• Volume 114, Issue 1

February 2004,   pages  1-97

• On some Vongruence with Application to Exponential Sums

We will study the solution of a congruence, $x≡ g^{(1/2)𝜔_g(2^n)}\mathrm{mod} 2^n$, depending on the integers 𝑔 and 𝑛, where $𝜔_g(2^n)$ denotes the order of 𝑔 modulo $2^n$. Moreover, we introduce an application of the above result to the study of an estimation of exponential sums.

• Representability of Hom Implies Flatness

Let 𝑋 be a projective scheme over a noetherian base scheme 𝑆, and let $\mathcal{F}$ be a coherent sheaf on 𝑋. For any coherent sheaf $\mathcal{E}$ on 𝑋, consider the set-valued contravariant functor $\hom_{(\mathcal{E},\mathcal{F})}$ on 𝑆-schemes, defined by $\hom_{(\mathcal{E},\mathcal{F})}(T)=\mathrm{Hom}(\mathcal{E}_T,\mathcal{F}_T)$ where $\mathcal{E}_T$ and $\mathcal{F}_T$ are the pull-backs of $\mathcal{E}$ and $\mathcal{F}$ to $X_T=X×_s T$. A basic result of Grothendieck ([EGA], III 7.7.9) says that if $\mathcal{F}$ is flat over 𝑆 then $\hom_{(\mathcal{E},\mathcal{F})}$ is representable for all $\mathcal{E}$.

We prove the converse of the above, in fact, we show that if 𝐿 is a relatively ample line bundle on 𝑋 over 𝑆 such that the functor $\hom_{(L^{-n},\mathcal{F})}$ is representable for infinitely many positive integers 𝑛, then $\mathcal{F}$ is flat over 𝑆. As a corollary, taking $X=S$, it follows that if $\mathcal{F}$ is a coherent sheaf on 𝑆 then the functor $T\mapsto H^0(T,\mathcal{F}_T)$ on the category of 𝑆-schemes is representable if and only if $\mathcal{F}$ is locally free on 𝑆. This answers a question posed by Angelo Vistoli.

The techniques we use involve the proof of flattening stratification, together with the methods used in proving the author's earlier result (see [N1]) that the automorphism group functor of a coherent sheaf on 𝑆 is representable if and only if the sheaf is locally free.

• On the Fundamental Group of Real Toric Varieties

Let 𝑋(𝛥) be the real toric variety associated to a smooth fan 𝛥. The main purpose of this article is:

• to determine the fundamental group and the universal cover of 𝑋(𝛥),
• to give necessary and sufficient conditions on 𝛥 under which 𝜋1 (𝑋(𝛥)) is abelian,
• to give necessary and sufficient conditions on 𝛥 under which 𝑋(𝛥) is aspherical, and when 𝛥 is complete,
• to give necessary and sufficient conditions for $\mathcal{C}_𝛥$ to be a 𝐾(𝜋, 1) space where $\mathcal{C}_𝛥$ is the complement of a real subspace arrangement associated to 𝛥.
• Cobordism Independence of Grassmann Manifolds

This note proves that, for $F=\mathbb{R},\mathbb{C}$ or $\mathbb{H}$, the bordism classes of all non-bounding Grassmannian manifolds $G_k(F^{n+k})$, with 𝑘 < 𝑛 and having real dimension 𝑑, constitute a linearly independent set in the unoriented bordism group $\mathfrak{N}_d$ regarded as a $\mathbb{Z}_2$-vector space.

• Height in Splittings of Hyperbolic Groups

Suppose 𝐻 is a hyperbolic subgroup of a hyperbolic group 𝐺. Assume there exists 𝑛 > 0 such that the intersection of 𝑛 essentially distinct conjugates of 𝐻 is always finite. Further assume 𝐺 splits over 𝐻 with hyperbolic vertex and edge groups and the two inclusions of 𝐻 are quasi-isometric embeddings. Then 𝐻 is quasiconvex in 𝐺. This answers a question of Swarup and provides a partial converse to the main theorem of [23].

• Rank-One Operators in Reflexive One-Sided $\mathcal{A}$-Submodules

In this paper, we first characterize reflexive one-sided $\mathcal{A}$-submodules $\mathcal{U}$ of a unital operator algebra $\mathcal{A}$ in $\mathcal{B}(\mathcal{H})$ completely. Furthermore we investigate the invariant subspace lattice Lat $\mathcal{R}$ and the reflexive hull Ref $\mathcal{R}$, where $\mathcal{R}$ is the submodule generated by rank-one operators in $\mathcal{U}$; in particular, if $\mathcal{L}$ is a subspace lattice, we obtain when the rank-one algebra $\mathcal{R}$ of $\mathrm{Alg} \mathcal{L}$ is big enough to determined $\mathrm{Alg}\mathcal{L}$ in the following senses: $\mathrm{Alg} \mathcal{L} = \mathrm{Alg} \mathrm{Lat} \mathcal{R}$ and $\mathrm{Alg} \mathcal{L} = \mathrm{Ref} \mathcal{R}$.

• On the Limit-Classifications of Even and Odd-Order Formally Symmetric Differential Expressions

In this paper we consider the formally symmetric differential expression $M[\cdot p]$ of any order (odd or even) ≥ 2. We characterise the dimension of the quotient space $D(T_{\max})/D(T_{\min})$ associated with $M[\cdot p]$ in terms of the behaviour of the determinants

$$\det\limits_{r,s\in N_n}[[f_r g_s](∞)]$$

where 1 ≤ 𝑛 ≤ (order of the expression +1); here $[fg](∞) = \lim\limits_{x→∞}[fg](x)$, where $[fg](x)$ is the sesquilinear form in 𝑓 and 𝑔 associated with 𝑀. These results generalise the well-known theorem that 𝑀 is in the limit-point case at ∞ if and only if $[fg](∞)=0$ for every $f, g \in$ the maximal domain 𝛥 associated with 𝑀.

• Inverse Solutions for a Second-Grade Fluid for Porous Medium Channel and Hall Current Effects

Assuming certain forms of the stream function inverse solutions of an incompressible viscoelastic fluid for a porous medium channel in the presence of Hall currents are obtained. Expressions for streamlines, velocity components and pressure fields are described in each case and are compared with the known viscous and second-grade cases.