Robert J. Valenza

Elasticity of factorization in number fields

Let R be a noetherian domain which is not a field. Define ϱ(R), the elasticity of R, to be sup(X), where X is the subset of Q defined by X={m/n: there exist irreducibles π1, π1, …, πm, t1, t2, …, tn ∈ R with π1π2⋯ πm = t1π2⋯tn} Thus ϱ(R) in part measures the failure of R to be a unique factorization domain. Let A denote the integers of K, a number field with class group C and class number h. By elementary number theory and a theorem of L. Carlitz (Proc. Amer. Math. Soc. 11 (1960), 391–392) one knows that ϱ(A) = 1 if, and only if, h ≤ 2. We show here that 1. (i) ϱ(A) is bounded from above by h2, provided that C is nontrivial. 2. (ii) ϱ(A) is bounded from below by an expression which depends only on the elementary divisor decomposition of C and which is greater than or equal to half the exponent of C; in particular, this agrees with the upper bound given in (i) in the special case that C is cyclic. 3. (iii) ϱ(A) → ∞ as h → ∞. The actual bound in (i) is somewhat sharper and involves a newly defined group invariant called sequential depth, which is fundamental to our discussion.

And Then There Were None: Winnowing the Shakespeare Claimants.

The Shakespeare Clinic has developed 51 computer tests of Shakespeare play authorship and 14 of poem authorship, and applied them to 37 claimed “true Shakespeares,” to 27 plays of the Shakespeare Apocrypha, and to several poems of unknown or disputed authorship. No claimant, and none of the apocryphal plays or poems, matched Shakespeare. Two plays and one poem from the Shakespeare Canon,Titus Andronicus, Henry VI, Part 3, and “A Lovers Complaint,” do not match the others.

Are the Thisted-Efron authorship tests valid?

We assess the validity of the Thisted-Efron author-ship tests in two stages. First, we construct simulated texts in accordance with the assumptions implicit in the underlying model and use these to validate the basic computations, to determine their range of applicability, and to evaluate their sensitivity to basic lexical parameters. Second, we experiment with actual texts from the Shakespearean canon and the plays of Christopher Marlowe. The results of the tests are mixed, showing good consistency for the Shakespeare plays (with some discrimination among early, middle and late works) but poor consistency between Shakespeares poems and plays, or among Marlowes plays.

A Touchstone for the Bard

We introduce an authorship identification test, called modal analysis, based on a new statistic derived from the Karhunen-Loeve transform. Application to the poems of the Shakespearean canon and to other contemporary poetry strongly supports the case for disqualification of most major claimants. Results also cast doubt that the recently discovered poems, Shall I Die and Elegy, were written by William Shakespeare, but do suggest that eight unascribed poems of The Passionate Pilgrim may have been his work.

So Many Hardballs, So Few Over the Plate

Fosterss critique of our work is overdrawn, has left our findings 99.9% intact.

Who Was Shakespeare

HUSL 6313 Shakespeare (3 semester hours) Study of the dramatic and/or poetic writings of William Shakespeare. (May be repeated for credit as topics vary to a maximum of 6 credit hours.) (3-0) T

A representation-theoretic approach to the DFT with noncommutative generalizations

It is known that both the one-dimensional and multidimensional DFTs (discrete Fourier transforms) can be constructed as transition matrices associated with the decomposition of finite-dimensional complex commutative group algebras into simple components. Two key attributes of these transforms, orthogonality and the convolution property, are inherent in such a description, suggesting the possibility of enlarging the class by extending the construction to noncommutative groups. In this context, one speaks of a noncommutative or generalized transform, the definition of which is based on the theory of semisimple rings. The author reviews the ring theory and representation theory fundamental to the existence and computation of group algebra decompositions and sketches the representation-theoretic construction of both the classical and noncommutative discrete Fourier transforms. The noncommutative transform associated with the class of dihedral groups is explicitly constructed and shown directly to exhibit both orthogonality and a noncommutative convolution property. >

Smoking guns and silver bullets : Could John Ford have written the Funeral Elegy?

American authorities, persuaded by Donald Fosters stylometric evidence, believe that Funeral Elegy by W.S. (FE) is at least possibly, and perhaps indisputably by Shakespeare. British authorities disagree sharply. Brian Vickers, Richard Kennedy, and Gilles Monsarrat argue that John Ford wrote the Elegy. We examine both ascriptions by applying to Fords poems the same kind of common-authorship, exclusionary-evidence tests that we previously applied to Shakespeares poems and play verse. We conclude that the odds are strongly against the Americans. If W.S. is either Ford or Shakespeare, Ford seems by far the more likely candidate. Counting firm rejections only, the Elegy fails sixteen of thirty-three Shakespeare tests and only one of twenty-nine Ford tests. If the distinguishing traits of both authors are Poisson-distributed - as some seem to be - the odds that the Elegys scores could have arisen by chance from one corpus or the other are about 3,000 times better for Ford than they are for Shakespeare.

Possibility, Actuality, and Freewill

I describe recent developments of Conway and Kochen on the physical meaning of freewill and their theorem that the assertion of freewill for human beings, in their specific sense, implies the same for elementary particles. This description is given in simplified metaphorical terms that nonetheless address the key physical axioms and essential analytic content of their argument. I then give points of contact of our metaphor with the full technical analysis of the cited authors and conclude with some associated metaphysical speculations. These include the implications of the freewill theorem for dual aspect theories and, in particular, for process metaphysics.

Adeles, Ideles, and the Class Groups

To do harmonic analysis on a global field F, one needs to introduce two locally compact abelian groups: A F , the adele group, and I F , the idele group.