Piecing Together the Past

Nearly fifty years ago, some Bedouin shepherds stumbled upon a cache of ancient texts in caves near the Dead Sea, thirteen miles east of Jerusalem. It soon became clear that this was the largest and most significant collection of manuscripts ever discovered in Palestine. Finds Included the oldest witnesses to the Hebrew, Aramaic, and Greek texts of Jewish scripture-the Christian Old Testament-along with nonbiblical manuscripts certain to illuminate the tumultuous period of the destruction of the Second Temple and the time of Christ. The singularity of these texts has brought about one of the most protracted and painstaking endeavors of contemporary scholarship on religious history and Scripture. One afternoon 1947, three Bedouin shepherds were herding their flocks in the vicinity of Wadi Qumran above the northwest shore of the Dead Sea. They casually tossed a rock in acave opening and heard something break. Returning later, they discovered ten large pottery jars, one of which contained three scrolls wrapped in protective linen coverings. Four additional scrolls were soon discovered in the cave. Neither the Bedouin nor the antiquities dealer whom they contacted had any idea what the documents contained. Thinking the script to be some form of Syriac, the antiquities dealer solf four of the scrolls to the Syrian Orthodox metropolitan at St. Mark\u27s Monastery in Jerusalem. For approximately one hundred dollars, the metropolitan unwittingly purchased the oldest extant Hebrew text of the Book of Isaiah, an ancient Hebrew commentary on Habakkuk, and two unknown texts. The antiquities dealer sold the other three manuscripts to Eleazar Sukenik, a professor at Hebrew University in Jerusalem. The scrolls were in such fine condition that they were all published almost immediately. The magnitude and antiquity of these finds soon became apparent, and the caves around Wadi Qumran were aggressively explored for additional scrolls. Of the many caves quarried, eleven near the Wadi yielded written material. Cave 11, the last to be discovered (1956), supplied several extensively preserved scrolls of Leviticus, Psalms, and other works whose state of preservation rivaled that of the original Cave 1 finds. Unfortunately, there were only about a dozen of these beautifully preserved scrolls. Most of the approximately eight hundred texts discovered in the Qumran caves were not scrolls but scraps from disintegrated scrolls. Cave 4 yielded its rich cache of more than 575 manuscripts in tens of thousands of pieces. The condition of the written material in the other caves was no better: Caves 2 and 3 and Caves 5 through 10 yielded only fragments of more than one hundred other texts. Lacking the protection of pottery jars and linen shrouds, these manuscripts had fallen prey to a host of aggressors over the centuries, from the moisture in the caves to the appetite of worms to the swords and sandals of the caves\u27 human visitors. The scrolls simply disintegrated over the centuries, with the result that rarely 5 percent of any individual manuscript survived. The few surviving pieces of discrete scrolls were separated from one another and jumbled indiscriminately in layers of dirt on the cave floors. The muddle of fragments was made all the more incomprehensible by the manner of their retrieval. The Bedouin had gathered and sold most of the initial fragments without any record of where they came from. Fortunately, subsequent scientific excavations of Cave 4 unearthed fragments that were manifestly part of the same scrolls represented by the Bedouin finds. This established that the Bedouin scraps had been removed from the floor of Cave 4 and thereby guaranteed the authenticity of the initial fragments

Divisibility by |G| for Powers of Ordered k-sets

It is shown that the number of ordered k-sets of a group G whose nth power contains exactly i elements is always a multiple of IGI. An elementary proof of the fact that the number of ordered pairs ( x , y ) such that x2 = y2 is equal to kr lGI is also given

Cubing Ordered 2-sets

Given a group G, we define pi as the probability that, given an ordered pair X = (x,y), there are exactly i elements in X3 = {x1x2x3 l xi in X}. We show that P2( G) = 0 if, and only if, IGI is odd, and that p3(G) = 0 if, and only if, IGI is not divisible by three. The groups for which p4 ( G) = 0 and p5 ( G) = 0 are also determined

Path-sequential labellings of cycles

AbstractWe investigate labelling the vertices of the cycle of length n with the integers 0, …, n − 1 in such a way that the n sums of k adjacent integers are sequential. We show that this is impossible for both n and k even, possible for n even and k odd, and that it is possible for many cases where n is odd. We conjecture that it is always possible when n is odd

Bounds on Squares of Two-Sets

For a finite group G, let pi(G) denote the proportion of (x,y) in GxG for which the set {x2,xy,yx,y2} has cardinality i. In this paper we develop estimates on the pi(G) for various i

Counting Nilpotent Pairs

In this paper , we consider the probability that two elements chosen at random from a finite group G generate a subgroup of a given nilpotency class. It is shown that in solvable non­-nilpotent groups, the probability that two elements generate a nilpotent subgroup is \u3c= l/p,, where p, is the smallest prime dividing the order of the group, and it is also shown that there exist groups such that the probability of two elements generating a subgroup of class i approaches one (and other groups for which it approaches zero) for all i =\u3e2. It is also shown that the number of pairs which generate a subgroup of a given class is always a multiple of the order of the group. Some preliminary results on the analogous problem for solvability are also given

Arithmetic and equidistribution of measures on the sphere

Motivated by problems of mathematical physics (quantum chaos) questions of equidistribution of eigenfunctions of the Laplace operator on a Riemannian manifold have been studied by several authors. We consider here, in analogy with arithmetic hyperbolic surfaces, orthonormal bases of eigenfunctions of the Laplace operator on the two dimensional unit sphere which are also eigenfunctions of an algebra of Hecke operators which act on these spherical harmonics. We formulate an analogue of the equidistribution of mass conjecture for these eigenfunctions as well as of the conjecture that their moments tend to moments of the Gaussian as the eigenvalue increases. For such orthonormal bases we show that these conjectures are related to the analytic properties of degree eight arithmetic L-functions associated to triples of eigenfunctions. Moreover we establish the conjecture for the third moments and give a conditional (on standard analytic conjectures about these arithmetic L-functions) proof of the equdistribution of mass conjecture.Comment: 18 pages, an appendix gives corrections to the article "On the central critical value of the triple product L-function" (In: Number Theory 1993-94, 1-46. Cambridge University Press 1996) by Siegfried Boecherer and Rainer Schulze-Pillot. Revised version (minor revisions, new abstract), paper to appear in Communications in Mathematical Physic