On the Star Class Group of a Pullback

For the domain $R$ arising from the construction $T, M,D$, we relate the star class groups of $R$ to those of $T$ and $D$. More precisely, let $T$ be an integral domain, $M$ a nonzero maximal ideal of $T$, $D$ a proper subring of $k:=T/M$, $\phi: T\to k$ the natural projection, and let $R={\phi}^{-1}(D)$. For each star operation $\ast$ on $R$, we define the star operation $\ast_\phi$ on $D$, i.e., the ``projection'' of $\ast$ under $\phi$, and the star operation ${(\ast)}_{_{T}}$ on $T$, i.e., the ``extension'' of $\ast$ to $T$. Then we show that, under a mild hypothesis on the group of units of $T$, if $\ast$ is a star operation of finite type, 0\to \Cl^{\ast_{\phi}}(D) \to \Cl^\ast(R) \to \Cl^{{(\ast)}_{_{T}}}(T)\to 0 is split exact. In particular, when $\ast = t_{R}$, we deduce that the sequence 0\to \Cl^{t_{D}}(D) {\to} \Cl^{t_{R}}(R) {\to}\Cl^{(t_{R})_{_{T}}}(T) \to 0 is split exact. The relation between ${(t_{R})_{_{T}}}$ and $t_{T}$ (and between \Cl^{(t_{R})_{_{T}}}(T) and \Cl^{t_{T}}(T)) is also investigated.Comment: J. Algebra (to appear

Towards Interactive Logic Programming

Linear logic programming uses provability as the basis for computation. In the operational semantics based on provability, executing the additive-conjunctive goal $G_1 \& G_2$ from a program $P$ simply terminates with a success if both $G_1$ and $G_2$ are solvable from $P$. This is an unsatisfactory situation, as a central action of \& -- the action of choosing either $G_1$ or $G_2$ by the user -- is missing in this semantics. We propose to modify the operational semantics above to allow for more active participation from the user. We illustrate our idea via muProlog, an extension of Prolog with additive goals.Comment: 8 pages. It describes two execution models for interactive logic programmin

Different Behavior of Magnetic Impurities in Crystalline and Ammorphous States of Superconductors

It has been observed that the effect of magnetic impurities in a superconductor is drastically different depending on whether the host superconductor is in a crystalline or an amorphous state. Based on the recent theory of Kim and Overhauser (KO), it is shown that as the system is getting disordered, the initial slope of the $T_{c}$ depression is decreasing by a factor $\sqrt{\ell/\xi_{0}}$, when the mean free path $\ell$ becomes smaller than the BCS coherence length $\xi_{0}$, which is in agreement with experimental findings. In addition, for a superconductor in a crystalline state in the presence of magnetic impurities the superconducting transition temperature $T_{c}$ drops sharply from about 50% of $T_{c0}$ (for a pure system) to zero near the critical impurity concentration. This {\sl pure limit behavior} was indeed found by Roden and Zimmermeyer in crystalline Cd. Recently, Porto and Parpia have also found the same {\sl pure limit behavior} in superfluid He-3 in aerogel, which may be understood within the framework of the KO theory.Comment: 7 figures, 20 pages, latex, to appear in Superconductor Science and Technolog