Equivalence Classes Of Functions Research Articles

Modelling residential location choice in an area with spatial barriers

A model is presented for residential location choice in rural areas with spatial barriers. We address the problem through comparative static analysis focusing on how residential location choices are affected by a new road link across the spatial barrier. We proceed through a probability theoretical approach: choose a family of utility functions representing every possible location, and equip this family with a probability measure. Then choose a representative within an equivalence class of utility functions, and represent the probability distribution by a parametrized family of distributions. Our analysis demonstrates that investments in new road links do not necessarily represent an adequate instrument for achieving ambitions in regional policy. We identify reasonable situations where a new road link could just as easily generate net migration from the area in which the investments are directed. In general, our analysis demonstrates how agglomeration and centralisation tendencies can be considerably affected by transportation infrastructure innovations.

A new approach to the formation of equivalence classes in pigeons.

Four pigeons were given simultaneous discrimination training with visual patterns arbitrarily divided into two sets, with the stimuli in one set designated A1, B1, C1, and D1 and those in the other set designated A2, B2, C2, and D2. In sequentially introduced training phases, the pigeons were exposed to a series of reversals to establish AB and then CD equivalences. In subsequent testing sessions, a subset of stimuli from one set served as positive stimuli and those from the other set as negative stimuli on training trials, and transfer of the reinforced relation to other members of the sets was tested with nonreinforced probe trials. The pigeons were trained further on AC and BD equivalences and then were tested for the emergence of untrained AD and BC equivalences. Two of the 4 pigeons exhibited the emergence of one of these untrained equivalences, evidence for the emergence of transitive relations. This finding suggests that the pigeons established three-member functional equivalence classes by incorporating separately trained multiple equivalence relations. Repeated reversal training and probe testing enabled us to explore the formation and expansion of functional equivalence classes in pigeons.

The development of emergent differential sample behavior in pigeons.

Three experiments attempted to replicate Manabe, Kawashima, and Staddon's (1995) finding of emergent differential sample behavior in budgerigars that has been interpreted as evidence of functional equivalence class formation. In Experiments 1 and 2, pigeons initially learned two-sample/ two-alternative matching to sample in which comparison presentation was contingent on pecking one sample on a differential-reinforcement-of-low-rate (DRL) schedule and the other on a fixed-ratio (FR) schedule. Later, two new samples were added to the task. Comparison presentation on these trials occurred after the first sample peck following a predetermined interval (Experiment 1) or after completion of either the DRL or FR requirement, whichever occurred first (Experiment 2). Experiment 1 found no evidence for emergent spaced versus rapid responding to the new samples as they established conditional control over the familiar choices. By contrast, differential responding did emerge for some pigeons in Experiment 2, with responding to each new sample coinciding with the pattern explicitly conditioned to the original sample occasioning the same comparison choice. This emergent effect, however, disappeared for most pigeons with continued training. Experiment 3 systematically replicated Experiment 2 using differential peck location as the sample behavior. Differential location pecking emerged to the new samples for most pigeons and remained intact throughout training. Our findings demonstrate a viable pigeon analogue to the budgerigar emergent calling paradigm and are discussed in terms of equivalence- and non-equivalence-based processes.

Rule following in functional equivalence classes

In Experiments 1 and 2 a match-to-sample procedure was used to establish the prerequisite conditional discriminations for two 3-member equivalence classes (A1, B1, C1 & A2, B2, C2) where stimuli A1 and A2 were written instructions. In Experiment 1 tests for transitivity confirmed the emergence of derived relations. Following this, an off-the-baseline test was given to verify the existence of functional equivalence classes. During off-the-baseline testing stimuli B and C from both classes controlled responding appropriate to the instructions contained in A1 and A2 respectively. In Experiment 2, the omission of tests for transitivity and the reversal of instructions contained in A1 and A2 did not impede the emergence of functional equivalence classes. In Experiment 3, classes A1, B1, C1 and E1, F1, G1 were linked via D1 to produce one class (A1, B1, C1, D1, E1, F1, G1); another two classes (A2, B2, C2 & E2, F2, G2) were linked with D2 to produce A2, B2, C2, D2, E2, F2, G2. The A and G stimuli in each class contained competing sets of instructions. When transformation of function was examined in the first class, the functions at C1 and E1 were controlled by the rules at A1 and G1 respectively; D1 controlled responding in accordance with the instructions at A1. Results from all experiments are discussed in the context of procedures designed to investigate transfer/transformation of function.

Relating articulation and acoustics through a sinusoidal description of vocal tract shape

A description of vocal tract shape for vowels is given in terms of a weighted sum of two sinusoids, displaced from each other along the length of the tract. The impetus for the description comes from uncovering a particularly simple relationship between F2 and a function on the parameters of a model of articulatory data. When the statistically derived articulatory model is simplified as displaced sinusoids, the relationship can be easily understood in terms of the physics of acoustic tubes. Thus, while the description is based on articulatory data, it relates simply to acoustics. The function (the sum of the weights of the two sinusoids) creates sets of articulatory shapes that can be said to trade in the amount of tongue front raising and the amount of tongue back raising to produce a constant F2. Alternatively, these equivalence classes of shapes can be said to trade in place of articulation and amount of constriction: this is so despite the fact that neither of these two variables are explicit parameters in generating the shapes. In embodying this compensatory property, it is also a description that is consistent with theories of motor control based on specifying functional equivalence classes, which are controlled with few degrees of freedom and allow context-conditioned variability.

A general differentiation theorem for multiparameter additive processes

Let $(L,\| \cdot \| _{L})$ be a Banach lattice of equivalence classes of real-valued measurable functions on a $\sigma $-finite measure space and $T=\{ T(u):u=(u_{1}, \dots,u_{d})$, $u_{i}>0$, $1\leq i\leq d\} $ be a strongly continuous locally bounded $d

Boundary Value Problems for the Stationary Schrodinger Equation on Riemannian Manifolds

We suggest a new approach to the statement of boundary value problems for elliptic partial differential equations on arbitrary Riemannian manifolds which is based on the consideration of equivalence classes of functions on a manifold. Using this approach, we establish some interrelation between the solvability of boundary value problems and solvability of exterior boundary problems for the stationary Schrodinger equation. Also we prove the comparison and uniqueness theorems for solutions to boundary value problems in this statement and obtain sufficient conditions for solvability of boundary value problems when the coefficient in the Schrodinger equation is changed.

The Effects of Response Topography on Functional Equivalence Class Formation

To examine the transfer of function phenomenon two 3member equivalence classes were established using the typical selection response of key pressing (A 1, 81, C1, & A2, 82, C2) . One 3-member class was then made functionally equivalent by training a topographically distinct response (clapping) to A 1 and obtaining clapping at 81 and C1. Five experiments examined the effects of adding a new function (stamping) to members of this original functional equivalence class. In Experiment 1 a new function was added to a three-member functional equivalence class by training it to a stimulus other than where the original function trained. The results indicated that for adults and 12-year­ old children the new function transferred to all original class members. However, for two 6-yr-old children the new function did not transfer to all class members. Experiments 2, 3, 4, and 5 were conducted exclusively with children who were approximately 6 years old and addressed issues arising from the findings of Experiment 1. Specifically, these experiments investigated the effects of the number of presentations of the stimuli in the transfer of function testing stage, where the new function was trained in relation to the original function, when the function was trained in relation to the actual equivalence training, and how the functions were trained. The general finding was that the new function rarely transferred to all members of an established three-member functional equivalence class for children who were approximately 6 years old, unless the new function was trained to the stimulus where the original function was trained. These findings may have relevance to the debate on both the constitution of class membership and contextual control of equivalence responding. Stimulus equivalence has attracted attention because of its potential contribution to the understanding of complex behavior including symbolic behavior, language (Hayes, 1989, 1991; Hayes & Hayes, 1989; Sidman, 1986, 1990), and the ability to behave appropriately in novel situations

Enumeration of linear threshold functions from the lattice of hyperplane intersections

We present a method for enumerating linear threshold functions of n-dimensional binary inputs. Our starting point is the geometric lattice Ln of hyperplane intersections in the dual (weight) space. We show how the hyperoctahedral group O(n+1), the symmetry group of the (n+1)-dimensional hypercube, can be used to construct a symmetry-adapted poset of hyperplane intersections Deltan which is much more compact and tractable than Ln. A generalized Zeta function and its inverse, the generalized Möbius function, are defined on Deltan. Symmetry-adapted posets of hyperplane intersections for three-, four-, and five-dimensional inputs are constructed and the number of linear threshold functions is computed from the generalized Möbius function. Finally, we show how equivalence classes of linear threshold functions are enumerated by unfolding the symmetry-adapted poset of hyperplane intersections into a symmetry-adapted face poset. It is hoped that our construction will lead to ways of placing asymptotic bounds on the number of equivalence classes of linear threshold functions.

An observational equivalence among [formula omitted]-control policies

This paper constructs an equivalence class of objective functions that produce the same linear state-feedback decision rule. Members of this class are distinguished by the relative weight placed on state variability and the minimized H ∞ norm of the system’s transfer function. This result is used to show that in models of dynamic policy formation a concern for robustness in the presence of model uncertainty can substitute for the ad hoc incorporation of adjustment costs.

Balanced Boolean functions

Many common logic circuits such as adders, parity checkers and multiplexers realise Boolean functions that are true for exactly half their input combinations, and false for the other half; we refer to such functions as balanced. Recently, these functions have been shown to be very useful for testing logic circuits, and for data encryption in cryptography. Here, we present a general theory of balanced Boolean functions. We derive a necessary and sufficient condition for balance by establishing an equivalence between a balanced function f(X) and a bijection from X to itself. We then analyse the conditions under which functional compositions preserve balance, and examine some specific balance-preserving decompositions. A new characterisation of functional completeness in terms of balance is presented. Finally, we address the problem of counting equivalence classes of balanced functions.

Degrees of parallelism in the continuous type hierarchy

A degree of parallelism is an equivalence class of Scott-continuous functions which are relatively definable by each other with respect to the language PCF (a paradigmatic sequential language). We introduce an infinite (“bi-dimensional”) hierarchy of degrees. This hierarchy is inspired by representing first order continuous functions as hypergraphs. We assume some familiarity with the language PCF and with its continuous model.

Boolean functions classification via fixed polarity Reed-Muller forms

In this paper, we present a new method to characterize completely specified Boolean functions. The central theme of the classification is the functional equivalence (a.k.a. Boolean matching). Two Boolean functions are equivalent if there exists input permutation, input negation, or output negation that can transform one function to the other. We have derived a method that can efficiently identify equivalence classes of Boolean functions. The well-known canonical Fixed Polarity Reed-Muller (FPRM) forms are used as a powerful analysis tool. The necessary transformations to derive one function from the other are inherent in the FPRM representations. To identify uniquely each equivalence class, a set of well-known characteristics of Boolean functions and their variables (including linearity, symmetry, total symmetry, self-complement, and self-duality) are employed. It is shown that all the equivalence classes of four-variable functions are uniquely identified where majority of the classes have a single FPRM form as their representative. The Boolean matching has applications in technology mapping and in design of standard cell libraries.

On sublinear functionals defined on the space of bochner integrable functions

UDC 513.88 It the article we study sublinear functionals that are defined on the space of Bochner integrable functions and possess the properties of decomposition and scalar compactness. We give an integral representation for the functionals. Such functionals are used in proving theorems on existence of continuous selections for a family of lower semicontinuous multivalued mappings with decomposable closed nonconvex values in the space of Bochner integrable functions. The article relates to the studies [1-3] and others. 1. We present basic definitions, notation, and a series of auxiliary results. Throughout what follows, X is a separable Banach space with norm H" ]l, T is a Hausdorff compact space with positive Radon measure/z and a-algebra ~ of/,-measurable subsets, and R stands for the real axis. We denote by L1 (T, X) the Banach space of equivalence classes of Bochner integrable functions u: T ---* X, with the norm I[u(.)HL1 = fJ[u(t)Hd#. T By X t and L ~ (T, X) we mean the topological dual spaces of X and L1 (T, X) with the strong (norm) topologies; (x, xl~ and (u(-), u'(.)) are the canonical bilinear forms establishing duality between X and X' and between LI(T, X) and UI(T , X) respectively. For the norm of X ~ we use the same notation as for that of X. The spaces X' and LI(T,X ) endowed with weak a(X~,X) - and a( L~ (T,X), Ll(T,X) )-topologies will be denoted by a-X' and a-L](T,X) (see [4, p. 6831). A function v : T ~ X I is called scalarly measurable if, given x E X, the scalar function (x, v(t)) is measurable. A function v : T ~ X ~ is called weakly measurable if, for every e > 0, there is a closed set Te C T, p(T \ Te) <_ e, such that the restriction of v to Te is a continuous function from T~ into a-X ~. We denote by Loo(T, X') the Banach space of equivalence classes of scalarly measurable, essentially bounded functions v : T --* X', with the norm IIv(.)Hoo = esssup Hv(t)H. Henceforth we identify lET equivalence classes in the spaces L~ (T, X) and L~ (T, X') with their representatives. It is well known (see [4, p. 807]) that U I(T,X) and Loo(T,X') are linearly isometric and that each element x'(.) e L~(T,X) is representable as (x(.),x'(.)) = f(x(t),v(t))d#, x(.) �9 LI(T,X), where T v(.) �9 Loo(T,X'). Therefore, we shall by convention regard Loo(T,X ~) as strongly dual to LI(T, X) and write (x(.),x~(.)) = f(x(t),x~(t))d#, x(.) �9 LI(T,X), considering x~(.) as an element of the space

Formation of Generalized Equivalence Classes

A category can contain either an infinite or limited number of stimuli. Categories with an infinite number of stimuli have been called open-ended; they contain stimuli that are physically similar. Categories with a limited number of stimuli include functional classes and equivalence classes; they contain stimuli that are physically disparate. The processes of equivalence class formation and primary generalization acting in combination yield a category containing stimuli that are physically similar as well as those that are physically disparate. Such a category is called a generalized equivalence class. A response trained to one member of such a class transfers to all remaining members of that class, yielding a generalized functional class. The extension of equivalence classes through generalization is predicted from primary generalization gradients obtained before class formation. Generalized equivalence classes and generalized functional classes bring together the study of stimulus equivalence and open-ended categories. They also provide a plausible account of the establishment of complex categories observed in natural settings.

The Search for Stimulus Equivalence in Nonverbal Organisms

A currently unresolved issue in stimulus equivalence research is the relation of equivalence and language competence. The current consensus is that equivalence according to widely adopted criteria proposed by Sidman has not yet been demonstrated in nonverbal experimental subjects. This judgment rests primarily on the failure of nonhuman subjects to display the equivalence outcome under conditions comparable to those used with human subjects. We discuss theoretical and methodological issues that arise in research seeking to delineate the “lower limits” of equivalence phenomena in studies with laboratory animals or nonverbal humans. We propose a behavior-analytic approach to the question, examine the relationship between functional stimulus classes and equivalence classes, and describe some preliminary studies.

Null-plane dynamics of particles and fields

The focus of this review of null-plane dynamics is the fundamental principle of quantum theory that states must form a linear manifold with a positive scalar product. The intent is to provide an integrated overview of diverse aspects of null-plane dynamics of particles and fields. Hamiltonian particle dynamics is based on the construction of nontrivial representations of the Poincaré group on finite tensor products of single-particle spaces, and on finite direct sums of such tensor products. The structure of the space of states and the representations of a kinematic subgroup are independent of the dynamics. The dynamics is specified by Hamiltonian operators which are the Poincaré generators outside the kinematic subgroup. Fock-spaces are infinite direct sums of tensor products of single particle spaces. Fock-space representations of Lagrangean field theories can be formulated as limits of Hamiltonian many-body dynamics. The required cutoff can preserve the symmetry of the kinematic subgroup but destroys the full Poincaré invariance. The nontrivial questions involve the existence of limits as the cutoff is removed. Covariant wave functions arise either as solutions of covariant wave equations, or as matrix elements of products of covariant field operators. The linear manifold of states is a manifold of equivalence classes of covariant functions. The dynamics appear in the nontrivial inner product and all Poincaré transformations are kinematic. Null-plane restrictions of the covariant functions may provide a unitary map of the covariant constraint dynamics into null-plane Hamiltonian dynamics.

On the upper and lower semicontinuity of the Aumann integral

Let ( T,τ,μ) be a finite measure space, X be a Banach space, P be a metric space and let L 1(μ, X) denote the space of equivalence classes of X-valued Bochner integrable functions on ( T,τ,μ). We show that if φ: T× P→2 X is a set-valued function such that for each fixed pϵ P, φ(·, p) has a measurable graph and for each fixed tϵ T, φ( t,·) is either upper or lower semicontinuous then the Aumann integral of φ, i.e.,∫ T φ( t, p)d μ( t)= {∫ T x( t)d μ( t): xϵS φ ( p)}, where S φ ( p)= { yϵL 1( μ, X): y( t) ϵφ( t, p) μ−a.e.}, is either upper or lower semicontinuous in the variable p as well. Our results generalize those of Aumann (1965, 1976) who has considered the above problem for X= R n , and they have useful applications in general equilibrium and game theory.

On the convexity coefficient of Orlicz spaces

In the whole paper it will be assumed that (T, X, #) is a space of a non-atomic infinite measure. The symbol 4~ stands for an Orlicz function, i.e. 9 is a convex, even, vanishing and continuous at zero function defined on the real line IR and with values in [0, + ov]. For a given Orlicz function q~ the Orlicz space Lr is defined as the set of all equivalence classes of N-measurable real functions f defined on T such that

Small Systems Convergence and Metrizability

The abstract notion of convergence of functions with respect to a small system has its roots in the concept of convergence of functions in measure. The second author has shown that convergence of functions with respect to a small system generates a Fréchet topology. In this paper we show that convergence with respect to a small system is equivalent to convergence with respect to a certain complete pseudometric (or metric if we consider equivalence classes of functions with respect to the small system).