Generalized hypergeometric series for Racah matrices in rectangular representations

One of the spectacular results in mathematical physics is the expression of Racah matrices for symmetric representations of the quantum group [Formula: see text] through the Askey–Wilson polynomials, associated with the [Formula: see text]-hypergeometric functions [Formula: see text]. Recently it was shown that this is in fact the general property of symmetric representations, valid for arbitrary [Formula: see text] — at least for exclusive Racah matrices [Formula: see text]. The natural question then is what substitutes the conventional [Formula: see text]-hypergeometric polynomials when representations are more general? New advances in the theory of matrices [Formula: see text], provided by the study of differential expansions of knot polynomials, suggest that these are multiple sums over Young sub-diagrams of the one which describes the original representation of [Formula: see text]. A less trivial fact is that the entries of the sum are not just the factorized combinations of quantum dimensions, as in the ordinary hypergeometric series, but involve non-factorized quantities, like the skew characters and their further generalizations — as well as associated additional summations with the Littlewood–Richardson weights.

Speech evolved from vocalization, not mastication

The segmentation of phonation by articulation is a characteristic feature of speech that distinguishes it from most nonhuman vocalizations. However, apart from the trivial fact that speech uses some of the same muscles and, hence the same motoneurons and motorcortical areas used in chewing, there is no convincing evidence that syllable segmentation relies on the same pattern generator as mastication. Evidence for a differential cortical representation of syllable segmentation (“frame”) and syllable “content” is also meager.

Double Generativity and Complete Effective Inseparability

§1. Complete Effective Inseparability. A disjoint pair (A1,A2) is by definition recursively inseparable if no recursive superset of A1 is disjoint from A2. This is equivalent to saying that for any disjoint r.e. supersets ωi and ωj of A1 and A2, the set ωi is not the complement of ωj —in other words, there is a number n outside both ωi and ωj. The disjoint pair (A1,A2) is called effectively inseparable—abbreviated E.I.—if there is a recursive function δ(x, y)—called an E.I. function for (A1, A2)—such that for any numbers i and j such that A1⊆ ωi and A2Í ωj. with ωi being disjoint from ωj, the number d(i , j) is outside both a;,- and ωj. We shall call a disjoint pair (A1, A2) completely E.I. if there is a recursive function δ(x, y)—which we call a complete E.I. function for (A1, A2)—such that for any numbers i and j, if A1⊆ ωi and A2Í ωj, then δ(i , j) Í ωi ↔ d(i , j)Í ωj (in other words, d(i, j) is either inside or outside both sets ωi and ωj.). [If ωi and ωj happen to be disjoint, then, of course, d(i, j) is outside both ωi and ωj, so any complete E.I. function for (A1,A2) is also an E.I. function for (A1,A2) In a later chapter, we will prove the non-trivial fact that if (A1, A2) is E.I. and A1 and A2 are both r.e., then (A1,A2) is completely E.I. [The proof of this uses the result known as the double recursion theorem, which we will study in Chapter 9.] Effective inseparability has been well studied in the literature. Complete effective inseparability will play a more prominent role in this volume—especially in the next few chapters. Proposition 1. (1) If (A1,A2) is completely E.I., then so is (A2,A1) —in fact, if d(x,y) is a complete E.I. function for (A1,A2), then d(y,x) is a complete E.I. function for (A2, A1).

Mandatory Retirement: Intergenerational Justice and the Canadian Charter of Rights and Freedoms

This study explores two conceptions of justice and their radically different implications for mandatory retirement. The author argues that the case against mandatory retirement rests on a conception of justice which ignores the fact that a society is composed of different generations. Yet the neglect of this seemingly trivial fact leads to serious problems of intergenerational justice; and the note considers both how these problems can be accommodated within a theory of liberal justice, and the implications of that theory for mandatory retirement. The note then considers which of these two conceptions of justice is embodied in the Canadian Charter of Rights and Freedoms. It argues that to ignore considerations of intergenerational justice in mandatory retirement cases amounts to a denial of the equal protection and the equal benefit of the law guaranteed by the Charter.

Multiplicative Groups Under Field Extension

Let K be a field and L an extension field. L. Fuchs [2, Problem 98] has suggested studying the change in multiplicative groups in going from K* to L*. We wish to indicate difficulties that arise in trying to relate the group theoretic structure of L* to that of K*, even when K* has particularly simple structure and the extension is quadratic.First let us note a trivial fact. If [L : K] = n < ∞ and K* has a free direct factor A, then L* has a free direct factor isomorphic to A. To see this, let ϕ be the composite L* → K* → A of the norm map followed by the projection map. Then L* has a free direct factor isomorphic to ϕ(L*). But the image of the norm map contains (K*)n, hence ϕ(L*) ≅ A.

On Maximal Sets of Mutually Orthogonal Idempotent Latin Squares

It is a well-known trivial fact that for a given integer n there exists at most n — 2 pairwise orthogonal idempotent latin squares. In the following note we prove that for n a prime power there always exists n—2 such squares.

Splitting and Decomposition by Regressive Sets, II

In (3), Dekker drew attention to an analogy between (a) the relationship of the recursive sets to the recursively enumerable sets, and (b) the relationship of the retraceable sets to the regressive sets. As was to be expected, this analogy limps in some respects. For example, if a number set α is split by a recursive set, then it is decomposed by a pair of recursively enumerable sets; whereas, as we showed in (6, Theorem 2), α may be split by a retraceable set and yet not decomposable (in a liberal sense of the latter term) by a pair of regressive sets. The result for recursive and recursively enumerable sets, of course, follows from the trivial fact that the complement of a recursive set is recursive.