A 2uFunction representation for non-uniform type-2 fuzzy sets: Theory and design

The theoretical and computational complexities involved in non-uniform type-2 fuzzy sets (T2 FSs) are main obstacles to apply these sets to modeling high-order uncertainties. To reduce the complexities, this paper introduces a 2uFunction representation for T2 FSs. This representation captures the ideas from probability theory. By using this representation, any non-uniform T2 FS can be represented by a function of two uniform T2 FSs. In addition, any non-uniform T2 fuzzy logic system (FLS) can be indirectly designed by two uniform T2 FLSs. In particular, a 2uFunction-based trapezoid T2 FLS is designed. Then, it is applied to the problem of forecasting Mackey–Glass time series corrupted by two kinds of noise sources: (1) stationary and (2) non-stationary additive noises. Finally, the performance of the proposed FLS is compared by (1) other types of FLS: T1 FLS and uniform T2 FLS, and (2) other studies: ANFIS [54], IT2FNN-1 [54], T2SFLS [3] and Q-T2FLS [35]. Comparative results show that the proposed design has a low prediction error as well as is suitable for online applications.     

Natural representations of some tilde and petersen type geometries

We determine the universal natural representations (embeddings) of certain sporadic geometries with three points per line.The final revision of this paper was done while the author held a temporary position at Michigan State University.     

An iterated logarithm law for families of Brownian paths

Summary If n is large, a plot of log n independent Brownian paths over [0, n] is nearly certain to give the appearance of a shaded region having square root boundaries.Research of this author was partially supported by the National Science Foundation under grant MCS 82-00786This work was undertaken while B. Schreiber, of Wayne State University, was a Visiting Professor of Mathematics at Michigan State University, whose hospitality he wishes to acknowledge with gratitude.     

Uniform approximation of vector-valued functions

Summary Uniform approximation of vector-valued functions is defined, together with analogs of extreme points and H-sets. Characterizations of best approximations are given in terms of these, and some applications are presented.Most of the work for this paper was done at Michigan State University, East Lansing; as a Ph. D. Thesis directed by D. G. MOURSUND. The author wishes to thank Prof. MOURSUND for his kind assistance.     

Quadrics of finite type

We show that the only quadrics of finite type in Euclidean 3-space are the circular cylinder and the sphere.This work was done while the second author was visiting the Michigan State University. He would like to thank MSU, and in particular the staff of the department of mathematics for their hospitality.Research Assistant of the National Fund for Scientific Research (Belgium)     

On bootstrap estimation of the distribution of the studentized mean

It is shown that bootstrap methods for estimating the distribution of the Studentized mean produce consistent estimators in quite general contexts, demanding not a lot more than existence of finite mean. In particular, neither the sample mean (suitably normalized) nor the Studentized mean need converge in distribution. It is unnecessary to assume that the sampling distribution is in the domain of attraction of any limit law.Now at Michigan State University     

On uniform convergence of measures

Summary A new and simpler proof is given in Section 3 for the sufficiency part of Theorem 3.1 in Ranga Rao [6] and its generalization by Billingsley and TopsØe [1]. Essential for the proof, which does not require the topological space X to be metric, is Lemma 2.1. As examples of possible wider application of this lemma, simple proofs are given for a well known result on uniformity in convergence of distribution functions (Example 2.3) and of Theorem 4.2 in Ranga Rao [6]. The derivation of the latter from the lemma is substantially simpler than the derivation from Theorem 3.1 in Ranga Rao [6]. Another result on uniformity, given in Rubin [7], is closely related to the Ascoli theorem, but outside the scope of applicability of our lemma.Most of the work on the paper was done while the author was with the Institute for Information Theory and Automation of Czechoslovak Academy of Sciences. After August 21, 1968 the author worked on it while a guest of the Forschungsinstitut für Mathematik, Eidgenössische Technische Hochschule, Zürich, and the paper was completed at Michigan State University.     

Recursive state and parameter estimation with applications in water resources

Hydrologic models, as well as measurements of hydrologic processes, are corrupted by noise. The Kalman filter is a convenient tool to estimate the true but unknown state of a hydrologic system. It is, however, difficult to specify the necessary error covariances. A procedure is proposed to estimate the error covariances recursively in a combined state and parameter filter. Applications of the procedure yield meaningful results for two hydrologic data series of very different character. A major benefit of the proposed algorithm seems to be its robustness against instability.     

The multiple server center location problem

In this paper, we introduce the multiple server center location problem. p servers are to be located at nodes of a network. Demand for services of these servers is located at each node, and a subset of nodes are to be chosen to locate one or more servers in each. Each customer selects the closest server. The objective is to minimize the maximum time spent by any customer, including travel time and waiting time at the server sites. The problem is formulated and analyzed. Results for heuristic solution approaches are reported. Paper was partially supported by a College of Business Administration, California State University San Marcos summer grant of the first author. Paper was partially supported by an NSERC grant of the second author.     

On Legendre curves in contact 3-manifolds

It is first observed that on a 3-dimensional Sasakian manifold the torsion of a Legendre curve is identically equal to +1. It is then shown that, conversely, if a curve on a Sasakian 3-manifold has constant torsion +1 and satisfies the initial conditions at one point for a Legendre curve, it is a Legendre curve. Furthermore, among contact metric structures, this property is characteristic of Sasakian metrics. For the standard contact structure onR ³ with its standard Sasakian metric the curvature of a Legendre curve is shown to be twice the curvature of its projection to thexy-plane with respect to the Euclidean metric. Thus this metric onR ³ is more natural for the study of Legendre curves than the Euclidean metric.This work was done while the first author was a visiting scholar at Michigan State University.     

Embedding weakly compact sets into Hilbert space

We give an example of a weakly compact set in a Banach space, which does not embed topologically as a weakly compact subset of Hilbert space. We also show that a weakly compact set embeds in a super-reflexive space iff it embeds in Hilbert space. This work was done while the authors were visiting Ohio State University. We wish to thank the Department of mathematics at Ohio State University for their kind hospitality. The work of the first author was partially supported by NSF Grant MPS-74-24249. The work of the second author is a part of his Ph.D. Thesis prepared at the University of California at Berkeley.     

Teaching for Understanding and Application of Science Knowledge

James Gallagher is Professor of Science Education at Michigan State University and currently co‐editor of Journal of Research in Science Teaching. He was a member of the Working Group on Teaching in the formulation of National Science Education Standards. His recent work has included the me of continuous assessment as a tool to aid teachers in teaching for understanding.     

The Wazewski topological method for contingent equations

Summary A Wazewski — type theorem for contingent equations is obtained using the fundamental theory of contingent equations. This work was done at the Istituto Matematico dell'Università di Firenze under the auspices of the Italian Research Council (C.N.R.). The resident addresses of the authors are, respectively, the University of Colorado, Boulder, Colo., U.S.A. and Michigan State University, East Lansing, Mich., U.S.A. Entrata in Redazione il 20 febbraio 1970.     

Sufficient conditions for variational problems with variable endpoints: Coupled points

In this paper we demonstrate that the notion of coupled points developed in [29] for the variable endpoints variational problems is the analog of that of conjugate points when the endpoints are fixed. We provide weak and strong local optimality criteria using the strengthening of necessary conditions involving both the coupled points and the regularity concepts.This research was supported by a grant from NSERC Canada and a summer support from Michigan State University.     

A foliated metric rigidity theorem for higher rank irreducible symmetric spaces

LetF be a foliation of a compact manifold with a transverse invariant measure of finite total mass. We prove that ifF admits a leafwise metric such that every leaf is an irreducible symmetric space of noncompact type and higher rank, then any other leafwise metric of nonpositive curvature is also symmetric along any leaf in the support of the transverse measure. A rank one version of this result is also exposed.The second author is partially supported, by a Seed Grant from The Ohio State University.     

Remarks on lower semicontinuity and lower closure

In the context of a recent geometric condition of Cesari, used in the reduction of seminormality requirements in lower closure theorems, this paper shows that the existence of a strongly convergent selection from the sequence of orientor fields, under Kuratowski property (K), is adequate to guarantee lower closure theorems. This generalization is justified through examples. Several related remarks are made.This work was done in the framework of Research Project AFOSR-71-2122 at the University of Michigan, Ann Arbor, Michigan. The author wishes to thank Professor L. Cesari for helpful discussions during the writing of this paper.     

Existence theorems for optimal control problems involving functional differential equations

This paper extends some existence theorems of Cesari for optimal control problems to systems whose dynamics is described by functional differential equations of finitely-retarded type. We show that the proper choice of state space is the spaceE ¹×C[–, 0], where >0 represents the time-lag of the system, and that it is necessary to choose initial conditions from a compact set inC[–, 0] as well as to employ the usual growth condition.This research was accomplished in the frame of research project AFOSR-942-65 at the University of Michigan. In particular, the author would like to thank Professor L. Cesari (University of Michigan) and Professor N. Chafee (Brown University) for many helpful remarks during the preparation of the research, which forms part of the author's doctoral dissertation written at the University of Michigan.     

N-person differential games governed by infinite-dimensional systems

This paper discussesN-person differential games governed by infinite-dimensional systems. The minimax principle, which is a necessary condition for the existence of open-loop equilibrium strategies, is proved. For linear-quadraticN-person differential games, global necessary and sufficient conditions for the existence of open-loop and closed-loop equilibrium strategies are derived.This work was supported by the Science Fund of the Chinese Academy of Sciences and the Research Foundation of Purdue University.The problems discussed in this paper were proposed by Professor G. Chen, during the author's visit to Pensylvania State University, and were completed at Purdue University. The author would like to thank Professors L. D. Berkovitz and G. Chen for their hospitality.     

Optimal ordering rule for a stochastic sequencing model

In this note, necessary and sufficient conditions are derived for the optimality of a sequencing rule for a class of stochastic sequential models. The optimal sequential rule generalizes the deterministic results, given in Refs. 1–2, for situations when some of the parameters of the problem are random variables. Two cases are given to demonstrate the usefulness of the results.The author would like to acknowledge the support provided by King Fahd University of Petroleum and Minerals, Dhahran, Saudi Arabia, during his sabbatical leave. Also, the Industrial and Operations Engineering Department, University of Michigan, is acknowledged for hosting the author.     

Stochastic semidefinite programming: a new paradigm for stochastic optimization

Semidefinite programs are a class of optimization problems that have been studied extensively during the past 15 years. Semidefinite programs are naturally related to linear programs, and both are defined using deterministic data. Stochastic programs were introduced in the 1950s as a paradigm for dealing with uncertainty in data defining linear programs. In this paper, we introduce stochastic semidefinite programs as a paradigm for dealing with uncertainty in data defining semidefinite programs.The work of this author was supported in part by the U.S. Army Research Office under Grant DAAD 19-00-1-0465. The material in this paper is part of the doctoral dissertation of this author in preparation at Washington State University.