Auxiliary Propositions Research Articles

Post-modern Interdisciplinarity: Kant, Diderot and the Encyclopedic Project

In the second part of the Critique of Judgment, Kant offers the following definition of how a science or discipline is established: "The principles of a science are either internal to it, and are then called indigenous (principia domestica), or they are based on principles that can only find their place outside of it, and are foreign principles (peregrina). Sciences that contain the latter base their doctrines on auxiliary propositions (lemmata), i.e., they borrow some concept, and along with it a basis for order, from another science" (Kant, Judgment 252). The second of these two cases, the borrowing of a concept from another science or discipline, is a practice that is easily discernible in recent critical history. One need only think of the borrowing from Saussurean linguistics that enabled the development of structuralism. Yet, as the history and the intentions of structuralism already show, such borrowing does not lead to the formation of a science or even a discipline we could call interdisciplinarity but rather, remains firmly within the practice of either a critical method or the idiosyncrasy of a particular critical interpretation. 2 Indeed, in such cases, the claim to interdisciplinarity has more to do with affirming the ability to borrow from one discipline or another as a central principle of modern humanistic study if not the history of the humanities in general.

Evidence Algorithm and Processing Formalized Mathematical Texts

The goal of a research programme Evidence Algorithm is a development of an open system of automated proving that is able to accumulate mathematical knowledge and to prove theorems in a context of a self-contained mathematical text. By now, the first version of such a system called a System for Automated Deduction, SAD, is implemented in software. The system SAD possesses the following main features: mathematical texts are formalized using a specific formal language that is close to a natural language of mathematical publications; a proof search is based on special sequent-type calculi formalizing natural reasoning style, such as application of definitions and auxiliary propositions. These calculi also admit a separation of equality handling from deduction that gives an opportunity to integrate logical reasoning with symbolic calculation.

The Realist Paradigm and Degenerative versus Progressive Research Programs: An Appraisal of Neotraditional Research on Waltz's Balancing Proposition

Several analysts argue that, despite anomalies, the realist paradigm is dominant because it is more fertile than its rivals. While the ability of the realist paradigm to reformulate its theories in light of criticism accounts for its persistence, it is argued that the proliferation of emendations exposes a degenerating tendency in the paradigm's research program. This article applies Lakatos's criterion that a series of related theories must produce problemshifts that are progressive rather than degenerating to appraise the adequacy of realist-based theories on the balancing of power advanced by neotraditionalists. This research program is seen as degenerating because of (1) the protean character of its theoretical development, (2) an unwillingness to specify what constitutes the true theory, which if falsified would lead to a rejection of the paradigm, (3) a continual adoption of auxiliary propositions to explain away flaws, and (4) a dearth of strong research findings.

The problem of the justification of variational formulations of the boundary element method

An investigation is made of the convergence of boundary element approximations of variational solutions of the boundary-value problems of linear elasticity theory, using double- and simple-layer potentials (DLP, SLP) /1, 2/. Auxiliary propositions are proved, concerning the basis property of the approximating sequence of potentials. The results may also be used to justify variational formulations of the boundary element method (BEM) is applied to the solution of other second-order elliptic boundary-value problems.

Fast probabilistic algorithms for hamiltonian circuits and matchings

We describe and analyse three simple efficient algorithms with good probabilistic behaviour; two algorithms with run times of O( n(log n) 2) which almost certainly find directed (undirected) Hamiltonian circuits in random graphs of at least cn log n edges, and an algorithm with a run time of O( n log n) which almost certainly finds a perfect matching in a random graph of at least cn log n edges. Auxiliary propositions regarding conversion between input distributions and the “de-randomization” of randomized algorithms are proved. A new model, the random access computer (RAC), is introduced specifically to treat run times in low-level complexity.

On Using Informants: A Technique for Collecting Quantitative Data and Controlling Measurement Error in Organization Analysis

Reviewing recent developments in the study of organizations and in the use of informants, this paper notes the compatibility of both and the need for further refinements. As an example, it presents methods used in a study of authority conflict in 136 Catholic dioceses in America. These include principles for choosing a small number of informants per unit and two sets of auxiliary methodological propositions. The first, an instrumental theory, was part of a strategy to control measurement error. It guided the data collection process, and the creation of control variables. The second set of propositions, devised for evaluating measurement error, presented expected mathematical properties of such error variables. These properties were expected to hold under the condition that the data were sound enough for substantive analysis. For these data, hypotheses were basically supported.

A Remark on C σ Spaces

We give a simple new proof of the following result, conjectured by Effros and proved by Fakhoury: Let E be a C, space and Z the set of extreme points of the unit ball of E*. Then Zu{O} = {p C E*: Kfgh, p)=Kf, p)(g, p)Kh, p) for all f, g, h in E}. Let C(X) be the Banach space of all continuous real valued functions on a compact Hausdorff space X, equipped with the supremum norm. If a: X--X is an involutory homeomorphism, then C(X) is the subspace of C(X) consisting of thosef in C(X) which satisfyf(ox)= -f(x) for all x in X. ex will stand for the point functional corresponding to a point x in X. If E is a Banach space, then we shall denote its unit ball by B(E) and its conjugate space by E*. The set of extreme points of a subset Q of E will be denoted by ext Q. The following result was conjectured by Effros [2, Remark 8.4] and proved by Fakhoury [4, Theorem 15]. THEOREM. If Eis a Cg space, then ext B(E*)U{O}={p E E*:(fgh,p)= (f, p)(g, p)(h, p) for allf, g, h in E}. Fakhoury's proof is measure-theoretic in nature. Since this theorem appears to be useful (see, for instance, [4, Theorem 25] and [3, Theorem 11]), a simple different proof might perhaps be of some interest. The purpose of this note is to present such a proof. We shall need a few auxiliary propositions. LEMMA 1 [1, p. 89]. If E=C,(X), then ext B(E*)={ex:x e X and x$ax}. LEMMA 2 [5, PROPOSITION 3.5]. Let E be a Banach space and J* the adjoint of the canonical map J: EE* *. Ify E ext B(E***), then J*y belongs to the wveak star closure of ext B(E*). Since the dual of a C, space is isometric to an L space, its second dual is isometric to C( Y) for some (extremally disconnected) compact Hausdorff space Y. Received by the editors November 9, 1972. AMS (MOS) slibject classiflcations (1970). Primary 46E15.

THE TOPOLOGY OF BASIS SETS FOR SMALE DIFFEOMORPHISMS

Diffeomorphisms of a closed surface are considered which satisfy Smale's axiom A and an acyclicity condition. It is shown that if one of its basis sets is one-dimensional, then there is also a zero-dimensional source or sink. As a preliminary, some auxiliary propositions of general character are established concerning sources and sinks of diffeomorphisms satisfying the axiom and the condition above. Bibliography: 10 entries.

The finite difference solution of the third boundary value problem for elliptic and parabolic equations in an arbitrary domain. Iterative solution of the difference equations. II

THE present paper is a direct continuation of [1] and is devoted to a study of the rate of convergence of the explicit scheme for the parabolic equation, and also to the justification of the method of alternating directions as an iterative method of solution. Below we retain all the notation of [1] without specially mentioning it. For convenience of numbering we begin the discussion with section 5. Therefore, a reference to the inequality (4.8) will mean inequality (4.8) of [1]. Section 5 contains auxiliary propositions. In section 6 an estimate is given of the rate of convergence of the explicit scheme for a parabolic equation, of order O( h 1.5), as also in the case of a problem for an elliptic equation. Section 7 is devoted to the justification of the convergence of the method of alternating directions to the solution of the difference scheme for an elliptic equation in the absence of mixed derivatives. For a domain which is a rectangular parallelepiped, the method of alternating directions with choice of the iteration parameters was considered in [2]. In section 8, following the method proposed in [3], we investifate the convergence of the method of compound iterations of the solution of difference equations when mixed derivatives are present. We notice that as distinct from [3], the concept of spectral equivalence is understood in the wider sense.

A finite difference method for solving the third boundary value problem for elliptic and parabolic equations in an arbitrary region. Iterative solution of finite difference equations. I

DIFFERENCE schemes are constructed on a uniform mesh for solving the third boundary value problem for a second order elliptic equation in a two-dimensional region; the schemes are suitable for solving such equations by simple iterations and by the method of alternating directions. An estimate in the norm of W 2 1 was obtained in [1] for the rate of convergence of finite-difference schemes of the variational type for solving the first and third boundary value problems for elliptic equations in a two-dimensional region; the estimate was of order 0(h 1.5). Irregular triangulation at the region boundary was used for the schemes. The present paper also used the variational approach and obtains a similar rate of convergence estimate for the solutions of difference schemes employing irregular triangulation, though the latter is more natural, see [2], for the third boundary value problem. Notice that the difference schemes thus obtained (see [2–4]) have the following drawback: there is no limit to how ill posed the matrices of the sets of difference equations are. In Section 1 difference schemes are constructed, suitable for solving the equations by a modification of a well-known iterative method such as simple iterations or the method of alternating directions. Section 2 presents auxiliary propositions. In Section 3, the convergence rate of the solutions of the difference equations to the solution of the differential problem is investigated. Section 4 proves the convergence of the method of simple iterations and finds the number of iterations required to achieve a given accuracy. The convergence of the iterative process is proved under conditions on the differential operator of the problem such as are almost, necessary for the method of simple iterations to be applicable.

Study of the rate of convergence of variational difference schemes for second-order elliptic equations in a two-dimensional field with a smooth boundary

IN this paper the rate of convergence of variational-difference schemes (VDS) is studied when the differential properties of the solution are varied, the estimates of the convergence rate being derived in different norms. The differential properties of the solution are determined by the properties of the coefficients and of the right side of the equation (we consider the boundary of the domain to be sufficiently smooth). We will vary the properties of the right side of the equation only, and with respect to the coefficients we will suppose that they have the smoothness which is required in the proofs. In obtaining the estimates we have attempted only to explain the order in h ( h is the step of the net) of the approximation to zero of the error, therefore we will not deduce in detail the values of the constants in the estimates. Estimates of the rate of convergence will be obtained in terms of the data of the problem. In the majority of cases the estimates obtained are exact with respect to order in the sense of section 4 of [1]. Section 1 contains auxiliary propositions. In section 2 a new estimate of the error in the norm of the space L 2 is obtained, and then using theorems of V. P. Il'in in [2, 3] a scale of error estimates is obtained for the right side from L 2. Section 3 is devoted to error estimates in net norms. As a corollary of the estimates obtained we derive estimates of the error in the norm of the space C. Almost all the results are derived for both the first and third boundary value problems. In the present paper we have considered only the case of a two-dimensional domain, but it seems to us that the transfer of all the estimates in the norms of Hilbert spaces to the multidimensional case taking the geometrical constructions into account [4], presents no difficulty.

Asymptotic Normality of Simple Linear Rank Statistics Under Alternatives

Results of Chernoff-Savage (1958) and Govindarajulu-LeCam-Raghavachari (1966) are extended from the two-sample case to the general regression case and, simultaneously, the conditions on the scores-generating function are relaxed. The main results are stated in Section 2 and their proofs are given in Section 5. Sections 3 and 4 contain auxiliary propositions, on which the methods of the present paper are based. Section 6 includes a counterexample, showing that the theorem cannot be extended to discontinuous scores-generating functions.

On the existence of a solution of Verigin's problem

On the existence of a solution of Verigin's problem

DIVERGENT FOURIER SERIES

CONTENTS § 1. Introduction § 2. Definitions and auxiliary propositions § 3. Theorems on summable series § 4. Zahorski' s construction § 5. Haar series § 6. Basic series § 7. Trigonometric series § 8. Walsh series § 9. Series in terms of complete orthonormal systems § 10. Weakly unconditionally convergent series § 11. Some problems References

THE REPRESENTATION OF MEASURABLE FUNCTIONS BY SERIES

CONTENTS Introduction § 1. Definitions and auxiliary propositions § 2. Proof of the fundamental lemmas § 3. Representation of measurable functions in series in terms of bases of the Lp spaces § 4. Null-series § 5. Representation of measurable functions in series in terms of Haar's functions § 6. Complete systems in the sense of convergence in measure § 7. Representation of measurable functions in series in terms of complete systems in the sense of convergence in measure § 8. Convergence of orthogonal series to +∞ § 9. Universal series § 10. Unsolved problems References