Showing papers in "Transactions of the American Mathematical Society in 1953"

Classes of recursively enumerable sets and their decision problems

Abstract: 1. Introduction. In this paper we consider classes whose elements are re-cursively enumerable sets of non-negative integers. No discussion of recur-sively enumerable sets can avoid the use of such classes, so that it seems desirable to know some of their properties. We give our attention here to the properties of complete recursive enumerability and complete recursiveness (which may be intuitively interpreted as decidability). Perhaps our most interesting result (and the one which gives this paper its name) is the fact that no nontrivial class is completely recursive. We assume familiarity with a paper of Kleene [5](2), and with ideas which are well summarized in the first sections of a paper of Post Í7].

Cohomology of group extensions

Abstract: This method is based on the Cartan-Leray spectral sequence, [3; 1 ], and can be generalized to other algebraic situations, as will be shown in a forthcoming paper of Cartan-Eilenberg [2]. Since the details of the Cartan-Leray technique have not been published (other than in seminar notes of limited circulation), we develop them in Chapter I. The auxiliary theorems we need for this purpose are useful also in other connections. In Chapter II, which is independent of Chapter I, we obtain a spectral sequence quite directly by filtering the group of cochains for G. This filtration leads to the same group E2=H(G/K, H(K)) (although we do not know whether or not the succeeding terms are isomorphic to those of the first spectral sequence) and lends itself more readily to applications, because one can identify the maps which arise from it. This is not always the case with the first filtration, and it is for this reason that we have developed the direct method in spite of the somewhat lengthy computations which are needed for its proofs. Chapter III gives some applications of the spectral sequence of Chapter II. Most of the results could be obtained in the same manner with the spectral sequence of Chapter I. A notable exception is the connection with the theory of simple algebras which we discuss in ?5. Finally, let us remark that the methods and results of this paper can be transferred to Lie Algebras. We intend to take up this subject in a later paper.

A theorem on holonomy

Abstract: Introduction. The object of this paper is to prove Theorem 2 of §2, which shows, for any connexion, how the curvature form generates the holonomy group. We believe this is an extension of a theorem stated without proof by E. Cartan [2, p. 4]. This theorem was proved after we had been informed of an unpublished related theorem of Chevalley and Koszul. We are indebted to S. S. Chern for many discussions of matters considered here. In §1 we give an exposition of some needed facts about connexions; this exposition is derived largely from an exposition of Chern [5 ] and partly from expositions of H. Cartan [3] and Ehresmann [7]. We believe this exposition does however contain one new element, namely Lemma 1 of §1 and its use in passing from H. Cartan's definition of a connexion (the definition given in §1) to E. Cartan's structural equation. 1. Basic concepts. We begin with some notions and terminology to be used throughout this paper. The term \"differentiable\" will always mean what is usually called \"of class C00.\" We follow Chevalley [6] in general for the definition of tangent vector, differential, etc. but with the obvious changes needed for the differentiable (rather than analytic) case. However if

again instead of d and 8<(> for the induced mappings of tangent vectors and differentials. If AT is a manifold, by which we shall always mean a differentiable manifold but which is not assumed connected, and mGM, we denote the tangent space to M at m by Mm. If W is a vector field at M we denote its value at m by W(m), and if Xi, • • • , xk is a coordinate system of M we always write X\\ ■ ■ ■ , Xk for the corresponding partial derivatives, i.e. Xi = d/dxi. We use the word \"diffeo\" for a 1:1 differentiable mapping of one manifold onto another whose inverse is also differentiable and call the manifolds diffemorphic. If W, W are vector fields we write [W, W] for WW'—W'W (opposite to [ó]). Rk will always denote ¿-dimensional Euclidean space of all ¿-tuples of real numbers and 8i, ■ • • , 8k will always denote the canonical unit elements there, i.e. 8, = (8lj, 82j, • ■ ■ , 8nj). Let G be a Lie group (i.e. a differentiable group) acting differentiably and effectively on a manifold F. It fGF and gGG we write gf for the image of/ under the action of gGG and if i is a tangent vector at/ we write gt for the image of i (which will be a tangent vector of gf) under g. If 0 is a manifold we denote by (0, F, G) the family of all transformations of OXF-^OXF of the form: (o,f)—>(o, t(o,f)) when i is any differentiable mapping of OXF—*F

On polya frequency functions. iii. the positivity of translation determinants with an application to the interpolation problem by spline curves(

Abstract: 1 A frequency function Δ(x), i e, a non-negative measurable function satisfying the inequalities $$ 0 < \int_{ - \infty }^\infty {\Lambda \left( x \right)dx < } \infty , $$ is called a Polya frequency function provided(2) it satisfies the following condition: For every two sets of increasing numbers $$ \begin{array}{*{20}{c}} {{x_1} < {x_2} < \cdots < {x_n},}&{amp;{y_1} < {y_2} < \cdots < {y_n},}&{amp;n = 1,{\mkern 1mu} 2,{\mkern 1mu} \cdots ,} \end{array} $$ (1) we have the inequality $$ D \equiv \det {\left\| {\Lambda \left( {{x_i} - {y_i}} \right)} \right\|_{1,n}}0 $$ (2)

The mathematics of second quantization

Abstract: quantum field theory. Although few nonspecialists have had opportunity to become familiar with the language of modern pure mathematics, quantum theory seems to have reached a point where it must use that language if it is to find a genuine escape from the divergence difficulties. Divergence can not be properly coped with when convergence itself has never been rigorously defined. In the classical analysis of real and complex numbers, results, even correct results, can be obtained by algebraic manipulation of formal power series; but these numbers are not just algebras, they are topological algebras, and only with Cauchy's introduction of the epsilon-delta treatment was mathematics provided an explicit method of separating sense from nonsense. Similarly, in the modern analysis of infinite-dimensional algebras results can be obtained by algebraic manipulation of formal expressions, but these results often require topological justification. One standard way of introducing a topology into the algebra of observables is to make them operators on a Hilbert space. This method, which does not seem to be extensively employed in quantum electrodynamics, can be used to construct a mathematically rigorous formalism the manipulation of which is directly followable by one's physical intuition. This construction requires the exercise of two dissimilar disciplines, mathematics and physics, so the exposition is divided into two parts upon which relative emphasis can be adjusted to suit individual tastes. In particular, physicists can greatly simplify the mathematics by ignoring: (1) operator-domain considerations (as is done here in the derivation of the Yukawa-potential); (2) discussions involving the group algebra of the symmetric group (since only the FermiDirac and Bose-Einstein cases have ever actually occurred); (3) material depending on the simply-connected covering group of the Lorentz group (since it is not needed to derive Maxwell's equations). However, Part I is empty, unmotivated mathematics without Part II; and Part II does not exist without Part I. The two are designed to be read, not consecutively, but in parallel. Sections are numbered accordingly. I would like to thank Professor I. E. Segal for liberal use of his time and advice in the preparation of this paper. It is to be submitted to the Depart

Convex bodies and periodic homeomorphisms in Hilbert space

Abstract: Introduction. We are going to discuss certain topological properties of infinite-dimensional normed linear spaces, the results being most complete for Hilbert space 'i. Let us begin by describing some results and questions of previous authors which are closely related to those of the present paper. From A. Tychonoff's fixed-point theorem ([31], 1935)(2) it follows that in the weak topology, the unit cell C= {x| |x|| < 1 } of 'i must have the fixedpoint property; that is, every weakly continuous map of C into itself admits at least one fixed point. In the norm topology, on the other hand, S. Kakutani ([13], 1943) described a homeomorphism without fixed points of C onto itself. He used this to show that the unit sphere S ={x x = 1 } is contractible and is a deformation retract of C. At the end of his paper, Kakutani raised several questions. Are any two of 'i, C, and S homeomorphic? Does C admit a periodic homeomorphism without fixed points? What is the situation in general Banach spaces? In partial answer to the last question, J. Dugundji ([7], 1951) proved that the unit cell of a normed linear space has the fixed point property only if the space is finite-dimensional. P. A. Smith had proved ([26], 1941) that each prime-period homeomorphism of Euclidean n-space En must have a fixed point and asked ([9, p. 259], 1949) whether 6i admits a period two homeomorphism without fixed points. 0. H. Keller ([14], 1931) proved that the infinite-dimensional compact convex subsets of are mutually homeomorphic and all homogeneous. W. A. Blankinship ([3], 1951) showed that if XC, and Cl X is compact, then i-i-X is contractible. In the present paper we answer the questions of Kakutani and Smith, strengthen the theorems of Keller and Blankinship, and establish some further topological properties of convex bodies and periodic homeomorphisms in 'i. The principal tools employed are (a) Mazur's homeomorphism [22] of the space (L') onto the space (L2); (b) the existence in every nonreflexive normed linear space of a decreasing sequence of bounded closed convex sets with empty intersection; (c) the existence in (L') of a one-parameter con-

On the algebra of networks

Abstract: The matrix elements Gij and the coordinates vi have a direct physical interpretation. E is related to energy or power. The state of the system is determined by a variational principle involving E. If the variations avi are arbitrary, then it is clear that G and its inverse would play a fundamental role. However, it often happens that the coordinates are subject to n-m homogeneous linear constraints. The usual procedure, in this case, is to express the vi in terms of a set of unconstrained coordinates qi. Hence

Group elements of prime power index

Abstract: The index [G:g] of the element g in the [finite] group G is the number of elements conjugate to -g in G. The significance of elements of prime power index is best recognized once one remembers Wielandt's Theorem that elements whose order and index are powers of the same prime p are contained in a normal subgroup of order a power of p and Burnside's Theorem asserting the absence of elements of prime power index, not 1, in simple groups. From Burnside's Theorem one deduces easily that a group without proper characteristic subgroups contains an element, not 1, whose index is a power of a prime if and only if this group is abelian. In this result it suffices to assume the absence of proper fully invariant subgroups, since we can prove [in ?2] the rather surprising result that a [finite] group does not possess proper fully invariant subgroups if and only if it does not possess proper characteristic subgroups. A deeper insight will be gained if we consider groups which contain "many" elements of prime power index. We show [in ?5 ] that the elements of order a power of p form a direct factor if, and only if, their indices are powers of p too; and nilpotency is naturally equivalent to the requirement that this property holds for every prime p. More difficult is the determination of groups with the property that every element of prime power order has also prime power index [?3]. It follows from Burnside's Theorem that such groups are soluble; and it is clear that a group has this property if it is the direct product of groups of relatively prime orders which are either p-groups or else have orders divisible by only two different primes and furthermore have abelian Sylow subgroups. But we are able to show conversely that every group with the property under consideration may be represented in the fashion indicated. In ?5 we study the so-called hypercenter. This characteristic subgroup has been defined in various ways: as the terminal member of the ascending central chain or as the smallest normal subgroup modulo which the center is 1. We may add here such further characterizations as the intersection of all the normalizers of all the Sylow subgroups or as the intersection of all the maximal nilpotent subgroups; and the connection with the index problem is obtained by showing that a normal subgroup is part of the hypercenter if, and only if, its elements of order a power of p have also index a powrer of p. Notation. All the groups under consideration will be finite. An element [group] is termed primary, if its order is a prime power;

Isolated singularities of solutions of non-linear partial differential equations

Abstract: A significant role in the theory of linear elliptic second-order partial differential equations in two independent variables has been played by the concept of the fundamental solution. Such a function is a single-valued solution of the equation, regular except for an isolated point at which it possesses a logarithmic singularity. For the special case of the Laplace equation =+OiSwthis solution can be uniquely characterized as the only (nonconstant) solution which exhibits radial symmetry. The requiirement of radial symmetry leads to an ordinary differential equation whose solution (up to a constant) is y log (x2+y2)"12. This function admits an important hydrodynamical interpretation as the velocity potential of an incompressible, nonviscous, two-dimensional source-flow. An amount 2w, of fluid is visualized as flowing in unit time out of a source at the origin (i.e. across every closed curve surrounding the source) and into a corresponding sink at infinity. The requirement of incompressibility may be relaxed by assuming a relation-called the equation of state-between the density p of the fluid and the velocity I V4 I. For adiabatic flows this relation takes the form p = [1((y1)/2) 1 V+| 2]1/(7-1), where y is the ratio of specific heats of the fluid. In this case the potential 4 satisfies not the Laplace equation but the nonlinear equation

The positive cone in Banach algebras

Abstract: This paper concerns Banach algebras which are real or * algebras and possess a unit. The principal method of at.tack is via an ordering of the algebra, the positive cone being the closure of the set of sums of squares (sums of elements xx*) in contrast to the positive open cone used by Raikov [9] (3) and others. An important role is played by an identity on norms, which together with a few preliminary lemmas is proved in ?1. In ?2 the real homomorphisms of a real commutative algebra are found to be the extreme points of the intersection of the dual cone and the unit sphere in the adjoint of the algebra, and the "real radical" is shown to consist of elements x such that -x2 is approximately a sum of squares. The theorem of Arens [1] characterizing real function algebras is derived. In ??3 and 4 these results are applied to * algebras. The new norm of an element x, which Gelfand and Naimark [3] introduced by means of positive functionals, is proved to be the square root of the distance from -xx* to the positive cone. Some results relating general * algebras to operator algebras, including the representation theorem of Gelfand and Naimark [2], are derived. In ?5, a refinement of the basic identity is established for the Fourier transform of a measure (discrete + absolutely continuous) on a locally compact Abelian group. R. V. Kadison [5 ] has recently investigated Banach algebras by means of an order relation. The positive cone he uses is identical with that used here only when 1 +xx* always has an inverse. The principal overlap with Kadison's work, outside of the deduction of certain known theorems by order methods, seems to be the geometric characterization of the real homomorphisms of a real algebra (see 2.1). Like Kadison's work, this paper is essentially self-contained. (Some notable exceptions occur in ?5.) 1. Preliminaries. 1.1. DEFINITIONS. A set C is a cone in a real Banach space R if it is closed, nonvoid, the sum of two members of C is a member of C, and non-negative scalar multiples of members of C are members of C. If C is a cone in R, then C', the dual cone, is the set of bounded linear functionals which are non

Some interconnections between modern algebra and mathematical logic

Abstract: Mathematical logic originated when mathematical methods were brought to bear on traditional questions of logic, especially the problem of what constitutes a valid proof. For the most part work in this field has remained close to foundational questions. Recent results, however, have indicated the possibility that the methods of mathematical logic may be of use in tackling specific problems in other branches of'mathematics. This possibility is illustrated in the present paper, which presupposes no knowledge of mathematical logic. 1. Some concepts of logic. We shall first focus our attention on algebraic structures consisting of a set D on which two binary operations are defined. Rings are familiar examples of such structures; but to begin with we do not make any restrictions on the nature of the binary operations. Instead, we proceed to construct a simple, specialized language, L, which can be used to refer to any one of our structures. First we provide two symbols, "+" and ".," to be used as names of the operations. Next we wish to have symbols (called individual constants) which can be used as names of particular elements in D. In the case of rings, for example, it is customary to employ the symbols "O" and "1" in this way; for our more general purpose, however, we shall reserve the letter "v," to be used with various distinguishing subscripts. In addition to referring to particular elements of D we wish to make general statements about all elements of D, and for this purpose we introduce further symbols "x," "y," "z," "X1," "Yi," (called individual variables), which will be used as variables whose range is D. Additional symbols are to include parentheses, an equality sign, and the following words: "not," "and," "or," "if," "then," "all," "exists." The symbols of L are used in constructing formulas and sentences as follows. An individual symbol (constant or variable) is called a term, and the result of putting a "+" or " " between two terms and enclosing the result in parentheses is also a term. Basic sentences are formed by placing an equality sign between two terms, and further sentences are built up from these basic ones in the following ways. If A and B are sentences, so are (not A), (A and B),

On $n$-dimensional Riemannian spaces admitting a group of motions of order $n(n-1)/2+1$

Abstract: Publisher Summary This chapter focuses on n-dimensional riemannian spaces admitting a group of motions of order n(n-1)/2+ 1. If an n-dimensional Rieinannian space admits a group of motions of the maximum order n(n+l)/2, then, the space is of constant curvature. An n-dimensional finslerian space for n>2, n ≠4, admits a group of motions of order greater than n(n-l)/2+ 1, then the space is Riemannian and of constant curvature(2). The largest group of motions in a space of constant curvature being of order n(n+1)/2, if it denote it by G, then Gr must be a subgroup of G. In an n-dimensional Euclidean space for n ≠4, n ≠8, any subgroup of the rotation group of order (n - 1) (n - 2)/2 fixes one and only one direction. If an n-dimensional Riemannian space for n ≠4 admits a group of motions of order n(n - 1)/2+ 1, then the group is transitive.

Some contributions to the theory of rings of operators. II

Abstract: The extension of von Neumann's work on factors to general rings of operators on Hilbert spaces of arbitrary dimension has been begun by Dixmier and Kaplansky in [1 ] and [3 ] (the numbers in brackets refer to the List of References at the end of the paper). It is the purpose of this paper to extend these results still further, in particular Chapter X of [5] and Chapters I to III of [8]. The general scheme of this paper is as follows: First, the constant C of von Neumann (Chapter X of [5 ]) is extended to an operator belonging to the center of a ring of semifinite type, such a ring being one with no type III part. Next, using techniques devised by Dye and von Neumann, this operator C (termed the coupling operator) is shown to be the chief invariant governing the spatial type of a ring. Finally, these results are applied to questions of topology in rings, yielding the fact that the strongest topology is purely algebraic along with the notion of semifinite subring. Besides these main results, we obtain various subsidiary results, in particular, conditions for strong and weak continuity of *-isomorphisms, continuity of the trace in various topologies, and conjugate isomorphism of a ring with its commutant. The notation of this paper is essentially that of [5] and [7], with but a few exceptions. Throughout the paper, the notation [Mx] will denote the closure in some Hilbert space H of the family of vectors I Ax } for A in a ring M and x a fixed vector. The symbol < between projections E and F (E < F) in a ring will denote the fact that E is equivalent to a subprojection of F belonging to M. This will denote a proper projection only when specifically stated. This particular notation is used because the printer does not have the symbol used in [51 for this relationship. The symbols U and n will be used in their usual sense of set theoretic union and intersection, the remaining ones being those standard in Hilbert space theory. In preparing this paper, we have received much valuable assistance from Professors I. E. Segal, I. Kaplansky, and P. R. Halmos, which we gratefully acknowledge. Note: Since this paper was written, Professor Segal has informed me that

Ideal lattices and the structure of rings

Abstract: It is well known that the set of all ideals(2) of a ring forms a complete modular lattice with respect to set inclusion. The same is true of the set of all right ideals. Our purpose in this paper is to consider the consequences of imposing certain additional restrictions on these ideal lattices. In particular, we discuss the case in which one or both of these lattices is complemented, and the case in which one or both is distributive. In §1 two strictly latticetheoretic results are noted for the sake of their application to the complemented case. In §2 rings which have a complemented ideal lattice are considered. Such rings are characterized as discrete direct sums of simple rings. The structure space of primitive ideals of such rings is also discussed. In §3 corresponding results are obtained for rings whose lattice of right ideals is complemented. In particular, it is shown that a ring has a complemented right ideal lattice if and only if it is isomorphic with a discrete direct sum of quasi-simple rings. The socle [7](3) and the maximal regular ideal [5] are discussed in connection with such rings. The effect of an identity element is considered in §4. In §5 rings with distributive ideal lattices are considered and still another variant of regularity [20] is introduced. It is shown that a semi-simple ring with a distributive right ideal lattice is isomorphic with a subdirect sum of division rings. In the concluding section a type of ideal, introduced by L. Fuchs [9] in connection with commutative rings with distributive ideal lattice, and which we call strongly irreducible, is considered. Some properties of these ideals, analogous to corresponding ones for prime ideals [19], are developed. Finally, it is observed that a topology may be introduced in the set of all proper strongly irreducible ideals in such a way that the resulting space contains the spaces of prime [19] and primitive [13] ideals as subspaces. I wish to take this opportunity to thank Professor M. F. Smiley for the encouragement and many helpful suggestions he has given me throughout the preparation of this paper. 1. Some lattice-theoretic preliminaries. In this section we state two results of a strictly lattice-theoretic nature with a view toward subsequent applications to rings. Our notation and terminology is that of Birkhoff [3]. In

Development of an extended exterior differential calculus

Abstract: The purpose of this paper is to set up an algebraic machinery for the theory of affine connections on differentiable manifolds and to demonstrate by means of several applications the scope and convenience of this mechanism. We shall associate with a manifold a series of spaces, best described as spaces of multivectors with exterior differential form coefficients, and shall exhibit the algebraic relations between these spaces. It is possible to consider, in a more general fashion, spaces of tensors with differential form coefficients; this is done, in fact, in Cartan [3, Chap. VIII, Sec. II]('), where their use is justified by means of geometrical considerations. We shall define an affine connection as a certain kind of operator on the space of ordinary vector fields to the space of vector fields with differential one-form coefficients. It will be seen that this is simply another formulation of the classical definition. One of our basic results (Theorem 7.1) is that an affine connection induces an operator on each of the series of spaces just mentioned. In Chapter I, we shall summarize the facts that we need about differentiable manifolds and introduce some notation. Chapter II is devoted to the algebraic structure of the series of spaces T;q that we introduce. In Chapter III, we give the calculus associated with an affine connection and applications to a number of identities. In the final chapter, we discuss some applications of our calculus to Riemannian geometry, in particular to the "curvatura integra" of S. Chern. We hope in the future to give applications of this theory to other parts of differential geometry, possibly to the theory of harmonic integrals and to the theory of Lie groups.

The representation of abstract integrals

Abstract: (suitably restricted) function of two real variables, and consider the function f'(x) =ff(x, y)dy. The mapping f-*f' is a linear, countably additive mapping from the space of (suitably restricted) functions of two variables to the space of functions of one variable; and it is "order-preserving" in the sense that positive functions are mapped into positive functions. The main result of this paper implies that this example is (roughly speaking) typical; under mild hypotheses, mostly of countability, every linear, countably additive, order-preserving mapping 4 from one function space Fo to another, Eo, can be obtained by coordinatewise integration in a product space. That is, there exist spaces X, Y such that Fo is "isomorphic," in a sense to be defined below (?1) to a certain space of functions on XX Y, and F' is isomorphic to a space of functions on X, and under these isomorphisms 4 corresponds to the mapping f->f' where f'(x) =ff(x, y)dy, the integral being formed with respect to an ordinary (countably additive) c-finite numerical measure on Y. (The exact theorem is stated in 5.3 below.) The formal properties of the mapping 0, and the representation theorem just mentioned, entitle 0 to be called an "abstract integral." The ordinary Lebesgue integral (over a fixed set with a a-finite measure) is included as a special case, in which F' consists of the functions defined on a single point.

Singular points of functional equations

Abstract: Hildebrandt and Graves [9](2) showed that if the partial differential dxb(0, 0; h), considered as a linear transformation of X, has a continuous everywhere-defined inverse, then there exists a unique continuous singlevalued function q defined on a neighborhood of the origin in 2) with values in X such that 4(0) =0 and b [k(y), y] =0 for all y in this neighborhood. Graves [8; 2, p. 408] showed that if dx4(0, 0; h) maps onto X, then there will be at least one solution corresponding to sufficiently small y. Cronin [3] recently considered a case in which dA need not map onto X and obtained, under suitable restrictions, theorems concerning the existence of solutions in terms of the topological degree theory. While our methods are closely related to hers, we focus our attention on the problem of studying the branching of the solutions that this situation allows. In a particular case we are able to apply Dieudonne's modification [5] of the Newton polygon method to obtain results exactly parallel to some for ordinary algebraic functions over the real or complex field. It is also seen that the work of E. Schmidt [17], L. Lichtenstein [14], and R. Iglisch [12] for a class of nonlinear integral equations hold valid for a general class of functions defined on Banach spaces. Also, in both their work and that of T. Shimizu [18], the assumption of analyticity can be replaced by that of the existence of a few continuous derivatives. Further, because of the simpler form for the equations we derive, it is possible to study particular cases in terms of initially given data. Our final part indicates briefly how these results can be applied to nonlinear differential equations with fixed end point boundary conditions. It is possible to treat questions of existence and uniqueness of solutions in the neighborhood of a given solution for a very general type of equation.

Representations of prime rings

Abstract: This paper is a continuation of the study of prime rings started in [2]. We recall that a prime ring is a ring having its zero ideal as a prime ideal. A right (left) ideal I of a prime ring R is called prime if abCI implies that acI (bCI), a and b right (left) ideals of R with b5O (aXO). We denote by r (i$1) the set of all prime right (left) ideals of R. For any subset A of R, Ar (Al) denotes the right (left) annihilator of A; Ar (A1) is a right (left) annihilator ideal of R. The set of all right (left) annihilator ideals of R is denoted by W, (1). For the prime rii'gs R studied in [2 ], it was assumed that there existed a mapping I-* of the set of all right (left) ideals of R onto a subset T? (3) of 1r ($3) having the following seven properties: