Universal Extension Research Articles

Complete models of differential fields

Introduction. This work constitutes a step in the direction of a global theory of differential algebra. The main result (Theorem 1 of ?2) is the following: Let K be an ordinary differential field of characteristic zero. Then there is a premodel of K with the property that every maximal differential homomorphism from a differential subring of K, whose differential field of quotients is K, induces a point on this pre-model. Under a certain condition these points will be uniquely determined. It is not yet known whether there exist differential fields of the form k , where the xl belong to some universal differential field extension of k, which do not admit complete models over k. Even in the case where x1, ..., x5 are differentially algebraically independent the answer is not apparent. For definitions, see the discussion preceding the above-mentioned Theorem 1. The author would like to thank Professors A. Seidenberg and E. R. Kolchin for their helpful comments and suggestions. NOTATION AND CONVENTIONS. We denote the integers, rationals, reals, and complex numbers by Z, Q, R, C, respectively, and usually consider Z as a differential ring, and Q, R, and C as differential fields of constants. If R is an integral domain, we write qf (R) for its quotient field. If S is a differential integral domain, S{y} will denote the ring of differential polynomials with coefficients in S. The letter y will represent a differential indeterminate, while x,, z will stand for differential quantities which may or may not be differential indeterminates. We begin by examining some of the elementary properties of the category D of differential rings and differential homomorphisms (these are ordinary differential rings, although many of the arguments remain valid in the partial case). For the moment, no restriction is placed on zero divisors or characteristic. First of all, D admits coproducts in the sense that if A, B E D we can define A 0 B by taking the usual tensor product of A and B (considered as Z-algebras) and setting (a 0 b)'=a' X b+a X b'. This makes A X B into a differential ring satisfying the conditions for a coproduct in a category. In fact, we have differential

Justice and the Individual in the Republic

THE INTERPRETATIONS of the Republic which limit justice to the philosopher rulers seriouLsly challenge Plato's renown as the first great ethical thinker. If he does, in fact, contend that only a philosophically gifted few can be just, he has surely violated the elementary principle of a universal extension among men of the two fundamental ethical concepts of the right and the good. Ethical theories as diverse as those of Kant and Hume agree on the universal or well-nigh universal extension of these two concepts. Since justice, the right, is both the necessary and sufficient condition for happiness (the good), Plato firmly believes no person can ever be happy without it. The vast numbers of mankind, therefore, cannot be happy in their lack of justice or, indeed, in their wickedness. Yet we need not plead that Plato must have intended a universal extension of the right and good or else he would not be the great philosopher that he supposedly is. As several scholars have recently called to our attention, Plato in the Republic has distinguished between the justice and the other virtues of the individual (which I shall refer to as personal justice and personal virtues) and the virtues of the polis 1. After presenting a broader perspective for this interesting reading of the Republic, I shall consider what kind of knowledge personal justice requires and how the citizens may attain such knowledge. Adequate justification of the personal virtues requires some consideration of their epistemological ground, for the necessary entailment of virtue and knowledge is as well established in the traditional interpretation of Plato as the fact that the citizens of the ideal polis are incapable of rational deliberation. Early in the Republic, Socrates lays down almost as a rudimentary law of nature the principle of a natural equality among all men:

Structure theory of faithful rings. II. Restricted rings

The first paper of this series(2) concerned itself mainly with closure operations on a lattice. This paper applies these results to the global structure theory of a faithful ring and its modules. A ring R is called faithful if aR = 0 implies a= 0. Let ? (?2) be the lattice of right (two-sided) ideals of R, and c={A; AC?', AnA =0, A=All}. The set i: becomes a Boolean algebra with the obvious definition of the union operation. In case 3Y is complete, R is called a restricted ring. Being complete, 9Y induces a closure operation f on ? and ?V. It is shown that if f is homogeneous, then there exist irreducible rings Ai such that i AiCR C As, where j ( *) designates the discrete (full) direct sum and 47 is a universal extension ring of Ai. A reducible ring is shown to be restricted, where R is reducible if and only if for every pair A, B of ideals of R with zero intersection, there exist ideals A'DA and B'DB also with zero intersection such that A'nB'I-0. Modules of a faithful ring are studied in the fourth section. It is shown that every suitably restricted closure operation on ? induces a closure operation on Y, the lattice of submodules of a R-module M. If 'A, A CR, designates the annihilator of A in M, then it is shown in the fifth section that 3c 'A; A EDY } is a Boolean algebra. In the final two sections, it is assumed that the ring R and the R-module M have the property that for every nonzero element x in R or M there exists a nonzero A G2 such that xa0 for every nonzero aEA. It is shown that R is restricted, aind that every A in ? (M1Z) has a unique maximal essential extension A, (At). For the closure operation s on ? so defined, 5 is proved to be the center of the lattice 28, and similarly for xC in St. Imbedding ll in its unique minimal injective extension M, it is proved that the lattice o't is isomorphic to the lattice of principal right ideals of the centralizer C of R over M. If s is atomic, e is a full direct sum of primitive rings with minimal right ideals. 1. Faithful rings. If R is a ring, then ?(R)(2'(R)) will designate the lattice of all right (two-sided) ideals of R. We shall upon occasion write S or ?' if the ring in question is obvious. For each subset A of R, AI (Ar) will

Realistic Independent-Particle Model of the Nucleus

A phenomenological central potential is explored which is characterized by a universal depth (${V}_{0}$), a universal surface extension ($d$), and an inner radius function $a={a}_{1}{A}^{\frac{1}{3}}+{a}_{0}$. The four parameters are adjusted to be in accord with experimental information obtained from low-energy neutron scattering and from neutron separation energies. The results ${V}_{0}=40$ Mev, $d=1$, ${a}_{1}=1.32$, and ${a}_{0}=\ensuremath{-}0.8$ (all in units of ${10}^{\ensuremath{-}13}$ cm), are in reasonable accord with what might be expected from other experimental and theoretical considerations. Since the experimental data used in the adjustment process bound the energy region in which the discreteness of nuclear energies is most evident, one might expect that the eigenvalues and eigenfunctions based upon this phenomenological model would furnish a highly realistic set for the theoretical study of low-energy nuclear transitions within the framework of the independent-particle model of the nucleus.The energy eigenvalues and the important parameters for the energy eigenfunctions are calculated by a procedure suitable for the restricted type of potential under consideration. A simple method is presented for handling small perturbations, particularly the Coulomb perturbation. In the latter case, it is shown that the derived proton eigenvalues agree with experimental proton separation energies, provided one introduces an anomalous attractive potential which cancels approximately one-half of the classical Coulomb potential acting upon an individual proton. The implications of these results are discussed.

On the Galois Theory of Differential Fields

1. Summary. In a preceding paper [7] there was presented a theory, for a certaini kind of differenitial field extension called strongly normal. The of a strongly normal extension is enidowed with a structure very much like that of a variety, as studied by Weil [14]. In the present paper the study of such groups is renewed for the purpose of clarifyinig their conniection with varieties on the onie hand and with strongly normal extensiolis on the other. Consider a differential field S of characteristic 0 with algebraically closed field of constants W, and a strongly normal extension A of 5; suppose given a universal extension 5* of X, and denote the field of constants of 5* by AV. For reasons of convenience we use the term Galois of & over 7 for the of all (ipso facto strong) isomorphisms of & over 5, and not for its subgroup conisisting of all automorphisms of & over J. The field R*, being algebraically closed anid of finite transeendence degree over A, may be used as a uniiversal domain for geometry (Weil [13], ch. I, ? 1); it is the varieties of this geometry, defilned over E and slightly generalized to permit varieties which are reducible (i. e. which have more thani one comiponent), whieh we consider. By an algebraic group we miean either a or a variety as above. Chapter I is primarily a study of rational homomorphisms of groups into groups; these seenm to be the homomorphismls appropriate to the consideratioin of groups. Ratioinal homomorphisms of varities inlto varieties, without restrictioin to fields of eharacteristic 0, were considered by Weil [14] (who omitted the adjective rational). Specializing the coneept of rational homomorplhism we obtaini that of birational isomorphisin. In Chapter II it is showni that every is birationally isomorphic to a variety, and conversely that every irreducible variety is birationallv isomorphic to a group; this

An existence theorem in the algebraic study of homogeneous linear ordinary differential equations

Introduction. The Picard-Vessiot theory is a Galois theory of homogeneous linear ordinary differential equations. Unlike ordinary Galois theory, however, in which fixing a universe automatically determines the zeros of a given polynomial, Picard-Vessiot theory is complicated by the fact that a given homogeneous linear ordinary differential polynomial L(y) of order ?1 possesses infinitely many zeros. To simplify matters, therefore, one wishes to choose those zeros of L(y) which are the simplest in a certain sense. More specifically, it is seen in a paper (to appear) generalizing the PicardVessiot theory as presented by Kolchin [1] that the algebraic and topological structures involved in that generalization are strongly dependent on the constants introduced when one adjoins zeros of L(y) to the ground field. It is desirable, therefore, to choose zeros of L(y) in such a manner that the constants introduced by these zeros are subject to certain restrictive conditions. This requires a theorem asserting the existence of zeros of L(y) for which these restrictive conditions are satisfied. The main result of this paper is a theorem on the existence of this special type of zeros of L(y). We fix an abstract ordinary differential field F of characteristic zero and with field of constants C (i.e. F is an abstract field of characteristic zero in which a derivation is defined, and C is the set of all elements of F which have derivative equal to 0). We shall assume throughout that we are given Q, a universal differential extension field of F, the existence of which is proved in [3, Chap. I, ?5]. This universal extension will contain all the elements which enter the discussion. In particular, Q will contain an algebraic closure C of C. We use the standard notation for derivatives; thus, for a EQ the successive derivatives of a are denoted a', a, , ( . . For the adjunction of elements to differential fields (for example, to the differential field F), we shall use F( . . . ) to denote ordinary field adj unction, F { ... } to denote differential ring adjunction, and F( ... ) to denote differential field adjunction: thus, for example,

Note on Cambrian Hyolithes Sandstones from Nuneaton

The fossiliferous rocks belonging to the now famous Olenelluz Zone or basement Cambrian series were practically unrecognised by the majority of geologists ten years ago. They are now, however, known to have an almost universal extension on both continents north of the Equator, from India to Lapland in the Old World, and from Virginia to British Columbia in the New. Fossiliferous beds belonging to this Lower Cambrian zone have been recently found to occur in their proper place among the well-known Cambrian quartzites of Nuneaton and Atherstone, and the fossils, which are many of them well-preserved, include Hyolithes, Coleoloides, Kutorgina, Sccnella , etc., of the same type as those which occur in the lowest Cambrian rocks of Scandinavia and Newfoundland. The first indications of fossils in this Midland zone were detected by Dr Stacy Wilson in July 1895, and the actual fossil-bearing beds were subsequently followed and worked out in the field by myself. It is exceedingly interesting to find the same general succession of Cambrian rocks in the English Midlands as in all those regions in Britain and abroad, where the well-known Cambrian quartzite band forms the lowest bed of the Cambrian system. The fossil-bearing layers in the Midlands come in, as usual, between two sets of arenaceous strata, namely between a lower sandy series—the well-known Nuneaton or Lickey quartzite—and a higher and thinner arenaceous series, which appears to be the local equivalent of the well-known Salterella Grits and Hollybush Sandstone of other districts. It thus becomes more clearly evident

Research in Education1

Research in Education1