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The present paper is devoted to the study of differential geometry of Kaehlerian homogeneous spaces. In section 1 we deal with the canonical decomposition of a simply connected complete Kaehlerian space and that of its largest connected group of automorphisms. We know that a simply connected complete Riemannian space V is the product of Riemannian spaces V0, V1, …, Vn, where V0 is a Euclidean space and V1, …,Vn are not locally flat and their homogeneous holonomy groups are irreducible [2]. Moreover, if V is homogeneous, so are all Vk [10]. We shall show that if V is Kaehlerian space with real analytic metric (resp. Kaehlerian homogeneous space), each factor Vk is also Kaehlerian (resp. Kaehlerian homogeneous) and that V is the product of V0, V1, …, Vn as Kaehlerian space. We call this decomposition the de Rham decomposition of the Kaehlerian space V. Although this result is supposedly known, there is no published proof as yet. Using this decomposition theorem we shall show that the largest connected group of automorphisms of a simply connected complete Kaehlerian space with real analytic metric is the direct product of those of the factors of the de Rham decomposition. In the Riemannian case this result has be been established in [3] by one of the authors of the present paper.

The purpose of this paper is to construct a theory of the amount of information provided by an experiment which does not rely on what Good (1962) has termed the modern Bayesian principle that it is legitimate to use the axioms of probability even when this involves the use of probabilities of hypotheses. In this respect the theory of this paper differs from the Lindley (1956), Mallows (1959) and Good (1960) each of which is written from a Bayesian viewpoint. Lindley (1956) expresses the opinion that Bayesian ideas would seem to be necessary to the development of a theory of the amount of information provided by an experiment and it is of interest therefore to determine how far such a theory may be developed without Bayesian ideas. There is further a need to use such a theory to examine how prior knowledge can be expressed quantitatively, and used in accordance with that theory.

Error-correcting codes are used in several constructions for packings of equal spheres in ^-dimensional Euclidean spaces En. These include a systematic derivation of many of the best sphere packings known, and construction of new packings in dimensions 9-15, 36, 40, 48, 60, and 2m for m g 6. Most of the new packings are nonlattice packings. These new packings increase the previously greatest known numbers of spheres which one sphere may touch, and, except in dimensions 9, 12, 14, 15, they include denser packings than any previously known. The density A of the packings in En for n = 2m satisfies log A ~ — \n log log n as n —* oo. 1.1. Introduction. In this paper we make systematic use of error-correct ing codes to obtain sphere packings in En, including several of the densest packings known and several new packings. By use of cross-section s we then obtain packings in spaces of lower dimension, and by building up packings by layers we obtain packings in spaces of higher dimension. Collectively, these include all of the densest packings known, and further new packings are also con­ structed. Part 1 of the paper is devoted to groundwork for the constructions. § 1.2 introduces sphere packings, and §§ 1.3-1.8 survey the error-correcting code theory used in the later Parts. Part 2 describes and exploits Construction A, which is of main value in up to 15 dimensions. Part 3 describes Construction B% of main value in 16-24 dimensions. Part 4 digresses to deal with packings built up from layers, while Part 5 gives some special constructions for dimen­ sions 36, 40, 48 and 60. Part 6 deals with Construction C, applicable to dimensions n = 2m and giving new denser packings for m ^ 6. We conclude with tables summarizing the results. Table I, for all n S 24, supersedes the tables of [18; 19], and Table II gives results for selected n > 24. The tables may be used as an index giving references to the sections of the paper in which the packings are discussed. Partial summaries of this work have appeared in [22; 23]. General references for sphere packing are [18; 19; 31] and for coding theory [4; 25].

A key to the genera of the Dacrymycetaceae and a taxonomic discussion of the various genera, including Dacrymyces, may be found in the preceding paper (Kennedy, 1958). There the methods employed in the study of these fungi and the terms used in the descriptions were explained. When Nees established the genus, which he spelled Dacryomyces, in 1816, he described the sporocarp as a fleshy or gelatinous substance, round or discoid in form, bright yellow or red in color and composed of interwoven filaments with spores. It included one species, Dacryomyces stillatus. Fries, in 1822, added to the description the characteristics of homogeneous consistency and spores formed on all sides of the sporocarp. Since then the boundaries of the genus have exhibited great flexibility, expanding or contracting according to the number of other genera recognized for the family. For example, in Neuhoff's (1936) treatment Dacrymyces, while not defined, includes at least species belonging to the genera Ditiola, Guepiniopsis and Femsjonia, and one gains the impression that he would place species of nearly all the genera but Calocera there. On the other hand Martin's (1952) concept is much narrower, because of the recognition of ten other genera in the family. The concept of the genus presented in this paper was arrived at by defining eight other genera for the Dacrymycetaceae. Such a procedure leads to a view of the genus Dacrymyces which is somewhat negative in nature, for one thinks of it as a group of species that does not fit the pattern of the other, more easily defined, genera. Thus we find that, except for allowance of considerable variation in external basidiocarp form, the present description differs little from that of Fries. Dacrymyces still includes the largest number of species but seems now to comprise a natural group. 1 Based on Part II of a thesis submitted in partial fulfilment of the requirements for the degree of Doctor of Philosophy in the State University of Iowa

In this paper further evidence is put forward in support of a previous paper on the subject. Very soon after the reading of that paper, A. E. Mitchell read a paper before the Chemical Society on a closely related subject. Working quite independently, and employing other methods of investigation, Mitchell arrived at conclusions in close agreement with those detailed in the former paper. H.L.J. Bäckström has criticized Mitchell's paper, raising some objections 'against the experimental evidence produced aud the … way of interpreting it'. Where Bäckström objections touch on anything in Mitchell's paper which is incidentally akin to any of the work described in this or the earlier paper, they will be discussed herein.

This paper applies the theory of semi-Markov processes to the construction of a stochastic model for interpreting data obtained from clinical trials. The model characterizes the patient as being in one of a finite number of states at any given time with an arbitrary probability distribution to describe the length of stay in a state. Transitions between states are assumed to be chosen according to a stationary finite Markov chain.Other attempts have been made to develop stochastic models of clinical trials. However, these have all been essentially Markovian with constant transition probabilities which implies that the distribution of time spent during a visit to a state is exponential (or geometric for discrete Markov chains). Markov models need also to assume that the transitions in the state of a patient depend only on absolute time whereas the semi-Markov model assumes that transitions depend on time relative to a patient. Thus the models are applicable to degenerative diseases (cancer, acute leukemia), while Markov models with time dependent transition probabilities are applicable to colds and epidemic diseases. In this paper the Laplace transforms are obtained for (i) probability of being in a state at timet, (ii) probability distribution to reach absorption state and (iii) the probability distribution of the first passage times to go from initial states to transient or absorbing states, transient to transient, and transient to absorbing. The model is applied to a clinical study of acute leukemia in which patients have been treated with methotrexate and 6-mercaptopurine. The agreement between the data and the model is very good.

Between 1956 and 1960, 20 ceramic figurines were discovered in Mound 42Utlll at the Hinckley site west of Utah Lake in northern Utah. Circumstances of the discovery are discussed, as well as two other finds previously made in Utah Valley. Comparisons are made with figurines from the Fremont and Sevier areas, particularly from sites just north and south of Utah Valley. On the basis of these comparisons and a consideration of other items, such as ceramics, house type, and ecological setting, it is hypothecated that Utah Valley was a cultural center for northwestern Utah. It is proposed that this area be called the Provo culture area. T HIS PAPER has two purposes: (1) to report the largest concentrated find of figurines in the Sevier area and compare these figurines with finds from other Sevier and Fremont sites, and (2) to formulate a working hypothesis concerning the position of Utah Valley within the Sevier culture. The eastern boundary of the Sevier area (Fig. 1) is the Wasatch Mountains, and the southern boundary borders the Virgin branch of the Anasazi in southern Utah. The northern boundary is unknown, but it is believed to lie in northern Utah or southern Idaho. The western boundary is equally obscure but probably lies in eastern Nevada. Utah Valley is at the foot of the Wasatch Mountains on the eastern border of the Sevier area, and it is very close to the southern edge of the northern extension of that area. Steward (1936, map) and Wormington (1955, Fig. 55) have indicated a line of separation that runs just south of Utah Lake. The field work with which this report is concerned was done in 1956, 1959, and 1960 under the auspices of Brigham Young University. The first of 20 figurines that were eventually dug from the damp soil of Site 42Utlll near Utah Lake was found by a student archaeologist in the fall of 1956. In the fall of 1959 three more were recovered, and in the summer of 1960 I excavated 16 additional specimens. Mound 42Utlll, known locally as part of the Hinckley site, is on the property of G. M. Hinckley west of Provo in central Utah and is * This paper was presented in shorter form at the 27th Annual Meeting of the Society for American Archaeology, May 3-5, 1962, Tucson, Arizona. separated from Utah Lake by the Provo Municipal Airport. This low mound is at the southwestern corner of the Hinckley farm (northwest quadrant of the southwest quarter-section of section 10, township 7 south, range 2 east, Salt Lake meridian). The mound may be reached by taking a farm road that leaves the airport highway shortly before the highway turns west to the airport. This farm road passes near Mound 42Ut112, which was excavated by Christensen between 1946 and 1949 (Christensen 1947; Christensen and Leigh 1950). In 1946 Christensen's party discovered the head of a small clay figurine at this site. At that time only one other such find had been reported from Utah Valley, a clay head illustrated by Steward (1936, Fig. 10 c). These two heads, along with the 20 specimens recovered from Mound 42Utlll, bring the total number of specimens discovered in Utah Valley to 22. Reagan (1935: 67) reports finding "fragments possibly of figurines" in his Mound H3 (Jones 1961: 49; Jones correlates Mound H3 with 42Ut109 under the present site designation system). Reagan gives no additional data, and we have not been able to find these "fragments" in the University collections. Although the 1956 crew was aware of these earlier finds, they were not prepared for the discovery of such a fine specimen (Jones 1958: 12; Green 1961: 55, Fig. 13 g). The discovery in 1960 of 16 additional specimens was even more surprising. Mound 42Utlll is only one of several low mounds on the Hinckley property (Green 1961, Map 2; Jones 1961). Through the generous permission of Hinckley, Brigham Young University was allowed the privilege of excavating these mounds as a part of its program for training student archaeologists. As a result, from 1946 to 1949 M. Wells Jakeman conducted field classes on what was then called Mound 1 but is now designated as Mound 42Utlll. In 1956 excavations were resumed at the Hinckley site under the direction of Ross T. Christensen. Mound 42Utlll11 was chosen for the first season's work because it was close to the access road in an area that was not under

In a previous paper attention was drawn to the close similarity in mineral and chemical composition of chondritic meteoric stones. In the case of about forty chondrites it was shown that, apart from variations in the amount of nickel-iron, not only are the constituent minerals present in very similar amounts, but their chemical compositions vary to no very great extent. In that paper sufficient importance was not attached to the variations in the amount of nickel-iron.

In the previous papers [1], [2] we have studied the prolongations of G-structures to tangent bundles of arbitrary order, and in [3] we have considered the prolongations of connections to the tangential fibre bundles of higher order. The purpose of the present paper is to study the liftings of tensor fields and affine connections to tangent bundles of higher order. In fact, most of results in [4], [5] will be generalized in a natural fashion and some of the formulas concerning vertical and complete lifts in [5] will be unified and generalized in our formulas concerning ‹Λ›-lifts (cf. §3).

History, Vol. 32, No. 2. (Publications, No. 526.) Chicago. POWERS, WILLIAM E. 1939. "Basin and Shore Features of the Extinct Lake San Augustin, New Mexico." Journal of Geomorphology, Vol. 11, No. 4, pp. 345-57. New York: Columbia University. RAY, CYRUS N. 1930. "Report on Some Recent Archaeological Researches in the Abilene Section." Bulletin of the Texas Archaeological and Paleontological Society, Vol. 2, pp. 45-58. Abilene. 1934. "Flint Cultures of Ancient Man in Texas." Bulletin of the Texas Archaeological and Paleontological Society, Vol. 5, pp. 107-11. Abilene. 1938. "The Clear Fork Culture Complex." Btdletin of the Texas Archaeological and Paleontological Society, Vol. 10, pp. 193-208. Abilene. ROBERTS, FRANK H. H., JR. 1935. "A Folsom Complex, Preliminary Report on Investigations at the Lindenmeier Site in Northern Colorado." Smithsonian Miscellaneous Collections, Vol. 94, No. 4. Washington. 1940. "Developments in the Problem of the North American Paleo-Indian." In "Essays in Historical Anthropology of North America." Smithsonian Miscellaneous Collections, Vol. 100, pp. 51-117. Washington. 1942. "Archaeological and Geological Investigation in the San Jon District, Eastern New Mexico," Smithsonian Miscellaneous Collections, Vol. 99, No. 1. Washington. ROGERS, MALCOLM J. 1939. "Early Lithic Industries of the Lower Basin of the Colorado River and Adjacent Desert Areas." San Diego Museum Papers, No. 2. San Diego. SAYLES, E. B. 1935. "An Archaeological Survey of Texas." Medallion Papers, No. 17. Globe: Gila Pueblo. SAYLES, E. B., AND E. ANTEVS 1941. "The Cochise Culture." Medallion Papers, No. 29. Globe: Gila Pueblo.

It is the purpose of the present paper to give a natural method of realizing bounded symmetric domains as matrix-spaces. Our method yields, as special cases, the well-known bounded models of irreducible bounded symmetric domains of classical type (I)-(IV), as were already described in the original paper of E. Cartan [1] (see §3; we follow in this paper the classification table in [14], not in [1]). A direct application of this method will be to determine the canonical bounded models of the irreducible bounded symmetric domains of exceptional type; it will be published in another paper (see [6], [7] for the summary of the results).

It has been noted by several investigators (Dobzhansky 1944; Novitski 1946; Pavan 1946) that in some species of Drosophila, variability in chromosome structure is virtually restricted to one chromosome. At the same time, they note, the variable chromosome has a large number of inversions distributed throughout the species. This paper shall be addressed to the former observation, i.e., the concentration of inversions in one chromosome. One possible explanation is that the third chromosome is intrinsically more subject to breakage and rearrangement. This was tested by 1Hlelfer (1941) who irradiated D. pseitdoobscitra and demonstrated that the induced breaks and rearrangements occurred at random throughout the chromosonme complement. It is the purpose of this paper to present evidence for an alternative explanation for this non-random variation. It has been shown by Cooper, Zimmering, and Krivshenko (1955) that female D. melanogaster, heterozygous for two unlinked inversions, produce an appreciable proportion of inviable eggs; whereas females heterozygous for only one inversion are fully fertile, or virtually so. To explain this phenomenon, they suggest that in the females heterozygous for two unlinked inversions, there is the possibility of non-homologous pairing in meiosis, with unsymmetrical disjunction, and production of unbalanced gametes. If this situation were to hold for a natural population in which one chromosome had

In this paper various concepts intrinsically defined by the differential equation1.1are interpreted geometrically by concepts analogous to those in the Minkowski plane. This is carried out in § 2. The point of such a development is that one may apply the techniques or transfer known results in the theory of curves (in particular, convex curves) to (1.1), thereby gaining an additional tool in the investigation of this equation. For an application of a result obtained in this way, namely (3.12), see (4).Throughout this paper,R(t)is a real-valued, continuous function ofton the real line (— ∞ < t < + ∞) and only the real solutions of (1.1) are considered.

The projective planes discovered in 1957 by Hughes [3] were originally described by means of a nearfield F satisfying the following conditions:(a) F is finite,(b) the centre and kernel of F coincide,(c) F is of rank 2 over its kernel.(The definitions of these terms will be given in § 2; the terminology used throughout the paper is that of [1].)Rosati [5] showed in 1960 that condition (a) is not necessary, thus constructing the first “infinite Hughes planes”. Condition (b), however, plays an essential part also in Rosati's work.The aim in this paper is to show that condition (b) is superfluous as well. For the finite case, this has been remarked by Ostrom [4] without proof; here we shall show that a “generalized Hughes plane” can be constructed over any nearfield satisfying condition (c) only.

Baron Karl Ludwig von Reichenbaeh (1788-1869) was born at Stuttgart, ennobled in 1839 by the King of Wiirttemberg, and died at Leipzig. But he spent most of his life in Austria, where he had iron works and estates in Moravia and Galicia and a castle near Vienna. Earning a livelihood as a copyist in the state archives, he was able to obtain a university education, and he started his career as a works chemist. He produced many papers on the distillation products of coal, tar, oil, &c, and was the discoverer of creosote and solid paraffin. He gained considerable notoriety with his ideas and books on what he called 'od' or 'odylic force', a combination of animal magnetism, light, heat, force of crystallization, &c., capable of producing hypnotism. The fall in 1833 of a shower of meteorites, which he witnessed near his works at Blansko in Moravia, fired his intense enthusiasm for the subject of meteorites, on which he wrote 29 scientific papers (1835—65) and many newspaper articles.

The present paper began as a natural outgrowth of our first paper, where we characterized the module homomorphisms from group algebras into a fairly restrictive class of group algebra modules. We now investigate module homomorphisms from group algebras into a more general class of group algebra modules. Although the two papers are thus related, they can be read quite independently.Section 2 contains our extension, Theorem 2.1, of P. J. Cohen's theorem on factorization in Banach algebras (1). Our extension is to Banach modules over Banach algebras equipped with an approximate identity. We should mention first that J.-K. Wang observed the existence of such a generalization, and secondly, that our proof requires no ideas different from those in Cohen's proof. Nevertheless, we include a proof that condenses the original proof considerably.

This paper applies the constant population approximation to the study of epidemics which involve more than a single type of infective. An example of this would be a situation in which both clinically infected individuals and subclinically infected individuals or carriers are present. We derive equations for the expected numbers of clinically infected individuals and carriers at any time t for the model with zero latent period and infectious periods having negative exponential distributions. From these equations we derive conditions under which a unimodal incidence curve can result, and expressions for the expected total epidemic size. The equations describing the course of an epidemic are analytically intractable except in certain simple cases, [1]. When one examines the reason for the mathematical difficulties that arise, one finds that the principal stumbling block is the assumption of a finite population of susceptibles. It can be argued, however, that in modern societies epidemics very rarely menace an entire population, and that observed epidemic sizes are usually much smaller than the total susceptible population. This is due to a variety of reasons, among which are public health measures and good communication facilities available to a modern society. In consequence it is reasonable to try to bypass some of the mathematical difficulties inherent in the theory of epidemics by assuming a susceptible population whose size does not change through the course of the epidemic. Bailey [2], Morgan [3], and Williams [4] have discussed in some detail the theory of epidemics using the constant population approximation, although such an approximation was used by several authors earlier, [5], [6], [7]. The theory developed so far has dealt mainly with epidemic processes in which there is only a single type of infective and a single type of susceptible. Recently Gart [8] has considered a model for epidemics involving more than one type of susceptible. It is the purpose of this paper to analyze the development in time of epidemics which involve more than a single type of infective individual. This Received in revised form 3 January 1967. 257 This content downloaded from 157.55.39.78 on Mon, 20 Jun 2016 07:20:57 UTC All use subject to http://about.jstor.org/terms 258 HUGH M. PETTIGREW AND GEORGE H. WEISS problem, in which there may be several types of infectives, is suggested by diseases in which carriers are important, [9]. In the present paper we derive equations for the expected numbers of carriers and clinically infected individuals at any time t, in the case of a zero latent period and negative exponential distributions of infectious periods. From these equations we will give the conditions under which an epidemic arises from the introduction of a bearer of the disease, i.e., the conditions under which the reporting curve increases initially. These conditions are of some theoretical interest since Bailey has shown that in the infinite population approximation to an epidemic with only one class of susceptibles and one class of infected individuals, no initial increase in the reporting curve can occur. Finally, we derive expressions; for the expected total epidemic size. Let us consider a homogeneously mixing population which, in the present approximation, can be characterized by four parameters, yi(t), y,(t), zi(t), zc(t). These are, respectively, the number of clinically infected individuals, the number of carriers, the cumulative number of removals of infectives, and the cumulative number of removals of carriers, all evaluated at time t. The probability that a single susceptible individual will become clinically infected in (t, t + dt) is assumed to be fli(y, + yc)dt, and the probability that a susceptible will become a carrier is assumed to be Ic(yi + ye)dt, where both l's are assumed to be constant and it is assumed for simplicity that carriers and infectives are equally infectious. The constancy of these p's is the principal assumption of the present theory; in the more detailed stochastic theory discussed by Bailey [1] the f's are proportional to the number of susceptibles as well. In the simplest model, we assume that the latent period is zero, i.e., that newly infected individuals are immediately capable of infecting others, and that the infectious period is a random variable with a negative exponential distribution. The rate parameter appearing in the distribution for clinically infected will be denoted by vi and that appearing in the distribution for carriers will be denoted by v,. In order to describe the stochastic process we introduce a set of probabilities p(r,,r2, r3, r4, t) defined by p(r, t) = Pr{y,(t) = r, yc(t) = r2, z(t) = r3, zc(t) = r4}. By our assumptions the p(r, t) satisfy ap = fl (r1 + r2 1)p(r, 1,r2,r3, r4, t) + fl(r, + r2 -1)p(r1,r2 1,r3, r4, t) (1) + vA(r1 + 1)p(r + 1,r2, r 1,r4)+ vc(r2 + 1)p(r, r2 + 1, r3,r4 1) [(fl + ic + vi)rl + (fl + flPc + ve)r2] p(r, t). This content downloaded from 157.55.39.78 on Mon, 20 Jun 2016 07:20:57 UTC All use subject to http://about.jstor.org/terms Epidemics with carriers: the large population approximation 259 The moment generating function (2) M(01, 02, 03 , 0,t) = M(O, t) = E{eo'Y'+2yYc+03zi+042C} therefore satisfies aM M = [f#,(ee I1) + flc(e02 1) + vi(ee, +03 1)] (3) aM + [fl(ee' 1) + flc(e02 1) + vc(e-02+4 1)] The interesting features of the development of the epidemic can be determined by analyzing the properties of the mean values Lpi(t) = E{yi(t)}, pc(t) = E{yc(t)} (4) ow(t) = E{z,(t)}, wc(t) = E{zc(t)}. The equations for these mean values are easily determined by equating coefficients of the O's on both sides of Equation (3). They are Pi = (/01v-i)it?+fliltc (5) Pc = flcpi+ (fc v)Pc

This paper attempts to establish the rules governing the recognition of names in the Maya script, particularly with regard to their glyphic context. These rules are applied to the Quirigua inscriptions; glyph clauses pertaining to three rulers are established with some probability and two others are suggested more doubtfully. The founder of the Quirigua dynasty was apparently related to an identified ruler at Copan, and it is shown that at least one and probably two later rulers at Copan share some names and titles with the Quirigua rulers. It is suggested that the family name of the rulers of Copan and Quirigua was caan, sky. THE DISCOVERY by Heinrich Berlin (1958) that certain glyphs are directly associated with, and appear to refer to, specific sites, and the evidence offered by Tatiana Proskouriakoff (1960) for the existence of a dynastic sequence at Piedras Negras must inevitably lead to a full reappraisal of most of the Mayan inscriptions. The present paper presents a preliminary statement of some of the associations in which personal names seem to be encountered most frequently, and an application of some of the tentative rules to the study of certain inscriptions at Quirigua and Copan. Proskouriakoff (1960) has shown that the monuments at Piedras Negras fall into a pattern which includes physical grouping, dating, repetition of glyph clauses, and a regular repetition of artistic motifs. Among these motifs, one which apparently shows the accession or inauguration of a ruler is accompanied by a contemporary date, which is emphasized thereafter, and by a date, not previously emphasized, referring to a time somewhat before the presumed inauguration. Thereafter, during a period which is consonant with the supposition that repetitions fall largely or entirely within a single individual's lifetime, both these dates tend to be repeated and emphasized. The earlier date she calls an initial date and the date contemporary with the accession motif she calls an inaugural date. The initial date is accompanied by the upended frog glyph and the inaugural date by the toothache glyph. Proskouriakoff (1960: 460) has suggested that the initial date, which falls between 12 and 31 years earlier, must represent an event that happened early in the ruler's life, possibly even his birth, though more probably some ceremony corresponding to baptism or to his initiation into formal training. There is, however, one important reason for thinking that the initial date is actually a birthday. This is the widespread use of a calendar date as a personal name for gods and humans throughout Mesoamerica. Among the Cakchiquels, the calendar name is the only one preserved in our sources for most of the rulers. The Mixtec manuscripts scrupulously give calendar names even when other names are not indicated. Among the peoples of the Valley of Mexico there are indications of a similar practice in secondary names of some of the rulers and deities, such as One Reed Quetzalcoatl. Zapotec monuments seem to indicate a similar practice extending well back into the Classic Period. Even some of the Yucatec Maya gods have calendar names. That the calendar name is actually a birthday name is indicated by dates in Codex Nuttall accompanying scenes of births (Nuttall 1902). This emphasis on the birth date is obviously an important part of the Mesoamerican cultural pattern. I have long speculated on the possibility that a systematic analysis might indicate that certain calendarround dates of the Maya inscriptions functioned as personal names. Whether or not these Maya dates functioned as personal names, the importance of the birth date throughout Mesoamerica seems to me to suggest that Proskouriakoff's initial date should be understood as probably a birth date. If they function, even partially, as personal names, one would expect that in inscriptions which contain only one calendarround date, and which seem to be historical, there is a fair chance that the date corresponds in function to an initial date. The fact that apparent birth dates and inaugural dates are repeated on a series of monuments, apparently during the lifetime of a particular ruler, and are then replaced by another set of dates on a different series of monuments seems to me more basic than the particular accompanying glyphs which distinguish initial dates from inaugural dates at Piedras Negras. The upended frog and toothache glyphs are rare or absent in a context of repeated dates at Quirigua and some other sites. Nevertheless, I

I t is straightforward, but tedious, to write down the integers whose representations in a given base do not have particular digits in certain positions. In the first section of this paper we give a computational scheme that enables us to carry out such operations in a rapid and simple fashion.In the second section of the paper we derive a general identity involving the digits of integers in arbitrary Cantor systems of notation.In the third section we apply this identity and deduce a number of results concerned with the splitting of integers into classes with equal power sums. The computational scheme of the first section leads us to an algorithm for the determination of such splittings.

A ring A has finite norm properly, abbreviated FNP, if each proper homomorphic image of A is finite. In [3], Chew and Lawn described some of the structural properties of FNP rings with identity, which they called residually finite rings. The twofold aim of this paper is to extend the results of [3] to arbitrary rings with FNP and to give characterizations of FNP rings independent of the results of [3].If A is a ring, let A+ denote A regarded as an abelian group. In the first section of this paper, we explore the effects of FNP upon the structure of A+. The following theorem is typical of the results in this section.

This paper is based on part of the thesis of one of the authors (5), submitted at the University of Toronto in 1963. In the first part of the paper a result on induced representations (2, 4, 9) is generalized slightly and a number of corollaries are derived. In the rest of the paper a special case of this result is applied to put the representation theory of the alternating group on a par with that of the symmetric group. A knowledge of the representation theory of Sn (7) on the part of the reader is assumed.

LetRbe the discrete group ring of the groupGover the ringA. In this paper we attempt to find necessary and sufficient conditions onGandAso thatRwill have some standard ring-theoretic property ; among the properties considered are those of being artinian, regular, self-injective, and semi-prime.The contents of this paper form essentially the author's doctoral thesis. The author would like to thank his supervisor Dr. J. Lambek for his generous encouragement and continued interest.

This paper consolidates and summarizes the mass of radiocarbon determinations that has accumulated in various books, journals, newsletters, and manuscripts between 1951 and the spring of 1965. The dates are derived from Paleo-Indian, Archaic, Early Ceramic, and Late Prehistoric occupations in the Canadian provinces of Alberta and Saskatchewan and the Plains areas of Montana, Wyoming, Colorado, North Dakota, South Dakota, Nebraska, Kansas, and Iowa. Included are 195 radiocarbon dates attesting to the presence of man on the Plains for the last 12,000 years. IN THIS REVIEW I shall endeavor to show to what degree certain of our archaeological typologies from Plains sites correlate with radiocarbon determinations. Since 1951, Plains archaeologists have been supplied with numerous radiocarbon assays; it is quite obvious that I will not, in this paper, be able to treat each date separately. Beginning with the earliest dates and working toward the Late Prehistoric period I have organized and sorted the material into categories named Paleo-Indian, Archaic, Early Ceramic, and the Central Plains-Middle Missouri traditions. The accepted validity of the dates is based, to a very large degree, upon their extent of association with certain diagnostic specimens, and in the case of the Paleo-Indian and Archaic categories these specimens are stone projectile points. For the Early Ceramic and Central Plains-Middle Missouri groupings pottery types have been the most diagnostic

The undertaking of constructing spaces which contain a given space as a subspace is by no means new: the extension of the complex number plane to the complex number sphere by the addition of the one point at infinity, the extension of the real line by adjoining the two infinities ∞ and -∞, and the construction of the space of real numbers from that of the rationals by means of Cauchy sequences or Dedekind cuts are 19th Century examples of this very thing. However, only the advent of general topology made it possible to raise the general question of space extensions. It appears that the first study of problems in this area was carried out by Alexandroff and Urysohn in the early twenties [l]. Another mile stone in the history of the subject was the 1929 paper by Tychonoff in which the product theorem for compact spaces is proved and used to identify the completely regular Hausdorff spaces as precisely those spaces which can be imbedded in a compact Hausdorff space [33]. During the same period, work on certain specific extension problems was done by Freudenthal [17] and Zippin [35]. However, the first large body of systematic theory, used for the investigation of a wide range of extension problems, was presented by Stone [31] in 1937. There, one also finds the remark that "one of the interesting and difficult problems of general topology is the study of all extensions of a given space", and it appears that Stone' s own work must have convinced many others of the truth of this observation, for since that time there has been a steady succession of papers in this field. But apart from that, the study of extension spaces clearly has a very particular attraction for some mathematicians.

As long ago as 1906 I had noticed a yellowish clay-like mineral occurring somewhat abundantly at both Cligga mine, Perranzabuloe, and at Bunny mine, St. Austell. From its appearance the mineral did not look particularly interesting, neither did it suggest being one containing a metal, and for these reasons no attempt was made to determine its character. In a paper read before this Society in 1937 I mentioned this yellowish clay-like substance as occurring associated with rashleighite at the Bunny mine. Soon after the publication of this paper in 1948 I received a letter from Monsieur N. Varlamoff in the Belgian Congo saying he had seen my reference to the 'yellowish clay-like substance' (p. 356) and suggested it might possibly be identical with the mineral varlamoffite which he had recently discovered in the Belgian Congo and which had been named after him, his description of the mineral with analysis, mode of occurrence, and localities having been published in 1948.

After the publication of the important paper by Rota [9] on Möbius functions a large number of papers have appeared in which the ideas are applied or generalized in various directions, the papers by Crapo [3], Smith [10] and Tainiter [11] are some of them. The theory of Möbius functions is now recognized as a valuable tool in combinatorial and arithmetical research.

This paper considers the properties of the representation of a Lie algebra when restricted to an ideal, the subduced* representation of the ideal. This point of view leads to new forms for irreducible representations of Lie algebras, once the concept of matrices of invariance is developed. This concept permits us to show that irreducible representations of a Lie algebra, over an algebraically closed field, can be expressed as a Lie-Kronecker product whose factors are associated with the representation subduced on an ideal. Conversely, if one has such factors, it is shown that they can be put together to give an irreducible representation of the Lie algebra. A valuable guide to this work was supplied by a paper of Clifford (1).

In a recent paper (7) Yood developed the beginnings of a theory of modular annihilator algebras. In this paper we extend his work on these algebras.The definition of modular annihilator algebra is algebraic in nature (see §4) ; in fact the algebra need not be assumed even topological. However, a significant number of important normed algebras are modular annihilator algebras. A list of examples is given in §8.The theory of modular annihilator algebras is related to the theory of certain important topological algebras. In §5 we consider the relationships between dual and annihilator algebras and modular annihilator algebras, and in §7, the relationship between completely continuous normed algebras and modular annihilator algebras.

Let Mm be a closed PL manifold of dimension m. Then a concordance between two PL-homeomorphisms h0, h1:M → M is a PL-homeomorphismH: M × I → M × I such that H|M × 0 = h0 and H|M × 1 = h. Concordance is an equivalence relation and in his paper [2], M. Kato classifies PL-homeomorphisms of Sp × Sq up to concordance. To do this he treats first the problem of classifying those homeomorphisms that induce the identity in homology, and then describes the automorphisms of the cohomology ring that can arise from homeomorphisms of Sp × Sq. In this paper we show that for sufficiently connected PL-manifolds that embed in codimension 1, one can extend Kato's classification of the homeomorphisms that induce the identity in homology.

For a formal definition of “map” the reader is referred to (7, §2). The maps in this paper are rooted by specifying an orientation for one of the edges. This also specifies a root vertex, the negative end of the root, and a root face, the face on the left of the root edge. Counting is, as usual, defined on isomorphism classes.Regular maps of even valence have been enumerated in a recent paper by Tutte. In this paper we determine the average number of trees in such maps, and include similar results for regular tri valent maps, that is, maps with three edges incident on every vertex. In the development for the latter, a formula for the number of trivalent maps with 2t vertices is produced.

In a paper published in 1956, Rund (4) developed the differential geometry of a hypersurface of n— 1 dimensions imbedded in a Finsler space of n dimensions, considered as locally Minkowskian.The purpose of the present paper is to provide an extension of the results of (4) and thus develop a theory for the case of m-dimensional subspaces imbedded in a generalized (Finsler) metric space.We consider an n-dimensional differentiable manifold Xn and we restrict our attention to a suitably chosen co-ordinate neighbourhood of Xn in which a co-ordinate system xi (i= 1, 2, … , n), is defined. A system of equations of the type xi = xi(t) defines a curve C of Xn the tangent vector dxi/dt of which is denoted by xi.

In an important paper (7), long overlooked, J. H. Redfield dealt with several aspects of enumerative combinatorial analysis. In a previous paper (1) I showed the relation between a certain repeated scalar product of a set of permutation characters of a symmetric group and Redfield's composition of his group reduction functions. Here I consider, from a group representational point of view, Redfield's idea of a range-correspondence and its application to enumeration of linear graphs. The details of the application of these ideas to more general enumerations are also given.

Since the pioneer studies by Barghoorn and Linder (1944), an increasing number of papers have been published dealing with taxonomy and, to a lesser extent, distribution and abundance of lignicolous marine fungi (Johnson and Meyers, 1957). Few investigations have been concerned specifically with the physiological aspects of these fungi. Barghoorn and Linder reported on physiological studies of selected species of marine fungi, and examined the effects of different media, including cellulose and wood flour, on vegetative growth. The mycelial growth rates and mode and rate of perithecial production on sea-water agar of seven different lignicolous Ascomycetes were studied at this laboratory (Meyers, 1954). Recently, Gustafsson and Fries (1956) investigated the nutritional requirements of various marine species of Ascomycetes and Deuteromycetes and recorded the rate of mycelial development on differential media. To our knowledge, the report from this laboratory (1954) was the first paper noting the development of perithecia in cultural studies of marine lignicolous Ascomycetes. During the past several years, we have been studying the physiology of these fungi, including nutritional studies, as well as tests in which wood and wood constituents are used to elucidate specific metabolic activities, such as reproduction. The latter work is reported here. The results of minimal-nutrition tests will be presented in a subsequent paper.

The theory of stochastic differential equations in a differentiate manifold has been established by many authors from different view-points, especially by R Lévy [2], F. Perrin [1], A. Kolmogoroff [1] [2] and K. Yosida [1] [2]. It is the purpose of the present paper to discuss it by making use of stochastic integrals.

In [5], Ikeda-Nagao-Nakayama gave a characterization of algebras of cohomological dimension ≦n In a subsequent paper [4] Eilenberg gave an alternative treatment of the same question. The present paper is devoted to the discussion of a number of questions suggested by the results of [4] and [5]. Among others it is shown that the conditions employed in stating the main results in [4] and [5] are equivalent, so that the main results of these two papers are in accord. Further, the cohomological dimension of a residue-algebra is studied in terms of that of the original algebra and the (module-) dimension of the associated ideal. The terminology and notation employed here are that of [3].

Let S denote the integral closure of a complete discrete rank one valuation ring R in a finite Galois extension of the quotient field of R, G the Galois group of the quotient field extension, and f an element of Z2(G,U(S)) where U(S) denotes the multiplicative group of units of S. A crossed product Δ(f, S, G) whose radical is generated as a left ideal by the prime element II of S is an hereditary order according to the Corollary to Thm. 2. 2 of [2], and we call such a crossed product a II-principal hereditary order. In previous papers the author has studied II-principal hereditary orders Δ(f, S, G) for tamely and wildly ramified extensions S of R (see [10] and [11]). The purpose of this paper is to study II-principal hereditary orders Δ(f, S, G) with no restriction on the extension S of R.

N many mineralogical problems it is desirable to obtain chemical information from extremely limited amounts of material. The technique now to be described is useful in this respect in that chemical data may be obtained from extremely small and precisely defined volumes within normal petrological thin sections. The advantage of studying chemical composition, at high resolution, within the fabric of a thin section will be apparent. In the present paper the technique of X-ray absorption microspeetroscopy is described and its usefulness illustrated in one particular ease. This concerns the determination of the calcium content of a hydrogel formed in microscopic amount from the mineral larnite (fl-Ca2Si04) in small, sealed hydration cavities in larnite-bearing rocks where the cavities also contain calcium hydroxide (portlandite). The present paper deals only with the determination of calcium. The technique can in practice be applied to the determination of other elements of higher atomic number. Theo~'y and method. The measurement of X-ray absorption at selected wavelengths provides an accurate method of determining the mass of an element present in a material of known thickness. A development of this method, using the fine focus X-ray tube devised by Cosslett and Nixon (1953), permits the analysis of small selected areas in a thin section without recourse to photographic recording (Long and Cosslett

CENTERING in Georgia and extending to the adjacent portions of the states of Tennessee, Alabama, North and South Carolina are a series of archaeological complexes characterized by pottery with a complicated stamped decoration. These complexes have not yet been sufficiently defined. This paper will attempt a partial statement of the resemblances and differences between the major divisions of the group. It will not attempt to evaluate the excavations in a detailed manner. This material is what Holmes (1903, pp. 130-137) called the South Appalachian Province in which he lumped all the stamped pottery from the southeastern area. The Swift Creek Complicated Stamped material as defined in a preliminary paper by Kelly (1938, pp. 25-31, P1. 11, a) will not be considered here. It has become evident in recent years that subdivisions of the rest of the material could be made and that they would have temporal or spatial significance. Various names have been used in the literature but the divisions have not been clear. Holmes' term of South Appalachian Province is perhaps still good for the whole area. The commonly used divisions of the later complicated stamped pottery materials are: Etowah, Savannah, and Lamar. The Etowah Complex is named for the Etowah site near Cartersville, Georgia, excavated by J. P. Rogan for the Bureau of American Ethnology (Thomas, 1887, pp. 96-107) and by Moorehead (1932). Specific names were not established for the pottery and subsequent examination has revealed a considerable mixture (Wauchope, 1948, pp. 204206). The Savannah Complex was named at the Irene site near Savannah on the Georgia Coast by Caldwell and McCann (1941, pp. 42-48). The Lamar Complex was first named by Kelly (1938, p. 50, P1. 12, a) from materials excavated at the Lamar site, a detached area of Ocmulgee National Monument near Macon in the central part of the state. Previously excavated sites showing Lamar materials included Nacoochee (Heye, et al, 1918) and Peachtree (Setzler and Jennings, 1941). In neither case was the complex named as such. Recent excavations by Sears (p. 137-142) in the Allatoona Basin at the Wilbanks site have further cleared up the stratigraphy suggested by Wauchope and should do much toward defining the three complexes. sta ped po tery materials are: Etowah, ah, and Lamar. The Etowah Complex ed for the Etowah site near Cartersville, 142 [2, 1950

1. This paper was motivated by some questions of Barnett and Lewis (1967) concerning extreme winter temperatures. The temperature during the winter can be hopefully regarded as generated by a stationary Gaussian process superimposed on a locally U-shaped trend. One is interested in statistical properties of the minimum of sample paths from such a process, and of their excursions below a given level. Equivalently one can consider paths from a stationary process crossing a curved boundary of the same form. Problems of this type are discussed by Cramer and Leadbetter (1967), extensively in the trend-free case and in less detail when a trend is present, following the method initiated by Rice (1945). While results on moments are easy to obtain, explicit results for the actual probability distributions are not usually available. However, in the important case when the level of values of interest is far below the mean, the asymptotic independence of up-crossing times makes it possible to derive simple approximate distributions. (See Cramer and Leadbetter (1967) page 256, Keilson (1966).) There is a dearth of particular examples of processes and trends for which the distributions of interest are known exactly. Such examples could give useful experience of the form of distribution to be expected in typical cases, and could serve as material on which to test out approximate methods. The object of the present paper is to provide an example of this kind. One process for which exact results are available in the trend-free case is the Ornstein-Uhlenbeck process, i.e., the stationary Gaussian Markov process X(t) generated by

This paper deals with the influence on the surface profile of the Greenland ice sheet, of an accumulation which increases between the ice divide and the coast, and of the thermal softening of the lowermost layers. It is concluded that the form of the surface of the profile measured by E.G.I.G. can be described by Glen's law with the exponent n = 3.5. The assumption is made that the bottom differs everywhere from the pressure melting point by a constant amount. This assumption is dropped in the second part of the paper. On the basis of the measured surface profile it is shown that the maximum increase of the bottom temperature is a few degrees within a range of 300 km. In view of the increasing surface temperature and heat of friction towards the outer edge it is concluded that, relatively close to the ice divide, that ice at the bottom must be temperate. Therefore we concluded that friction forces are preventing the ice from slipping on the bedrocks.

In considering the origin of reproductive isolation, two theories, which are not mutually exclusive, have been advanced. Muller (1939) considers that reproductive isolation between two subspecies is just a consequence of their genetic divergence following selection in geographic isolation. In this way, as the divergence proceeds, genes that prevent free interbreeding will be incorporated into the genotypes of the two subspecies. On the other hand, Dobzhansky (1940) holds that divergence does not necessarily imply reproductive isolation. In becoming more adapted to their particular environments, each subspecies may diverge far enough so that hybrids between them are less well adapted to the available habitats than either parental subspecies. Natural selection then will work to build up a gene system for further reproductive isolation. The formation of ill-adapted hybrids is prevented, avoiding in this way a wastage of reproductive potential and food resources. Grant (1966) proposed that the process of selection for reproductive isolation be designated as the Wallace effect. In animals, there are some interesting experiments to test the origin of reproductive isolation. Koopman (1950) was able to demonstrate a marked intensification of the reproductive isolation between Drosophila pseudaoobscura and D. persimilis as a result of selection. The experiment was conducted below 16 C where little isolation between these two species prevails. Using stocks marked with different recessive genes, all hybrid offspring could be readily detected since they carried both dominant genes. All hybrid progenies were then eliminated in every generation, and in a few cycles of selection the isolation increased substantially. Using a similar technique, Knight et al. (1956) attempted to develop sexual isolation between two stocks within a species. Experiments were conducted with D. melanogaster for 33 to 38 generations and a partial sexual isolation was obtained. Experimental studies on the origin of reproductive isolation in plants are almost completely lacking. The present paper deals with the results obtained after selecting for intraspecific reproductive isolation between two varieties of maize. Different recessive marker genes were used in the two populations. Hybrids, resulting from heterogamic fertilization, were readily recognized and discarded in every generation. Since Zea mays L. has, perhaps, more genetic variability, it seems that except for the generation interval, maize is better material for such a study than is Drosophila. In fact, within the single species Zea mays L. over 250 races have been described as occurring in different areas of the Americas (Wellhausen et al., 1952, 1957; Hatheway, 1957; Roberts et al., 1957; Brieger et al., 1958; Brown, 1960; Grobman et al., 1961; Ramirez et al., 1961; Timothy et al., 1961, 1963). Some of these races are quite ancient, about 4000 years old, and some are of recent origin, being the result of hybridization among more ancient ones. Many kinds of variation in 1 Supported in part by grants from the Program in Agricultural Sciences of The Rockefeller Foundation. 2 This paper is part of the homage the Brazilian Society of Genetics pays to Prof. Th. Dobzhansky on his 70th birthday in acknowledgment of his effective role in developing genetics in Brazil.

A Brief note in Science in the spring of 1943 reported the find of an association of human burials and artifacts with the remains of Camelops, Equus, and Bison, near Tranquillity, Fresno County, California. The present paper is a more detailed description of the cultural remains and a fuller report of the writer's findings at the site prior to 1944. Excavations in that year by the University Museum, University of Pennsylvania, are mentioned below. They will be described in a forthcoming paper by Lloyd and Satterthwaite, to include a critical examination of the evidence for and against the contemporaneity of the human and extinct mammal remains.

The definition and some of the properties of what may be called a Perron second integral (P2-integral) were given in a previous paper [4]. This integral starts with a function f(x) defined in an interval (a, c) and goes directly to a second primitive F(x) with the property that the generalized second derivative D2F is equal to f(x) for almost all x in (a, c). In the present paper the definition is changed slightly and further properties are deduced.

This paper arose out of an effort to present a result which simplifies the application of the theory of blocks to what is often called the theory of exceptional characters. In order to obtain the most effective use of this result, it is necessary to reformulate part of this theory. Thus we present a development which explains an application of R. Brauer’s main theorem on generalized decomposition numbers. In particular, we improve a result of D. Gorenstein and the author [8; Proposition 25]. These results are needed in a forthcoming paper and will simplify somewhat the use of this theory in existing papers. We are interested principally in determining the values of certain irreducible characters on trivial intersection subsets. The organization of the theory presented here is influenced by an exposition of M. Suzuki [13] and uses concepts introduced by W. Feit and J. G. Thompson [7]. Also it is hoped that this exposition will serve as an introduction to the theory.

The paper considers how to calculate the budget history of a glacier from the historical record of its advance and retreat. The linearized differential equations developed in previous papers are used to compute a series of coefficients g(n). The g(n) operate directly on a record of annually spaced observations of the position of the glacier terminus, to give the time variation of net budget. The g(n) are calculated for South Cascade Glacier, Washington, U.S.A. and for Storglaciären, Kebnekaise, Sweden, and are applied to the terminus records of these two glaciers. The results indicate that the figures for the retreat during individual years are not sufficiently reliable to deduce annual net budget changes with confidence. But the coarser features of the terminus records give net budget figures that agree well with the 9 yr. or 10 yr. means of recent observations. The theory extends the generalized budget record of Storglaciären back by several decades. and shows an increase in net budget in the 1930’s that is not immediately apparent in the terminus record.

HNN groups have appeared in several papers, e.g., [3; 4; 5; 6; 8]. In this paper we use the results in [6] to obtain a structure theorem for the subgroups of an HNN group and give several applications.We shall use the terminology and notation of [6]. In particular, if K is a group and {φi} is a collection of isomorphisms of subgroups {Li} into K, then we call the group1the HNN group with base K, associated subgroups { Li,φi(Li)} and free part the group generated by t1, t2, …. (We usually denote φi(Li) by Mi or L–i.) The notion of a tree product as defined in [6] will also be needed.

Throughout this paper, Z is the ring of integers, ƒ*(t) (ƒ(t)) is an integer monic (co-monic) polynomial in the indeterminate t (i.e., each coefficient of ƒ* (ƒ) is in Z and its highest (lowest) coefficient is 1 (5, p. 121, Definition) and M* (M) is the multiplicative semigroup of all integer monic (co-monic) polynomials ƒ* (ƒ) having no constant term. In (3, Theorem 2), Herstein proved that if R is a division ring with centre C such that1then R = C. In this paper we seek a generalization of Herstein's result to semi-simple rings. We also study the following condition:(1)*Our results are quite complete for a semi-simple ring R in which there exists a bound for the codegree ofƒ (ƒ*) (i.e., the degree of the lowest monomial of ƒ(ƒ*)) appearing in the left-hand side of (1) ((1)*).

Sanderson (Canad. Math. Bull., 8 (1965), 505–513), considering a nonempty collection Σ of left ideals of a ring R, with unity, defined the concepts of “Σ-injective module” and “Σ-essential extension” for unital left modules. Letting Σ be an idempotent topologizing set (called a σ-set below) Σanderson proved the existence of a “Σ-injective hull” for any unital left module and constructed an Utumi Σ-quotient ring of R as the bicommutant of the Σ-injective hull of RR. In this paper, we extend the concepts of “Σinjective module”, “Σ-essentialextension”, and “Σ-injective hull” to modules over arbitrary rings. An overring Σ of a ring R is a Johnson (Utumi) left Σ-quotient ring of R if RR is Σ-essential (Σ-dense) in RS. The maximal Johnson and Utumi Σ-quotient rings of R are constructed similar to the original method of Johnson, and conditions are given to insure their equality. The maximal Utumi Σquotient ring U of R is shown to be the bicommutant of the Σ-injective hull of RR when R has unity. We also obtain a σ-set UΣ of left ideals of U, generated by Σ, and prove that Uis its own maximal Utumi UΣ-quotient ring. A Σ-singular left ideal ZΣ(R) of R is defined and U is shown to be UΣ-injective when Z Σ(R) = 0. The maximal Utumi Σ-quotient rings of matrix rings and direct products of rings are discussed, and the quotient rings of this paper are compared with these of Gabriel (Bull. Soc. Math. France, 90 (1962), 323–448) and Mewborn (Duke Math. J. 35 (1968), 575–580). Our results reduce to those of Johnson and Utumi when 1 ∊ R and Σ is taken to be the set of all left ideals of R.

Soon after the publication of Tutte's paper [5] on m-cages, H. S. M. Coxeter asked in a letter to the author whether one-regular graphs of degree 3 exist. The purpose of the following paper is to show by an example that the answer is in the affirmative.

This discussion deals with the theoretical reasoning that underlies the type-variety method of ceramic analysis. Not only do pottery "types" and "varieties" embody sets of recognizably distinct attributes and impart particular cultural, areal, and temporal connotations, but they are also meaningful entities of cultural interpretation. 1. "Varieties" that have been tested by continued close study and by increases in knowledge concerning their nature and range of variation are close approximations and reliable indicators of original ceramic manifestations due to individual or small social group variation in a society. 2. Ceramic "types" represent the combining of a number of attributes into abstract conceptions which, when executed in clay by potters, are acceptable to them and a majority of others within their cultural configuration. Types are summations of individual or small social group variation consistent with boundaries imposed by the interaction of individuals on a societal level and determined by the operative value system present in any society. Pottery types are therefore representative of cultural phenomena. 3. Ceramic types are cultural derivatives and can be related one to another through space and time. Relationships of this kind are recognized as "ceramic systems" and "ceramic sequences." O NE OF THE primary reasons why the archaeologist finds it advantageous to analyze pottery in terms of typological concepts is that a pottery type, once recognized and described, will inevitably repay the analyst by providing him with certain information otherwise unavailable or only available in part. Over thirty years ago Vaillant (1930: 9) observed "that the backbone of most of the New World chronologies is variation in pottery types and that the arrangement of a tribal ceramic into chronological divisions is not only very technical but also highly interpretive and impressionistic." The minimum elemental information now held to be usually bound up in a type is that the ceramic entity, such as Benque Viejo Polychrome, a type, is a specific kind of pottery embodying a unique combination of recognizably distinct attributes, and that this pottery, as well as being the product of a certain cultural *This paper was presented at the 24th Annual Meeting of the Society for American Archaeology, May 1, 1959, Salt Lake City, Utah. configuration, has a definable areal distribution and temporal significance. When the analyst has set down a pottery type description, then the taxonomic device, or pottery type name, he attaches to the described ceramic entity, comes to stand for the entire set of recognizably distinct attributes plus a particular cultural affiliation, areal distribution, and temporal connotation. In addition anthropological schemes of classification such as this should be means to define and weigh Ithe relative importance of variation and regularity in human society. Most classifications used in anthropology find their rationale in that part of the scientific method which subscribes to the basic assumption that out of what may appear to be a mass of variation, regularities or laws of (cultural) process may be discerned, delineated, and described. A scheme of ceramic classification has been proposed in three recent papers that is geared especially to the analysis of material aspects of human culture and that also provides a foundation upon which theoretical cultural considerations can be based. These three papers outline and define what is called the "typevariety concept" (variety, Wheat, Gifford, and Wasley 1958; type, Phillips 1958; type-variety concept, Smith, Willey, and Gifford 1960). In he following discussion whenever the term "type" is used, it is intended strictly as defined by Phillips. His usage corresponds to the "type cluster" in the Wheat-Gifford-Wasley nomenclature. For present purposes the terms "type," " ariety," and "type-variety concept" are words with a special meaning not to be construed in settings other than those set forth in the three interpretations cited. Within the context of the type-variety concept, variation as a recognizable reality can be thought of as the product of the individual or of relatively small social groups in human society. "Individual and small social group variation" is meant to encompass not only (at a minimum) actual individual variation, but also variation due to the activities of potters in a village segment, a village, a group of small villages, a community, or a series of communities (at a maximum). When entire cultural configurations are taken into account certain regu