Finite Means Research Articles

Semantic compositionality: still the only game in town

It is standardly assumed that theories of meaning for natural languages will be compositional. That is, a theory of meaning for a language, E, will consist of a characterization of the semantic features of some set of primitive components of E, along with rules for recursively determining the semantic features of the more complex expressions of E. The main line of support for the compositionality of a theory of meaning, and more importantly for the existence of compositional theories of meaning, arises from the following considerations. First, speakers can comprehend an infinite or at least an indefinitely large number of sentences. In order to explain how we manage to understand indefinitely many sentences we must assume that we have a compositional understanding. That is, we are able to understand indefinitely many sentences because our understanding of sentences is a matter of our understanding parts of sentences. Second, it is generally accepted that a theory of meaning should characterize the explicit or implicit knowledge that is applied in understanding. So a semantic theory will explain the meanings of sentences by explaining the meanings of parts of sentences. That is, if we are going to explain (semantic) language understanding, we will need a compositional theory of meaning. As part of an ambitious attempt to persuade us that a theory of meaning is not feasible, Stephen Schiffer [2] has challenged this 'productivity' argument for a compositional theory of meaning by presenting a hypothetical case of a language understander whose means of understanding languages supposedly fails to support compositionality. He has also provided an argument that is supposed to show that it is possible for us to explain a (hypothetical) mentalese (i.e. language of thought) without the need for a compositional theory of mentalese. As I will show, though, his cases fail to address the central concern of the productivity argument, namely the explanation of infinite understanding through finite means, and are therefore unsuccessful attacks on our commitment to a compositional theory of meaning.

Regenerative closed queueing networks

Consider a closed queueing network of single-server stations, as introduced by Jackson, Gordon-Newell and Whittle, but with service times that are arbitrarily distributed with finite means. We verify that such a network has a unique stationary distribution if one of its service times is unbounded. The proof is carried out by modeling the network as a piecewise-deterministic Markov process which is shown to be regenerative. If, furthermore, the unbounded service-time distribution is non-lattice, then the stationary distribution is also the unique equilibrium. When the service-times enjoy finite second moments, and the number of customers in the network grows indefinitely, this equilibrium (properly normalized) converges to the equilibrium of a reflected Brownian motion on a simplex

On the Deskins index complex of a maximal subgroup of a finite group

We analyze the influence of the maximal subgroups in the structure of a finite group G G by means of the index complex introduced by Deskins in Proc. Sympos. Pure Math. vol. 1, Amer. Math. Soc. 1959, pp. 100-104.

A Note on the Large Sample Properties of Linearization, Jackknife and Balanced Repeated Replication Methods for Stratified Samples

Krewski and Rao consider inference for a (nonlinear) function of a vector of finite population means $\theta = g(\bar{Y}).$ For a sequence of finite populations with increasing number of strata, they demonstrate that $\hat{\theta} = g(\bar{y})$ is asymptotically normal, where $\bar{y}$ is the usual unbiased stratified estimator of $\bar{Y}.$ Additionally, they demonstrate that $(\hat{\theta} - \theta)/\upsilon^{1/2}(\hat{\theta})$ is asymptotically a standard normal distribution, where $\upsilon(\hat{\theta})$ is a variance estimator obtained using linearization, jackknife or balanced repeated replication (BRR) methods. In this note we extend their results to when the partial first derivatives $(g_1(\mu), g_2(\mu),\ldots, g_p(\mu)) \equiv 0,$ where $\mu$ is the limit of $\bar{Y}$ with increasing number of strata. We explore the asymptotic distribution of $(\hat{\theta} - \theta)/\upsilon^{1/2}(\hat{\theta})$ and show (1) that it is no longer normal and (2) that it depends upon which variance estimator is used. We describe an application of these results to hypothesis testing using complex survey data.

Hierarchical baxes predictive means and variances with application to sample survey inference

Samples are observed from k populations having means ,(i and distributed according to a natural exponential family with quadratic variance function (NEF-QVF). Assuming an exchangeable two-stage prior, conjugate at the first stage, formulas for posterior means, variances and covariances of the μi are expressed in a unified way for all NEF-QVF models in terms of integrals over the posterior density of the two hyperparameters. This posterior density is expressed simply in terms of the normalizing function for the conjugate prior.Posterior predictive means and variances for averages of new observations are also obtained. These formulas are applied, with very little effort, to the finite population situation where the samples are independent simple random samples from the k populations, and where an exchangeable NEF-QVF superpopulation model is assumed. Posterior (predictive) means and variances for finite population totals, means and proportions are obtained. These results are illustrated with an application ...

Optimal estimation of linear functions of finite population means in rotation sampling

Abstract Within a general class of linear design-unbiased estimators of a linear function of means from successive occasions in sampling in time, it is shown that under a general one-cycle rotation design, a weighted average of the rotation group means or elementary estimates is the optimal estimator. The determination of the weights depends upon the underlying correlation structure. The correlation structure studied here allows a correlation between different sampling units.

Quantum theory, the Church–Turing principle and the universal quantum computer

It is argued that underlying the Church–Turing hypothesis there is an implicit physical assertion. Here, this assertion is presented explicitly as a physical principle: ‘every finitely realizible physical system can be perfectly simulated by a universal model computing machine operating by finite means’. Classical physics and the universal Turing machine, because the former is continuous and the latter discrete, do not obey the principle, at least in the strong form above. A class of model computing machines that is the quantum generalization of the class of Turing machines is described, and it is shown that quantum theory and the 'universal quantum computer’ are compatible with the principle. Computing machines resembling the universal quantum computer could, in principle, be built and would have many remarkable properties not reproducible by any Turing machine. These do not include the computation of non-recursive functions, but they do include ‘quantum parallelism’, a method by which certain probabilistic tasks can be performed faster by a universal quantum computer than by any classical restriction of it. The intuitive explanation of these properties places an intolerable strain on all interpretations of quantum theory other than Everett’s. Some of the numerous connections between the quantum theory of computation and the rest of physics are explored. Quantum complexity theory allows a physically more reasonable definition of the ‘complexity’ or ‘knowledge’ in a physical system than does classical complexity theory.

Asymptotic properties of multistate random walks. I. Theory

A calculation is presented of the long-time behavior of various random walk properties (moments, probability of return to the origin, expected number of distinct sites visited) for multistate random walks on periodic lattices. In particular, we consider inhomogeneous periodic lattices, consisting of a periodically repeated unit cell which contains a finite number of internal states (sites). The results are identical to those for perfect lattices except for a renormalization of coefficients. For walks without drift, it is found that all the asymptotic random walk properties are determined by the diffusion coefficients for the multistate random walk. The diffusion coefficients can be obtained by a simple matrix algorithm presented here. Both discrete and continuous time random walks are considered. The results are not restricted to nearest-neighbor random walks but apply as long as the single-step probability distributions associated with each of the internal states have finite means and variances.

A numerical procedure for the generation of orthogonal body‐fitted co‐ordinate systems with direct determination of grid points on the boundary

AbstractThe procedure proposed is based on the solution by finite difference means of a set of Laplace's equations, by the application of a relaxation method.The curvilinear orthogonal grid so generated is fitted to a 2‐D physical domain with closed boundary and the contribution of the present work consists in the arbitrary choice of grid points on two adjacent boundaries, in order to achieve the desired density of grid points where the geometry of the boundaries varies rapidly.The method proposed is rapid and stable. Some characteristic examples are finally presented.

Languages and grammars of action and interaction: Some further results

This paper continues the work reported in Skvoretz and Fararo (1980) by advancing proofs for results that were only previously conjectured. The proofs show that under conditions of relatively wide generality, actions and interactions can be seen to have a language-like structure; that is, generated by grammars of action and interaction. From a systems viewpoint, the concern is with the temporal structure of the activity of living systems–organisms, groups, and/or organizations. The theory is applicable at any level of analysis to the extent that the actions of units at each level can be appropriately specified. The theory shows how a potentially infinite number of distinct strings of action and interaction can be produced by finite means available to an acting unit. Substantively, the theory offers a more sophisticated basis for the study of the structure of the activity of living systems, even when that activity contains apparently “creative” elements that would seem to argue against a structural analysis (Turner, 1962).

Finite Dimensional Compensators for Parabolic Distributed Systems with Unbounded Control and Observation

It is proved that for a class of parabolic distributed parameter systems with unbounded control and observation there exists a finite dimensional compensator using dynamic output feedback. Finite dimensional means here that the dynamics relating the input to the output is finite dimensional. A constructive design algorithm is presented and several examples are considered.

Another round with nonlinear acoustics

Early work in nonlinear acoustics largely followed the behavior of one or two idealized beams of high intensity sound. More recent work has centered on attempts to make the beams more realistic in nature, and to follow up on possible ways in which nonlinear acoustics can be applied to other problems of acoustics. In the former case, such analyses as that of Zabolotskaya‐Khokhlov on a tapered beam, or the use of truncated equations, especially in the study of nonlinear interaction of sound with air bubbles in water come to mind. There has also been considerable attention to the physical meaning of the nonlinear parameter. On the experimental side, much work has been done on the sound‐bubble interaction. In addition, the existence of higher harmonics in a search beam of finite amplitude means that some of the limitations presented by the frequency of the fundamental beam can be favorably modified. Examples will be discussed.

The topology of complete minimal surfaces of finite total Gaussian curvature

RECENTLY GREAT progress has been made in the classical theory of minimal surfaces in R3. For example, the proof of the embedding of the solution to Plateau’s problem for an extremal Jordan curve [lo] and the embedding of a solution to the free boundary value problem [ll], the proof of the bridge theorem [12], the regularity and finiteness of least area oriented surfaces bounding a smooth Jordan curve [5], the uniqueness and topological uniqueness of certain minimal surfaces 18, 9, 121, the proof that a complete stable minimal surface is a plane [l, 31, the existence of a collection of 4 Jordan curves which bound a continuous family of compact minimal surfaces [141, the existence of a complete minimal surface contained between two parallel planes [13] and a theorem [18] which states that if the Gauss map of a complete minimal surface misses 7 points on the ‘sphere, then the minimal surface is a plane. The aim of this work is to begin the exploration of the topological properties of complete minimal surfaces of finite total curvature. First we recall the fundamental classical result of Chern-Osserman [6] that states that such surfaces M are conformally equivalent to compact Riemann surfaces iii punctured in a finite number of points. Furthermore, they prove that the Gauss map on the surface M extends conformally to a. In particular, the theorem of Chern-Osserman implies that such surfaces have finite topological type with a finite number of topological ends and that the normal vectors at infinity on these ends are well defined. (Finite topological type means in this case that M is diffeomorphic to the interior of a compact surface with boundary.) Our first theorem gives a nice description of the topological placement in IX3 of a complete surface of finite topological type which has well defined normal vectors at infinity. We prove that the ends of such surfaces behave like the ends of the catenoid at infinity. More precisely, Theorem 1 shows that if f: M + R3 is an immersion of such a surface, then f is proper and the image f(M) viewed from infinity looks like a finite collection of flat planes (with multiplicity) that pass through the origin. For example, the catenoid viewed from infinity looks like two oppositely oriented copies of a plane passing through the origin. Theorem 1 has a natural generalization to submanifolds of arbitrary codimension in R”. This generalization which is Theorem 2 is discussed in

A Note on Product-Form Solution for Queuing Networks with Poisson Arrivals and General Service-Time Distributions with Finite Means

Product form was first introduced by Jackson [Operations Research, 1957; Management Science, 1963] who considered only negative exponential service distributions with queue length dependent service rates. Posner and Bernholtz [Operations Research, 1968] showed that one has product form in the case where there are many classes of customers, service times are exponential with class queue length dependent service rates, and lag times (the time for a customer to travel between two servers is a random variable with continuously differentiable distribution function). Baskett, Chandy, Muntz and Palacios [JACM, 1975] extended these results to networks with multiple classes of customers and service time distributions which have rational Laplace transforms vanishing at ∞. Chandy, Howard and Towsley [JACM, 1977] extended this further to the case where the service time distribution functions are continuously differentiable. They also introduced the concept of station balance and established some important relationships between station balance, local balance and product form. In particular, for the closed queueing network they showed that if each queue in the network satisfies local balance in isolation with Poisson arrivals then one has product form

Existence of Limiting Distributions in the GI/G/s Queue

A new proof is given for the Kiefer-Wolfowitz (Kiefer, J., J. Wolfowitz. 1955. On the theory of queues with many servers. Trans. Amer. Math. Soc. 78 1–18.) theorem: In the GI/G/s queue the sequence of workload vectors, and thus also the sequence of waiting times, has a proper limiting distribution independent of the initial conditions if and only if p < 0.1 (assuming the interarrival times and service times have finite means). From stochastic monotonicily, convergence is easy to establish if the system starts off empty. The limit is shown lo be proper here by bounding the standard FCFS system by the corresponding s-server system with a cyclic assignment discipline, using a recent construction of Wolff (Wolff, R. W. 1977. An upper bound for multi-channel queues. J. Appl. Probab. 14 884–888.). The general initial condition case is treated by appropriately bounding the original sequence above and below by sequences of countable-state Markov chains for which zero is always a regenerative state. This paper thus illustrates how recent stochastic comparison and continuity results can be applied to prove limit theorems. The bounding systems are also of special interest in regenerative simulation because they provide a way lo increase the number of regeneration points.

On the Noetherian property in endomorphism rings of certain comodules

In the above (and throughout this paper) Noetherian means right and left Noetherian and finite dimensional means of finite k dimension. By an R comodule we understand a left R comodule, that is, a pair (V, r) where V is a k-space and r: V-+R Ok V is a k map satisfying (1 @ 7)7= Q @ 1)7, (E @ I)7 = 1, the identity map. The terms injective and essential extension have the appropriate meanings in the category of left comodules. We refer the reader to [7] for the basic representation theory of comodules. All rings in this paper have an identity and we insist that the identity of a subring coincides with that of the overring. This work arose from question about the representation theory, in characteristic zero, of polycyclic-by-finite groups: the reader who feels that practical examples are lacking in the present work is advised to consult the companion paper [5]. Our results also apply to representations of finite dimensional Lie algebras and rational representations of algebraic groups in characteristic zero. At the heart of our proof of Theorem A lie certain results on completion in noncommutative Noetherian rings. These are dealt with in the first three 394 002 l-8693/8 l/060394-26%02.00/O

Stochastic Dominance Tests for Decreasing Absolute Risk-Aversion II: General Random Variables

Necessary and sufficient conditions are given for stochastic dominance over the class of decreasing absolute risk-averse utility functions. The random variables being compared may be continuous as well as discrete but are assumed to be bounded from below, to have finite means, to have only finitely many mass points in finite intervals, and to have cumulative distribution functions which cross one another only finitely many times, or touch one another only finitely many times in finite intervals. The precise forms of the dominance tests depend upon the number of times the distribution functions cross. An example of the DSD test is presented, and a dynamic programming algorithm for carrying out the general test is given.

An alternative proof of order statistics moment problem

AbstractA necessary and sufficient condition that two distributions having finite means are identical is that for any fixed integer r > 0, the expected values of their rth (n ≫ r) order statistics are equal [or the expected values of their (n‐r)th (n > r ≫ 0) order statistics are equal] for all n where n is the sample size.

Langue and Parole Since Saussure

Summary It is not possible and in fact not necessary to review everything that has been written since the publication of Saussure’s Cours de linguistique générale concerning the langue / parole diachotomy. What is attempted in the present article is first of all a concrete analogy for langue and parole, so that these two notions, which have been so often misunderstood, can be presented in as concrete a manner as possible, and secondly to follow certain interesting developments of the basic Saussurian notions. The concrete analogy which is used is that of the child’s construction set, such as a Meccano set for example. The set itself is an analogy for langue, and the models, such as planes, cars, machines, etc., that the child makes are a model for parole or discourse. This concrete analogy also serves to illuminate ‘the infinite use of finite means’, the finite means being langue, and the infinite use parole. Next an examination is made of certain of the Saussurean notions that have provoked discussion over the years: (1) langue as a pure ίοrm; (2) psychologism; (3) the sociological notions; (4) synchrony as static linguistics. Subsequent to (4) the notions of process and generation are examined, and this leads first of all to a critical examination of the terms competence and performance, along with their underlying notions, and then to a discussion of the psychomechanics of Gustave Guillaume. It is pointed out that Guillaume’s theory is an interesting development’ of the Saussurian notions and that some of the criticisms by Wunderli in HL 1:1/1974 are based upon a fundamental misapprehension of the notion of parole in Guillaume.

The limiting density of a nonlinear system

The RC filter-hard limiter-RC filter nonlinear system is the subject of this paper. Because of computational difficulties implicated in the analysis of the above system, only its response to the zero mean Gaussian system input has been analytically investigated. An approximate output density has also been found for nonzero mean Gaussian, while verified to be “close” to the real one for finite means. In the present paper, a close form of the system output density is obtained when the input mean tends to infinity. For that, ε-upcrossing methods were used.