When the first edition of the Encyclopedic Dictionary of Mathematics appeared in 1977, it was immediately hailed as a landmark contribution to mathematics: "The standard reference for anyone who wants to get acquainted with any part of the mathematics of our time" (Jean Dieudonné, American Mathematical Monthly). "A magnificent reference work that belongs in every college and university library" (Choice), "This unique and masterfully written encyclopedia is more than just a reference work: it is a carefully conceived course of study in graduate-level mathematics" (Library Journal).The new edition of the encyclopedia has been revised to bring it up to date and expanded to include more subjects in applied mathematics. There are 450 articles as compared to 436 in the first edition: 70 new articles have been added, whereas 56 have been incorporated into other articles and out-of-date material has been dropped. All the articles have been newly edited and revised to take account of recent work, and the extensive appendixes have been expanded to make them even more useful. The cross-referencing and indexing and the consistent set-theoretical orientation that characterized the first edition remain unchanged,The encyclopedia includes articles in the following areas: Logic and Foundations; Sets, General Topology, and Categories; Algebra; Group Theory; Number Theory; Euclidean and Projective Geometry; Differential Geometry; Algebraic Geometry; Topology; Analysis; Complex Analysis; Functional Analysis; Differential, Integral, and Functional Equations; Special Functions; Numerical Analysis; Computer Science and Combinatorics; Probability Theory; Statistics; Mathematical Programming and Operations Research; Mechanics and Theoretical Physics; History of Mathematics.Kiyosi Ito is professor emeritus of mathematics at Kyoto University.

3099317. [ 101 J. Lindenstrauss and L. Tzafriri, On complemented subspaces problem, Israel J. Math., 9 (1971) 2633269. 38 (XXI.1 4) Bernoulli Family The Bernoullis, Protestants who came originally from Holland and settled in Switzerland, were a signitïcant family to the mathematics of the 17th Century. In a single Century, the family produced eight brilliant mathematicians, ah of whom played important roles in the development of calculus. The brothers James (1654-1705) and John (1667- 1748) and

function f satisfying If(z)1 < 1 for IzI < 1 is 1 c, . . . c, ~ 1 . Cl cn-1 30, CPI n=1,2,... q-1 . . . 1 (C. Carathéodory). Furthermore, for n = 1,2, , when we regard (cl, . , c,) as a point of complex n-dimensional Euclidean space, we cari determine the domain of existence of points satisfying this criterion by Carathéodory. This result is generalized for coefficients of the Laurent expansion of a function that is holomorphic and single-valued in an annulus. Next, there are some results for

ý2(t) be a tseparable modification of Y*(t). Then the discontinuities of almost a11 sample functions of F1(r) are of the first kind. If we set Y$(t)= y2(t +), Y;(t) is a +modifïcation of Y2(t). and almost all sample functions of Y;*(t) are right continuous and have a left-hand limit at every t. In the study of the process Y2(t), it is always convenient to take such a modification. Thus we give the following general definition: An additive process is called a Lévy process if it is continuous in

K/k is a relative algebraic number field. Let 0 be the principal order of K. For a (fractional) ideal a of k, Da is an ideal of K. We Write Dn=E(a) and call E(a) the extension of a to K. For ideals a, b of k, we have E(ab)=E(a)E(b)and E(a)nk=a. Let ‘Pi : K + C be k-isomorphisms (i = l,...,n),wheren=[K:k].WewriteK’= ‘V(K) an dA (il =Y-(A)for AcK.Fora’i idéal 9I of K, 91”’ I {A”’ 1A E 9I) is an ideal of K”‘, and 9I(‘) is called the conjugate ideal of ‘11 in K”‘. Let L be the composite lïeld of K”‘,

obtained further results in this direction. M. Noether and geometers of the Italian school, such as F. Enriques, G. Castelnuovo, and F. Severi, studied algebrogeometric properties of algebraic surfaces. In particular, the Italian school geometers recognized the importance of irregularity and thoroughly investigated its geometric meaning. In the early 20th Century they succeeded in constructing the great editïce of the theory of algebraic surfaces. Though some of their results lack rigorous proof,