Converse Question Research Articles

Human Security or National Defense: The Question of Conversion

How can we convert the enormous human, financial and technological resources currently committed to military illusions of "national security" to programs and institutions which provide real human security? That is the central question of this paper.

Human Security or National Defense: The Question of Conversion

How can we convert the enormous human, financial and technological resources currently committed to military illusions of "national security" to programs and institutions which provide real human security? That is the central question of this paper.

Extensions of totally projective groups

It is an unpublished observation of L. Fuchs and E. A. Walker that if L is a fully invariant subgroup of the totally projective p-group G, then both L and G/L are totally projective. In this note we treat the more difficult converse question.

Mapping Cylinder Neighborhoods of Some ANR's

Let X be a closed subset of a manifold without boundary Q. A codimension 0 submanifold M of Q is a mapping cylinder neighborhood of X if it is a closed neighborhood of X in Q and if there is a proper map r: AM-P X such that M is homeomorphic to the mapping cylinder of r, fixing AM and X in the natural way. For example, regular neighborhoods of locally finite complexes in PL manifolds are mapping cylinder neighborhoods. We do not insist that M be collared in Q since this can always be arranged by an isotopy of M into itself along the structure lines. Edwards ([5]) contains a lot of information about mapping cylinder neighborhoods. It follows immediately from the definition that subsets possessing mapping cylinder neighborhoods are retracts of the neighborhoods, hence are finite dimensional ANR's. The converse question, whether finite dimensional ANR's have mapping cylinder neighborhoods, at least for some embeddings in manifolds, is interesting. If they do, then compact, finite dimensional ANR's have finite homotopy type, since these neighborhoods are compact manifolds, and so have finite homotopy type by Kirby-Siebenmann (17]). We prove a partial result in the direction of the converse, namely, if X is a locally compact ANR, embedded as a closed, 1-LC codimension 4 subset of a finite dimensional manifold without boundary Q, then X x M, has a mapping cylinder neighborhood in Q x M, where either M. M= SI, or (M, Mj) = (R', [0, ao)). Our proof also works in infinite dimensions and yields the fact that if X is a locally compact ANR which is embedded as a closed Z-set in a Hilbert cube manifold Q, then X x M. has a cylinder neighborhood in Q x M. Taking suitable compactifications in the case (M, Mj) = (Ri, [0, ao)), one finds that the cone on X is the image of the cone on Q by a map with contractible point inverses. J. West ([10]) has taken this last form of our result and shown that in the infinite dimensional setting it implies X actually has a mapping cylinder

Aspects on Applications of IR Reflectivity- and Raman Scattering-Experiments on Polaritons in Solid State Chemistry and Biophysics

Abstract The coupled phonon-photon state called “polariton” is an elementary excitation which exists at all temperatures in single crystalline materials with well defined translational symmetries. In the view of this aspect polaritons turn out to be more important for the dynamics of crystal lattices in general than e.g. magnons or plasmons which normally can be excited only in certain low temperature ranges. The present article summarizes experimental results which all might be applied as analytical methods. An improved method for the determination of fundamental mode frequencies is described. Polariton interactions with localized modes, and second order phonons and the relation to the ferroelectric phase transition mechanism is reviewed. Furthermore experiments concerning parametric light scattering, the TM-reflection technique, the ATR-method, and nonlinear optical experiments are summarized. We finally discuss the question of mechanical-electromagnetic energy conversion and some tentative future aspects.

Mapping cylinder neighborhoods of some ANR’s

Let X be a closed subset of a manifold without boundary Q. A codimension 0 submanifold M of Q is a mapping cylinder neighborhood of X if it is a closed neighborhood of X in Q and if there is a proper map r: AM-P X such that M is homeomorphic to the mapping cylinder of r, fixing AM and X in the natural way. For example, regular neighborhoods of locally finite complexes in PL manifolds are mapping cylinder neighborhoods. We do not insist that M be collared in Q since this can always be arranged by an isotopy of M into itself along the structure lines. Edwards ([5]) contains a lot of information about mapping cylinder neighborhoods. It follows immediately from the definition that subsets possessing mapping cylinder neighborhoods are retracts of the neighborhoods, hence are finite dimensional ANR's. The converse question, whether finite dimensional ANR's have mapping cylinder neighborhoods, at least for some embeddings in manifolds, is interesting. If they do, then compact, finite dimensional ANR's have finite homotopy type, since these neighborhoods are compact manifolds, and so have finite homotopy type by Kirby-Siebenmann (17]). We prove a partial result in the direction of the converse, namely, if X is a locally compact ANR, embedded as a closed, 1-LC codimension 4 subset of a finite dimensional manifold without boundary Q, then X x M, has a mapping cylinder neighborhood in Q x M, where either M. M= SI, or (M, Mj) = (R', [0, ao)). Our proof also works in infinite dimensions and yields the fact that if X is a locally compact ANR which is embedded as a closed Z-set in a Hilbert cube manifold Q, then X x M. has a cylinder neighborhood in Q x M. Taking suitable compactifications in the case (M, Mj) = (Ri, [0, ao)), one finds that the cone on X is the image of the cone on Q by a map with contractible point inverses. J. West ([10]) has taken this last form of our result and shown that in the infinite dimensional setting it implies X actually has a mapping cylinder

A fixed point criterion for linear reductivity

Let G be a linear algebraic group over an algebraically closed field. If for all actions of G on smooth schemes, the fixed point scheme is smooth, then G is linearly reductive under either of the additional assumptions: (a) the ground field is characteristic zero, or (b) G is connected, reduced, and solvable. Let K be an algebraically closed field and G a linear algebraic group over K. Say G has the smooth fixed point property if for all actions of G on a smooth scheme, the fixed point scheme is also smooth. Fogarty [1I has shown that if G is linearly reductive, then G has the smooth fixed point property. One can ask the converse question: If G is not linearly reductive, is there an action of G on a smooth scheme with a nonsmooth fixed point scheme? In this note we show how to construct such an action for any G that is a split extension by a unipotent subgroup. This gives an affirmative answer to the question for any class of groups where G being not linearly reductive implies G is a split extension by a unipotent subgroup. In particular this includes all groups of characteristic zero and in arbitrary characteristic connected reduced solvable groups. Let G be a linear algebraic group over K which is a split extension by the unipotent subgroup U. We first show that we may assume that U is the direct sum of copies of the additive group. If G modulo a normal subgroup has an action with a nonsmooth fixed point scheme, then certainly G does. Using this we can replace G by G modulo the commutator subgroup of U, and hence we may assume G is commutative. In characteristic zero this already implies U is the direct sum of copies of the additive group. In characteristic p there are truncated Witt groups; however, if we take G modulo p * U we may assume G is the direct sum of additive groups by a theorem of Serre [2]. We now construct an action of G on affine space with a nonreduced fixed point scheme. Using our assumption that G is a split extension, we Received by the editors April 16, 1974. AMS (MOS) subject classifications (1970). Primary 20G15. Copyright (C 1975, American Mathematical Society

Random walk point processes

Suppose there is known the set of points visited by a transient random walk on the real line, but that there is no information concerning the order in which these points are visited. Such a set of points may aptly be called a random walk point process (RWPP), and the object of this paper is to study the class of such processes. This class arises as a generalization of certain one-dimensional clustering processes, and while this is how one of us (D.J.D.) originally came to consider them, it is not the only reason for persevering in this study. Like renewal processes, which are a special sub-class of RWPP's, any random walk point process can be specified by means of the step distribution of the underlying random walk. On the other hand our study does not show whether or not the converse question of determining the step distribution from the RWPP is always feasible. The more demanding question of determining the random walk itself from the RWPP is obviously possible only for very special step distributions. Those results which we obtain below in closed form appear to depend on assuming that at least one tail of the step distribution is exponential. However such results have immediate relevance to the problem of inferring the step distribution from the point process. In the simplest case considered (both tails exponential) we are led to a new model for a stationary point process on the real line that is typically highly over-dispersed. And a final raison d'etre for our study is to exhibit connections with other stochastic processes, notably Poisson cluster processes and, perhaps surprisingly, Markov branching processes and birth and death processes. In Section 2 we introduce our notation and quote and discuss results needed in the sequel from the theory of stationary point processes and the theory of random walks. Integral equations for the one-dimensional counting distribution of the RWPP are written down in Section 3, and their solution is derived when the step distribution has a double exponential form. In Section 4 the second order counting properties are considered and an explicit solution for the second order intensity is derived for the double exponential case. The inter-point properties of the process are considered in Section 5, explicit results being obtained when the dominant tail of the step distribution is exponential. The connection with birth and death processes is shown in that Section, and in the concluding Section 6, the connections with Poisson cluster processes and Markov branching processes are mentioned.

Random walk point processes

Suppose there is known the set of points visited by a transient random walk on the real line, but that there is no information concerning the order in which these points are visited. Such a set of points may aptly be called a random walk point process (RWPP), and the object of this paper is to study the class of such processes. This class arises as a generalization of certain one-dimensional clustering processes, and while this is how one of us (D.J.D.) originally came to consider them, it is not the only reason for persevering in this study. Like renewal processes, which are a special sub-class of RWPP's, any random walk point process can be specified by means of the step distribution of the underlying random walk. On the other hand our study does not show whether or not the converse question of determining the step distribution from the RWPP is always feasible. The more demanding question of determining the random walk itself from the RWPP is obviously possible only for very special step distributions. Those results which we obtain below in closed form appear to depend on assuming that at least one tail of the step distribution is exponential. However such results have immediate relevance to the problem of inferring the step distribution from the point process. In the simplest case considered (both tails exponential) we are led to a new model for a stationary point process on the real line that is typically highly over-dispersed. And a final raison d'etre for our study is to exhibit connections with other stochastic processes, notably Poisson cluster processes and, perhaps surprisingly, Markov branching processes and birth and death processes. In Section 2 we introduce our notation and quote and discuss results needed in the sequel from the theory of stationary point processes and the theory of random walks. Integral equations for the one-dimensional counting distribution of the RWPP are written down in Section 3, and their solution is derived when the step distribution has a double exponential form. In Section 4 the second order counting properties are considered and an explicit solution for the second order intensity is derived for the double exponential case. The inter-point properties of the process are considered in Section 5, explicit results being obtained when the dominant tail of the step distribution is exponential. The connection with birth and death processes is shown in that Section, and in the concluding Section 6, the connections with Poisson cluster processes and Markov branching processes are mentioned.

Embedding the complement of an oval in a projective plane of even order

A configuration D with parameters ( v, b, r, k) is an incidence structure ( P , B , I ), where P is a set of v “points”, B is a set of b “blocks” and I is an “incidence relation” between points and blocks such that each point is incident with r blocks, and each blok is incident with k points. A block may be regarded as the subset of those points of p with which it is incident. A regular two component pairwise balanced design with parameters v, k 1, k 2 r 1, r 2 ,1 is an incidence structure D = ( P , B 1 ∪ B 2, I ), where D i = ( P , B i , I ) is a configuration with parameters ( v, b i , r i , k i ), i = 1,2, b i = vr i/k i , and such that any two distinct points of P are contained in exactly one block of B 1 ∪ B 2. The configuration D i is called the i th component of D , i = 1,2. Hence by deleting the points of an oval S from a projective plane П of even order q, we obtain a regular two component pairwise balanced design for which v=q 2−1, b 1= 1 2 q(q−1), b 2= 1 2 (q+1)(q+2), r 1= 1 2 q, r 2= 1 2 (q+2), k 1=q+1, k 2q−1. In this paper, we investigate the converse question and prove: Given a regular two component pairwise balanced design with the above parameters, then if q ≠ 6, the design can be embedded in a projective plane of order q.

The semigroup of endomorphisms of a Boolean ring

The family (R) of all endomorphisms of a ring R is a semigroup under composition. It follows easily that if R and T are isomorphic rings, then (R) and (T) are isomorphic semigroups. We devote ourselves here to the converse question: ‘If (R) and (T) are isomorphic, must R and T be isomorphic?’ As one might expect, the answer is, in general, negative. For example, the ring of integers has precisely two endomorphisms – the zero endomorphism and the identity automorphism. Since the same is true of the ring of rational numbers, the two endomorphism semigroups are isomorphic while the rings themselves are certainly not.

Some Elementary Converse Problems in Ordinary Differential Equations*

In studying differential equations, the usual task is to determine properties of the solutions of such equations from a knowledge of the coefficient functions. The converse question, namely, of determining the coefficient functions from properties of solutions, also has significance. It has been studied especially in the case of Sturm-Liouville equations.A discussion of the inverse Sturm-Liouville problem can be found in [8, Chapter 8], where references are given to the work of W.A. Ambarzumiam, G. Borg, I.M. Gelfand, M.G. Krein, B.M. Levitan, N. Levinson and W.A. Marchenko on this problem. Work of a quite different character, but dealing also with questions of a converse type arising from Sturm-Liouville equations, has been done by O. Boruvka and his colleagues and students [2].

A “metric” characterization of reflexivity

I*=sup(x,f)' xeB I XI is finite for each fEB*. We call iff * the dual norm of jx|. It is clear from the above inequalities that if IfI * is a dual norm on B*, then f I * is equivalent to the natural norm |flJ * on B*. Thus, in general, each dual norm on B* is an equivalent norm on B*. The converse question is answered by the following theorem. THEOREM 1. Let {B, j | } be given with dual space {B*, || Then every norm on B * which is equivalent to II II* is a dual norm if and only if B is reflexive. The proof will make use of the following characterization of dual norms:

Estimators of a Location Parameter in the Absolutely Continuous Case

In the last decade there have been a number of papers dealing with the admissibility of translation invariant estimators of a location parameter. Blyth [2] treated sequential procedures in the case of normally or rectangularly distributed random variables. If $d$ is estimated and $\theta$ is the actual parameter value, for Blyth, op. cit., loss was measured by $W(|d - \theta|)$ where $W(\cdot)$ was a nondecreasing function on $\lbrack 0, \infty)$. In the same year Blackwell [1] treated the fixed sample size problem in the case of discrete random variables taking only a finite number of values. For Blackwell, op. cit., loss was measured by $W(d - \theta)$ where $W(\cdot)$ was assumed continuous and bounded from below but otherwise arbitrary. Blackwell showed that if the discrete random variables (which could be vector valued) took values only on the integer lattice points and if there was a unique minimax translation invariant estimator then it was admissible. Later papers by Karlin [7] and Stein [11] discuss the admissibility of Pitman's estimator for square error. In reviewing these results we discovered that if the loss satisfied \begin{equation*}\tag{0.1}0 \leqq x < y \text{implies} W(x) < W(y),\quad y < x \leqq 0 \text{implies} W(x) < W(y),\end{equation*} and if there were several minimax translation invariant estimators then no translation invariant estimator could be admissible. This and a related result constitutes Section 4. It was logical to ask the converse question, does uniqueness imply admissibility? In the case of square error Pitman's estimator (except for changes on sets of measure zero) is necessarily unique since the loss function is strictly convex. In the case of normally distributed random variables and symmetrical loss functions as considered by Blyth, op. cit., the sample mean is the uniquely determined minimax translation invariant estimator. In this paper we have restricted the discussion to random variables which have density functions relative to Lebesgue measure. Since it proved possible to deal with the question of admissibility for a larger class of estimators than the translation invariant estimators, we define generalized Bayes estimators as the solution of a minimization problem. Section 2 deals with the question of the existence of measurable solutions to the minimization problem. Sections 5 and 6 deal with the question, does uniqueness imply admissibility? Section 3 deals with the question of whether generalized Bayes estimators are strongly consistent and shows that under mild restrictions this is the case. Section 8 is a generalization of the results of Katz [8] for minimax estimators of $\theta \varepsilon \lbrack 0, \infty)$. We show how to construct such estimators whenever loss is measured by $W(d - \theta), W(\cdot)$ strictly convex, non-negative, $W(0) = 0$. Blyth, op. cit., proved admissibility only within the class of continuous risk functions. In Section 9 we remove this restriction. We then show that if the loss function $W(\cdot)$ is strictly convex and symmetrical then the sample mean based on $n$ observations is admissible, in the nonparametric context of estimating the mean of an unknown density function, within the class of all sequential procedures having expected sample size $\leqq n$.

On a special class of regular 𝑝-groups

The concept of a finite p-group being regular was first defined and studied by P. Hall in two papers [3; 4]. Besides proving theorems for such groups and showing they had several properties in common with abelian groups, Hall gave a few sufficient conditions for a p-group to be regular. These criteria, in a vague sense, showed that if the commutator structure or power structure of a p-group was suitably restrictive then the group was regular. For example, if the class of a finite p-group, as a nilpotent group, is less than p, then the group is regular. This result evokes the converse question, namely, if a p-group is regular, is its class necessarily bounded by some function of p. This is, however, false for odd primes p, since there are metacyclic p-groups which are regular and of arbitrarily high class. Nevertheless, we can weaken the query and ask whether, for any regular p-group G, is there necessarily any bound on the derived length of G as a solvable group? This is true if p = 2, since every regular 2-group is abelian. However, for p 2 5, P. Hall, in yet unpublished work, has constructed regular p-groups of arbitrary derived length. This leaves only the prime p = 3 to be considered. We shall answer the question raised above by the following result:

Political Behavior and Politics

THROUGHOUT the history of the study of politics, theories of political society have been based on theories of human nature. Thus the study of the psychology of political behavior lies firmly within the tradition of political thought. But a gap has developed between our theories of individual political behavior and our theories of how political society (or, to use the current term, the political system) operates. If we are to develop a science of politics-that is, a science that deals with the total political process on both the individual and the system level-we shall have to close this gap. Studies of the psychology of political behavior have asked a number of the right questions about the nature of Political Man: Is he rational, or does he behave the way he does because of social and psychological pressures he knows little about? Does he vote the way he does because of his social class, because his friends vote that way, or because his father voted (or didn't vote) that way? Does he know anything about the issues of politics? Does Political Man lust unceasingly after power? Or is his political lust merely the sublimation of some primitive childhood lust? These are important questions, and studies of the psychology of politics have given us important and fascinating answers to some of them. But though much is said' on the various social and psychological influences that lead to a particular voting choice, there seems to be little concern over who won the electioni.e., what the effect of all this is on the political system. Moreover, the converse question of the impact of the political system on individual political behavior has received less attention than it deserves. By sticking too close to the non-political forces that impinge on the individual and by ignoring for the most part the interrelationship between an individual's political behavior and the political system in which he lives, the study of political behavior has divorced itself somewhat from the study of politics.

A note on the location of the zeros of polynomials

Certain relationships exist between the location in the complex plane of the ratios (1) and the zeros of p(z). Takahashi has proved1 that every sector of the plane, with vertex at the origin, which contains all the zeros of p(z) must also contain all the ratios (1). The present note gives an answer to the converse question, that of finding restrictions on the location of the zeros of p(z) when all the ratios (1) are known to lie in a given sector of the plane.

GERMAN READING IN ELEMENTARY COURSES

THE German reading lesson is a subject which should receive more careful consideration. Too often teachers feel that they have performed their whole duty, when they have taught the elements of German grammar to a beginning class. In fact, they devote their whole attention to better methods of presenting the subjunctive mood, the passive voice, the modal auxiliaries and equally interesting topics, while the question of German reading is left to take care of itself. Such a state of affairs is deplorable, for really nothing is more vital in successful language teaching than that the proper reading habits be formed at the outset. The character of the material read and its mode of treatment during a student's first two years of language study not infrequently determine his whole attitude toward a particular language. For after all, the amount of language instruction given in class is slight compared with the amount of reading and independent study which the student must do, who is to have pleasure and success in his language pursuits. All that we can hope to do in the classroom is to awaken interest in the foreign language by the wise selection of reading material and by our showing the student how to read by himself intelligently and with increasing mastery and appreciation of the language. The teacher can supply inspiration, incentive and method. The student must do the rest. Accordingly, the right selection of the reading material in the earlier stages of language instruction is all important. The stories should be easy, full of human interest, abounding in action and devoid of many abstract terms. Reading that is too difficult will discourage the average student at the start and make a subject distasteful to him which otherwise might develop into a real joy. If the plot is clever and there is plenty of action, his attention is held; he is eager to do some extra reading in order to follow the thread of the story, and when he is called upon in class to answer conversational questions in the foreign tongue, he has the plot and incidents so well in mind that he easily understands the teacher's