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The word or is a member of a small class of words (or, if… then, and, and not) that are used in both natural language and formal logic. The natural language and formal logic meanings of and and not are sufficiently similar that a person without training in formal logic will experience little difficulty in understanding how these terms are used in formal logic. In contrast, if… then and or have sufficiently different meanings in the two domains that without special instruction a speaker of English may experience difficulty in understanding the specialized meanings these “familiar” terms assume in formal logic.6 Failure to respect the distinction between the formal logic and ordinary language meanings appears to underlie the conclusion, reached by some investigators, that young children do not understand the meaning of or.

It has been proposed ( Johnston & Diven, 1969a ) that it is valid to use semilogarithmic (first-order) plots of the extent of reaction versus time for graphic determination of initial velocities of enzyme-catalyzed reactions. This proposition suggests the assumption that the initial velocity error expected to be introduced by the proposed procedure is smaller than or comparable to the error introduced by the customary graphic procedure. The latter is based on the assumption that the progress curves of enzyme-catalyzed reactions have an initial linear segment of sufficient duration to permit accurate determination of slope. The validity of the procedure proposed by Johnston and Diven is examined in this report. It is concluded that the procedure is applicable to a very small class of enzyme-catalyzed reactions and only under certain experimental conditions.

The operations of a homomorphic equality and an inverse homomorphic equality are introduced. These operations are obtained from n-tuples of homomorphisms, incorporating the notion of an equality set. For one-tuples they are a homomorphism and an inverse homomorphism. Homomorphic equality and inverse homomorphic equality operations provide simple and uniform characterizations of the recursively enumerable sets in terms of the regular sets, and of classes H(L λ MR) in terms of L. These characterizations resemble the Chomsky-Schutzenberger theorem for context-free languages.

The title of this talk is still an open question. Very few positive answers will be provided here. We will give, however, a survey of solutions of the class number one and/or class number two problem for various types of fields.

A brief review of the electronic and vibrational structure of amorphous As is presented. Using the random network model as a proto‐type for the structure of the amorphous phase, theoretical electron and phonon densities of states are calculated. The influences of topology on these state densities are examined by using ’’cluster‐Bethe‐lattice’’ techniques. These results are then used to interpret the experimental X‐ray photomeission and infra‐red absorption measurements on amorphous As. It is shown that As is one of a small class of unique materials which exhibit an isomorphism between their electronic and vibrational excitations.

Spatially discrete difference approximations of hyperbolic initial-boundary-value problems (IBVPs) require numerical boundary conditions in addition to the analytical boundary conditions specified for the differential equations. Improper treatment of a numerical boundary condition can cause instability of the discrete IBVP even though the approximation is stable for the pure initial-value or Cauchy problem. In the discrete IBVP stability literature there exists a small class of discrete approximations which are called borderline cases. For nondissipative approximations, borderline cases are unstable according to the theory of the Gustafsson, Kreiss, and Sundstrom (GKS) but they may be Lax-Richtmyer stable or unstable in the L 2 norm on a finite domain. We show that borderline approximations can be characterized by the presence of a stationary mode for the finite domain problem. A stationary mode has the property that it does not decay with time and a nontrivial stationary mode leads to algebraic growth of the solution norm with mesh refinement. We give an analytical condition which makes it easy to detect a stationary mode and we investigate several examples of numerical boundary conditions corresponding to borderline cases.

In the previous chapter we obtained a complete characterization of efficient models Σ=(St)teN where eacn St belonged to a small class of well-behaved sets.

Optimality properties of multiway block designs are deduced from the general results of J. Kiefer's approximate-design theory. In the model with additive effects these optimality properties solely depend on the two-dimensional marginals of the designs. Uniform designs, and designs whose two-dimensional marginals are products of the one-dimensional marginals, are shown to be optimal. Approximate Youden designs are introduced for the case when the support sets of the two-dimensional marginals are prescribed in advance. They are optimal in a relatively small class of competing designs only. On se sert de la theorie des plans approximatifs de Kiefer pour deduire certaines proprietes ďoptimalite des plans de blocs pluridimensionnels. Dans le cadre ďun modele a effets additifs, ces proprietes optimales ne dependent que des marges bidimensionnelles des plans ďexperience. On montre que les plans uniformes sont optimaux, de měme que les plans dont les marges bidimensionnelles sont des produits de marges a une dimension. On introduit des plans approximatifs de Youden dans le cas ou les supports des marges bidimensionnelles sont prescrits a ľavance. Ces derniers ne sont optimaux que par rapport a une classe relativement petite de plans rivaux.

A small class of normal cellular genes has the potential to transform cells malignantly. These cellular oncogenes have been identified by their similarity to the transforming genes of oncogenic RNA viruses (retroviruses), and by their ability, under certain circumstances, to transform cells into which they have been transferred. Cellular oncogenes are not normally tumorigenic and, indeed, appear to be critically involved in normal cellular processes, possibly at the level of growth regulation. However, it appears that a number of mechanisms exist whereby these oncogenes can become malignantly activated, leading to uncontrolled cellular proliferation. The evidence implicating cellular oncogenes in human malignant disease is reviewed.

Let C(δ), δ>0, be the class of sequences of covariance matrices K = (Kt(u, v), u, v=1,...,t)t=1 2,... such that $${{\sum\limits_{u=1}^{t}{\sum\limits_{v=1}^{t}{K}}}_{t}}\left( u,v \right)=0\left( {{t}^{2-\sigma }} \right)ast\to \infty $$ Consider discrete-time stochastic process Xt=μ + ξt, t=1,2,..., where μ is an (unknown) constant and (ξ t) is a Gaussian sequence such that Eξt=0 and K=(Eξuξv, u,v=1,...,t)t=1,2.. is an(unknown) covariance structure of (ξt). Let P μ,K)be the probability measure induced by (Xt).Suppose that for every k=1,2,... we can observe simultaneously k independent copies of (Xt).The result is: for every δ>0, e>0 and γ Є(0,1) there exist a sequential estimate \({{\overset{\wedge }{\mathop{\left( {{\mu }_{t}} \right)}}\,}_{t}}=1,2\ldots \) and a finite stopping rule τ such that $${\text{P}}_{{\text{(}}\mu {\text{,K)}}} \{ |\hat \mu _\tau - \mu | > \varepsilon \} < \gamma \,for\,all\,\mu \in R^1 \,and\,K \in C(\delta ).$$

Accelerated by the advent of time-sharing, on-line systems of all types have come into being during the past five years. All of these systems are designed to facilitate the interchange of information between the user and the machine. The applications to which such systems have been put are numerous and extremely varied, yet those systems which attempt to provide a general facility for mathematical computation form a relatively small class. This paper attempts to survey and analyze some members of this restricted class in order to illustrate their common features, strong points, and areas of weakness, as well as to indicate some of the features which are desirable in any system designed to provide on-line mathematical assistance. We believe that these systems, when implemented on third generation computers, will serve large numbers of users, including numerical analysts, scientists and engineers.

Even in this day of enlightenment as regards etiology and pathology, there are thousands, very many thousands, of patients being systematically treated for functional diseases of the body that are due to diseases of the eyes. There are other thousands being treated for organic diseases of the body which originally were caused by ocular diseases. In both cases neither the patient nor the doctor has the remotest suspicion of the ocular origins of the functional and organic abnormalities. Sneering and top-lofty contempt of the specialist "hobby-rider" will not change the facts, nor lessen the hobbyriding. Moreover, the assertion that the patient has been in the hands of an ophthalmologist of good reputation does not in the least lessen the truth for which I contend. Out of perhaps a hundred classes I wish to-day to choose one small class of cases illustrative of the general thesis. This group is composed of

Project TACT (Technological Aids to Creative Thought), in the Aiken Computation Laboratory of Harvard University, has been designing, implementing, and using on-line systems since 1964. Our investigation has concentrated on a rather small class of interactive systems which are "dedicated," that is, tailored to provide a coherent set of tools for mathematical assistance, and which use fairly powerful terminal facilities: display scopes and natural input.

Important as is the operative treatment of squint, it is less widely applicable than the non-operative treatment. A large proportion of cases may be cured without operation. The number so cured will increase, as it becomes customary to resort to the proper measures, as soon as the deviation is noticed. But in all cases some non-operative treatment is essential to effect a perfect cure. It may not be required only in that small class of operative cases in which binocular vision and normal ocular movements are not hoped for; but in which by operation we may lessen a deformity that we can not wholly remove. The non-operative treatment for strabismus includes the entire treatment of many cases, and the earliest treatment for all. The indications to be met by it are: 1, to bring about normal innervation of the muscles concerned in ocular movements, by the removal and exclusion of

Conventional MPC uses quadratic programming (QP) to minimise, on-line, a cost over n linearly constrained control moves. However, stability constraints often require the use of large n thereby increasing the on-line computation, rendering the approach impracticable in the case of fast sampling. Here, we explore an alternative that requires a fraction of the computational cost (which increases only linearly with n), and propose an extension which, in all but a small class of models, matches to within a fraction of a percent point the performance of the optimal solution obtained through QP. The provocative title of the paper is intended to point out that the proposed approach offers a very attractive alternative to QP-based MPC. © 2002 Published by Elsevier Science Ltd.

Downward collapse (a.k.a. upward separation) refers to cases where the equality of two larger classes implies the equality of two smaller classes. We provide an unqualified downward collapse result completely within the polynomial hierarchy. In particular, we prove that, for k < 2, if P^{\sum^p_k[1]} = P^{\sum^p_k[2]} then \sum^p_k = \prod^p_k = PH. We extend this to obtain a more general downward collapse result.

The biological world is studded with an immense variety of sugar molecules. These sugars, affixed to other molecules such as proteins and lipids, play many major roles and bit parts in living cells. Exactly what they perform, how they are synthesized to do so, and some newer research techniques to probe into such matters were the focus of a three-day conference on complex carbohydrates in biological recognition, held at the National Institutes of Health in Bethesda, Md., earlier this month. The group of proteins called lectins is probably at center stage. First defined as proteins that "like" and therefore bind specifically to certain sugars, lectins now have taken on a more general meaning. For instance, except among stalwart traditionalists, the term lectin no longer is reserved for the small class of plant-derived proteins originally described as having that property. Indeed, the spotlight has shifted to several lectins that are found on the surfaces of cells ...

Upon reading Christopher Goodnow's recent Review article in Cell, which emphasized the overlapping role of somatic mutation in autoimmunity and cancer (Goodnow, 2007), I was reminded of a graduate lecture course given by Professor Roderick MacLeod in 1995 at the University of Illinois. While introducing F.M. Burnet's clonal selection theory to our small class, Professor MacLeod mentioned Burnet's idea that germline and somatic mutations provide the inherited and stochastic mechanisms that disrupt tolerance and result in autoimmunity.

The Thayer Concept of teaching emphasizes small class size, ability grouping, and competition among students; summary of its benefits, requirements, and application.

To the Editor. — Personal watercraft or personal water vehicles are small class A inboard-powered, water-jet-propelled craft that an operator rides on or behind as opposed to inside, as with conventional motorboats. 1 Several models are manufactured, including the Jetski, SeaDoo, Wetbike, Wetjet, Surfjet, Waverunner, Wavejammer, Fastrac, and Fazer. 1,2 Consumers are attracted to these vessels because they are small, fast, and maneuverable. As with other motorized vehicles that can attain high speed, there is a risk of injury associated with their use. Study.— Cases were identified from the Dade County Trauma Register by reviewing all entries coded as sports, water accident, or drowning from September 1985 through August 1988. We included individuals injured on and injured by personal watercraft. These cases represent all fatalities and survivors who sustained injuries severe enough to necessitate transport to a trauma center in Dade County, Florida. Six people were injured riding a personal

The laboratory program described here is intended for a small class of mainly honor students who have a good background in general and analytical chemistry and who are taking other courses in chemi...

The paragraphs that follow do not provide definitive answers to any of these questions. They do provide some clues and some data which may prompt others to gather information based on wider samples or other types of courses. The setting was a small class in the Literature of the Humanities offered during the Winter Quarter of 1970. During the ten weeks of the quarter, the class met once a week for two hours and fifteen minutes. In the eleventh week, there was an examination which lasted one hour and forty-five minutes. Thus total class time for the quarter was twenty-four and one quarter hours. Several students volunteered to keep logs recording the amounts of time spent on various projects. At the end of the quarter, three logs were submitted one from a student who received an "A" grade for the course and two from students with "B" grades. The time spent (exclusive of class sessions) ranged from a low of 107 hours to a high of 138 hours. On balance, the "A" student spent 30 hours more time on the assignments than did the two "B" students. The assignments included: use of individual titles on the lists to answer reference questions; a religious encyclopedia comparison; three dialogs portraying reference interviews (one of which was videotaped)

AS we all know, the "new" inquiry learning actually rests on a very old Socratic premise: people learn by asking questions. The Italian scholar, Benedetto Croce, put the idea most succinctly for the history teacher: "Every line of history should ask a question."' Recently I attempted to evolve a course structure that would follow Croce's imperative and enable my students and I to raise as many stimulating questions as possible about the cultural history of seventeenth and eighteenth-century America. The course I developed was neither novel in its theoretical assumptions nor strikingly unique in its inquiry technique; it was simply an attempt to apply the principles of the historian's craft to a historian's classroom. In the Spring of 1972, I taught History 330, Studies in the Cultural History of Seventeenth and Eighteenth-Century America, at Grinnell College (Grinnell, Iowa) to a small class of twenty-two students. The course was considered an advanced departmental offering, designed principally for junior and senior history majors and scheduled to meet for two-hour sessions, two afternoons a week, over a fourteen-week semester.

An old undergraduate grumble, that professors won't teach, has lately become a shout. Rebellious students on a number of campuses are demanding changes in the system that leaves the classroom to ill-prepared graduate assistants, while professors devote their time to research. Meanwhile, the graduate assistants, who have long been irked by their neither-fish-nor-flesh position in the academic hierarchy, want clarification of their status and opportunities to improve their teaching ability. According to Drs. Robert Dubin and Frederic Beisse of the University of Oregon, an undergraduate's chances of seeing a professor are not very good. They give the following odds: * One in three that an undergraduate will get a graduate assistant as teacher in any course he may choose to register for; * In the first two years the odds go up to two in five; * If the undergraduate seeks a small class for intimate contact with a professor, the chances are only one in three that he will find a professor in the class he registers for. Concerned by the growing picture of the professorless classroom, the American Chemical Society's Committee on Chemistry and Public Affairs invited Prof. Frank Westheimer of Harvard to see if there really has been a flight from teaching on the part of research chemists and scientists generally. In chemistry, particularly, and in other sciences, Prof. Westheimer finds no evidence of a faculty flight. On the contrary he finds distinguished research chemists lining up to teach undergraduate courses. When the University of Chicago years ago tried to separate research and teaching by setting up a series of research-only institutes, Enrico Fermi would accept the chairmanship of the Institute for Nuclear Studies only on condition he be allowed to continue to teach freshman physics. His example was followed by other members of his institute, in what Prof. Westheimer calls "a spectacular example of the flight to teaching." Research professorships are rare. In a survey of 13 large universities Prof. Westheimer says he could find only six research professors among 200 faculty chemists. Data on absenteeism-one of the charges often made against senior faculty-for 20 of the professors at universities surveyed by Prof. Westheimer show the number of undergraduate lectures missed for reasons other than illness is between one and two a year per lecturer. Prof. Westheimer marshalls other evidence, including the names of people teaching undergraduate courses in various universities, to show that research scientists not only teach but like it. In general, he says, good researchers make the best teachers and students prove it by lining up for their courses. But even if the senior professors in the sciences are not running from the classroom, graduate assistants are coming in in larger numbers. And here Prof. Westheimer, like others, sees a problem: "The undergraduate should feel the excitement that research brings to most graduate students and this excitement should permeate the laboratory. Somehow," he finds, "it doesn't." And that itself is a problem. Many are the complaints that teaching by graduate assistants is dull, routine and listless, and a number of Prof. Westheimer's colleagues are trying to see how they can change the situation. What is really the best thing to do is a matter of some disagreement. Proposals that graduate teaching assistants simply be abolished are unrealizable, says Prof. K. B. Wiberg of Yale. There just aren't enough senior faculty members to cover the laboratory sections traditionally taught by assistants. Whatever it may mean for the assistants' own education-and this part