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A convenient form of graph paper is described that allows the frequency constant (f c ) and the MTF index ( n ) of many devices to be easily determined.

The purpose of this short paper is to give an exact theory of quarter-wave coupled circulators with Chebyshev characteristics. The synthesis starts by replacing the lumped-element equivalent shunt resonator of the circulator by a distributed one that has the same susceptance slope parameter as the original circuit. In this way the overall network involves commensurate transmission lines only. The bandwidth over which the assumed form of the equivalent circuit applies is carefully discussed in terms of the two split frequencies of the magnetized junction. Tables for the required circulator parameters and transformer admittances for one and two transformer sections as a function of VSWR and bandwidth are included. The realizable solution for the latter arrangement is severely restricted by the equivalent circuit of the basic junction. Experimental results on an octave-band stripline circulator, with a two-section transformer, are also included.

It is proposed to construct a simple "crossoverless" Iumped-element circulator, which can be made without sophisticated thin-film processing. The circulator can be described by a "delta connected" equivalent circuit. A simple capacitor arrangement can be used to influence the three eigenvalue phases of the circulator independently, thus permitting this circulator to be maximized systematically. A set of computer-generated eigenvalues gives insight into the behavior of the device under varying operating conditions. Preliminary measurements using a very simple pattern on a 0.650-in-diam ferrite substrate show a 20-dB bandwidth of 10 percent and an insertion loss < 1 dB (0.3 dB/min) at L band.

This paper presents a method of encoding geometric line-drawings in a way which allows sets of such drawings to be interpreted as formal languages. A characterization of certain geometric predicates in terms of their properties as languages is obtained, and techniques usually associated with generative grammars and formal automata are then applied to the geometric framework. Section 1 of the paper specifies the geometric framework; section 2 develops the background material on formal languages, grammars, and automata; section 3 is concerned with geometric predicates; section 4 covers invariance theorems; and section 5 indicates areas for further investigations. (DT)

In 1964 World Book wanted to be able to say that more than 1,000 maps being made for the encyclopedia were designed to be appropriate and useful for children between the ages of nine and fourteen. However, there was virtually no research on which such a program could be based. This paper briefly reviews research projects in this area started in 1964 by Field Enterprises Educational Corporation and continued through 1967. In summary, this work has provided two major things: 1) it has shown the inadequacy of existing research relating to children and their characteristics as map users, and of the theoretical framework in which the research has been done; and, 2) it has provided an approach to map design that consists essentially of matching an intellectual array of map use tasks to a visual array of graphic elements. (SBE)

A symmetrical 3-dB quadrature hybrid, consisting chiefly of a bifilar pair of twisted wires, is described. A cascade of two such hybrids can achieve an octave bandwidth with a 0.7-dB coupling error. Since this class of hybrid is simple, compact, and low in cost, its use may be preferred over the more common coaxial line or printed-circuit types in the frequency region below 1 GHz.

The influence of the T-junction effect on the microstrip 3-dB branchline and rat-race directional couplers design has been described. In the proposed method, the T-junction equivalent circuit similar to the circuit used by Franco and Oliner for the symmetric stripline junction is applied. Simple design expressions that combine the modified characteristic impedance and electrical length of the coupler arm with the equivalent parameters of the T-junction are given. These expressions are helpful in the determination of the microstrip coupler dimensions. The experimental verification of an improvement of the design procedure is also included.

A test was designed to measure whether or not the child's conceptualization of space would demand that two topological properties of figures (connectedness, and openness and closedness) remain invariant when the figures were subjected to transformations. Ninety children, 30 from each of ages four, six, and eight, were tested. Six items were given to each child in four modes; for each item the child indicated a "most like" and a "worst" preference. Results showed there was no significant interaction between age and mode, that age had a significant effect on the "worst" preference scores but not on "most like" scores, that test mode affected both preference scores, and that for two of the test modes the mean scores based on number of times copies not topologically equivalent to the model were selected as being "most like" the model were significantly above chance means of all age levels. The conclusion saw that the results do not support the theory that topological concepts develop prior to Euclidean and projective concepts in the child's representational space. (DT)

Efforts to create instructional materials which would be cognitively appealing to students demonstrating aptitude for figurally, verbally, or symbolically oriented materials are described. Students enrolled in preservice freshman level mathematics content courses for prospective elementary school teachers were given a battery of tests designed to measure their figural, semantic (verbal), and symbolic aptitudes; they then studied a short unit on network tracing that supposedly was written in the figural, verbal, or symbolic mode. Criterion variables were scores on tests of immediate retention, retention after one week, and retention after four weeks. Analysis showed that students with high figural aptitude did significantly better on the figural lesson than did students with low figural aptitudes. High verbal and high symbolic subjects did not perform significantly higher on the figural lesson than did low verbal and low symbolic students respectively. There were corresponding results for the verbal lesson. No definitive results were derived from the symbolic lesson. (Author/DT)

The research reported dealt with the ability of first, second, and third grade students to understand the topological concepts of solids, sheets, lines, networks, and of order and betweenness. The concepts were presented through the use of manipulative materials. The instruction was conducted by the regular teachers for a period of three weeks after which test items in workbooks completed by the students, were used for obtaining data. The achievement of each class on each item was tabulated, and differences in performance are discussed. The author concludes that the material presented seems suitable for children in the age range studied. (JG)

This study investigated the effect an advanced organizer, a post organizer, or knowledge of a behavioral objective has on learning the mathematical concept of network tracing. Self-instructional booklets on network tracing were given to eighth grade students. The booklets were designed so that the instruction and posttest could be completed within a regular class period. Pages for the advanced organizer, post organizer, and knowledge of a behavioral objective varied according to the experimental treatment in a 2x2x2 factorial design. A retention test was given one week later. Analysis of variance was used to examine the data. A significant negative main effect for post organizers was found on the retention test. Similar but not significant results were found on the learning test. (Author/FL)

where K(X) is the Grothendieck group (2) of complex vector bundles over X. The general theory of these K-groups, as developed in [1], has found many applications in topology and related fields. Since the periodicity theorem is the foundation stone of all this theory it seems desirable to have an elementary proof of it, and it is the purpose of this paper to present such a proof. Our proof will be strictly elementary. To emphasize this fact we have made the paper entirely self-contained, assuming only basic facts from algebra and topology. In particular we do not assume any knowledge of vector bundles or K-theory. We hope that, by doing this, we have made the paper intelligible to analysts who may be unacquainted with the theory of vector bundles but may be interested in the applications of K-theory to the index problem for elliptic operators [2]. We should point out in fact that our new proof of the periodicity theorem arose out of an attempt to understand the topological significance of elliptic boundary conditions. This aspect of the matter will be taken up in a subsequent paper.(3) In fact for the application to boundary problems we need not only the periodicity theorem but also some more precise results that occur in the course of our present proof.