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Showing papers in "Pacific Journal of Mathematics in 1955"


Journal ArticleDOI
TL;DR: In this paper, the authors formulate and prove an elementary fixpoint theorem which holds in arbitrary complete lattices, and give various applications (and extensions) of this result in the theories of simply ordered sets, real functions, Boolean algebras, as well as in general set theory and topology.
Abstract: 1. A lattice-theoretical fixpoint theorem. In this section we formulate and prove an elementary fixpoint theorem which holds in arbitrary complete lattices. In the following sections we give various applications (and extensions) of this result in the theories of simply ordered sets, real functions, Boolean algebras, as well as in general set theory and topology. * By a lattice we understand as usual a system 21 = (A 9 <) formed by a non-empty set A and a binary relation <; it is assumed that < establishes a partial order in A and that for any two elements a f b E A there is a least upper bound (join) a u b and a greatest lower bound (meet) an b. The relations >L, <, and > are defined in the usual way in terms of <. The lattice 21 = (A, <) is called complete if every subset B of A has a least upper bound ΌB and a greatest lower bound Πβ. Such a lattice has in particular two elements 0 and 1 defined by the formulas 0 = ΓU and 1 = 11,4. Given any two elements a 9 b E A with a < b, we denote by [a 9 b] the interval with the endpoints a and b, that is, the set of all elements x E A for which a < x < b; in symbols, [ a,b] = E x [x E A and a .< x .< b ]. The system \[α,6], <) is clearly a lattice; it is a complete if 21 is complete. We shall consider functions on A to A and, more generally, on a subset B of A to another subset C of A. Such a function / is called increasing if, for any 1 For notions and facts concerning lattices, simply ordered systems, and Boolean algebras consult [l].

2,873 citations







Journal ArticleDOI
TL;DR: In this article, the partial differential operator L = L(x, d\{dx), A) on functions of two independent variables is constructed, which is independent of n and is commutative with A.
Abstract: Substituting A=ydj(dy) for n, supposing the left member a polynomial in n, we construct the partial differential operator L=L(x, d\{dx), A) on functions of two independent variables. This operator is independent of n and is commutative with A. A solution of the simultaneous equation Lu=Q, Au=nu, where n is a constant, has the form u=vn(x)yn, where v=vn(x) is a solution of (1.1). Conversely, if v=vn(x) is a solution of (1.1), then ιt=vn(x)yn is a solution of the equations Lu = 0, Au = nu. Now suppose that, independently of the preceding considerations, we have obtained an explicit solution u=g(x, y) of Lu = 0, and that from the properties of this function we know that it has an expansion in powers of y of the form

122 citations









Journal ArticleDOI
TL;DR: In this article, the authors considered the case where an ~ 0 for n. t >_ 0 a strongly continuous semi-group of operators acting either on the space of bounded functions (M) or integrable functions (L ) with | | Γ ( ί ). | | < 1.
Abstract: Σ, " M l I _oo un exist. Similar results are valid for the circumstance where T" does not exist. Then we deal only with the case where an ~ 0 for n . t >_ 0 a strongly continuous semi-group of operators acting either on the space of bounded functions (M) or integrable functions ( L ) with | | Γ ( ί ) . | | < 1. Let A denote the infinitesimal generator of Tit) and let df it) define a nonlatt ice distribution with finite first moment on [0, col. If u belongs to iU) we consider u ( t ) \ (°° T ( t ) d f ( t ) ] u = v L J o J where v belongs to ( L ). The linear operator ί°° Tit)dfit) Jo is well defined either over iM) or ( L ) into itself. Put rit) = 1 f it), then r G L and the Fourier transform of r never vanishes . Since r is monotonic decreasing and in L it can be easi ly shown that












Journal ArticleDOI
TL;DR: In this paper, the authors introduce the notion of non-commutative group under composition, i.e., a · b 6 = b · a. When are two groups the same?
Abstract: Most interesting groups arise as a group of transformations. For example the set of “rigid motions” of 2-dimensional space R forms a group denoted Isom(R). There are three kinds of transformations: reflection about a line, rotation about a point, and translation in a direction. These form a group under composition (i.e., do one transformation and then do the other). For example, the composition of two reflections is either a rotation (if the two lines intersect) or a translation (if the two lines are parallel). If the two lines intersect, notice that the direction of rotation depends on which reflection you do first. Thus this group is non-commutative, i.e., a · b 6= b · a. When are two groups the same? To address this, we must first introduce the

Journal ArticleDOI
TL;DR: In this article, the discriminant is defined as an element of K which is not entirely defined by / ; however, it is entirely determined when in addition a basis of E is chosen, and when the basis is changed, when the difference is multiplied by a square in K.
Abstract: 1. Let E be a vector space of finite dimension over a field K. To a bilinear symmetric form f(x, y) defined over ExE is attached classically the notion of discriminant: it is an element of K which is not entirely defined by / ; however, it is entirely determined when in addition a basis of E is chosen, and when the basis is changed, the discriminant is multiplied by a square in K. More precisely, let u be a linear mapping of E into E, and let fx(x, y)=f(u(x), u(y)) the form "transformed" by ^ if Δ(f), Δ(f1) are the discriminants of / a n d f1 with respect to the same basis of E, and D(u) the determinant of u with respect to that basis, then one has the classical relation