$. For $S^n$ we consider $r_1 < 𝜋$. Let 𝑢 be a solution of the problem $-𝛥 u = 1$ in $𝛺: = B_1\ \overline{B_0}$ vanishing on the boundary. It is shown that the associated functional 𝐽(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 of $L^p(G, X)$

Let 𝐺 be a locally compact group with a fixed right Haar measure and 𝑋 a separable Banach space. Let $L^p(G, X)$ be the space of 𝑋-valued measurable functions whose norm-functions are in the usual $L^p$. A left multiplier of $L^p(G, X)$ is a bounded linear operator on $L^p(G, X)$ which commutes with all left translations. We use the characterization of isometries of $L^p(G, X)$ onto itself to characterize the isometric, invertible, left multipliers of $L^p(G, X)$ for 1 ≤ 𝑝 < ∞, 𝑝 ≠ 2, under the assumption that 𝑋 is not the $l^p$-direct sum of two non-zero subspaces. In fact we prove that if 𝑇 is an isometric left multiplier of $L^p(G, X)$ onto itself then there exists $a y \in G$ and an isometry 𝑈 of 𝑋 onto itself such that $Tf(x) = U(R_y f)(x)$. As an application, we determine the isometric left multipliers of $L^1 \cap L^p(G, X)$ and $L^1 \cap C_0(G, X)$ where 𝐺 is non-compact and 𝑋 is not the $l^p$-direct sum of two non-zero subspaces. If 𝐺 is a locally compact abelian group and 𝐻 is a separable Hilbert space, we define $A^p(G, H)=\{f\in l^1(G, H):\hat{f}\in L^p(𝛤, H)\}$ where 𝛤 is the dual group of 𝐺. We characterize the isometric, invertible, left multipliers of $A^p(G, H)$, provided 𝐺 is non-compact. Finally, we use the characterization of isometries of 𝐶(𝐺,𝑋) for 𝐺 compact to determine the isometric left multipliers of 𝐶(𝐺,𝑋) provided 𝑋* 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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