For an ideal I of O(D), we will denote its image in the ideal class monoid by [I]. In particular, composing the map f 7→ I(f) with the map which sends an ideal to its class, we get a map which sends a primitive, positive definite quadratic form f of discriminant D to the ideal class [I(f)]. This has remarkable properties. Indeed, we have the following important and beautiful result (unlike some...

We discuss a simple definition of conditional mutual information (CMI) for fields and σ-fields. The new definition is applicable also in nonregular cases, unlike the well-known but more restricted definition of CMI by Dobrushin. Certain properties of the two notions of CMI and their equivalence for countably generated σ-fields are established. We also consider an application, which concerns the...

In his polemic composition al-Radd ‘alā Ibn al-Naghrīla al-Yahūdī, Abū Muḥammad ‘Alī b. Aḥmad Ḥazm al-Andalusī (994/1064 CE) attempts to demonstrate the Jews’ taḥrīf (falsification) of Torah by citing several examples. This article intends shed some light on an important aspect Ḥazm’s rhetoric as reflected in al-Naghrīla. I argue, however, that juxtaposing arguments with relevant Jewish sources...