Diameter of a (0, 1)-Matrix*

The experimenter asked if documented equivalence class membership would influence the development of shared discriminative stimulus function established through simple successive discrimination training. In Experiment 1, equivalence classes were established with two sets of 9 stimuli. Common stimulus functions were then trained within or across the equivalence classes. Greater acquisition rates of the simple discriminations with stimuli drawn from within the equivalence classes were observed. In Experiment 2, a third stimulus set was added with which no equivalence relations were explicitly trained. The findings of Experiment 1 were replicated, but the Set 3 results were inconsistent across subjects. The outcomes of the two experiments demonstrate that equivalence classes have an effect on other behavioral relations which requires further investigation.

Two experiments investigated possible interactions between existing equivalence relations among stimuli and the acquisition of simple discriminations involving the same stimuli. During the first phase of Experiment 1, two adult humans learned two sets of six conditional relations among arbitrary visual stimuli designed to produce two independent sets of three, three-member equivalence classes (Set 1 and Set 2). Subjects whose performance met accuracy criteria on tests for equivalence relations proceeded to a second phase in which subjects were taught, via reinforcement, to make the same response to a cluster of three stimuli (e.g., response 1 in the presence of A1, B1, and C1). The clusters of three stimuli were initially consistent with documented equivalence relations for both Set 1 and Set 2 stimuli. In the second half of this phase, the contingencies were realigned such that the simple discriminations being established were consistent with documented equivalence relations for Set 1 stimuli but inconsistent for Set 2 stimuli. Results showed that the simple discriminations were acquired faster for Set 1 stimuli than for Set 2 stimuli. Experiment 2 introduced a third set of stimuli as a control set and systematically replicated Experiment 1 with four subjects. This study found that three of four subjects acquired simple discriminations faster when the cluster of stimuli were drawn from within equivalence classes rather than across equivalence classes. The development of equivalence relations with Set 3 stimuli constrained the utility of the control condition in this experiment. The results are best interpreted as suggesting that existing equivalence relations among stimuli can interact with the development of analytic units such as simple discriminations to either facilitate or retard their development.

The purpose of this study was to examine the transfer of consequential (reinforcement and punishment) functions through equivalence relations. In Experiment 1, 9 subjects acquired three three-member equivalence classes through matching-to-sample training using arbitrary visual forms. Comparison stimuli were then given conditioned reinforcement or punishment functions by pairing them with verbal feedback during a sorting task. For 8 of the 9 subjects, trained consequential functions transferred through their respective equivalence classes without additional training. In Experiment 2, transfer of function was initially tested before equivalence testing per se. Three of 4 subjects showed the transfer without a formal equivalence test. In Experiment 3, 3 subjects were given training that gave rise to six new three-member conditional equivalence classes. For 2 of the subjects, the same stimulus could have either a reinforcement or punishment function on the basis of contextual cues that defined its class membership. Experiment 4 assessed whether equivalence training had established general or specific consequential functions primarily by adding novel stimuli in the transfer test. Subjects treated even novel feedback stimuli in the transfer test as consequences, but the direction of consequential effects depended upon the transfer of specific consequential functions through equivalence relations.

Similarity is an important way of categorizing mathematical objects. In this paper we consider how similarity between undirected graphs, or networks, can be defined. We do this by reducing any undirected graph, G = (V,E) to its irreducible spine, I. First we show that the reduction process, ω, is a unique transformation and thus is a well-defined function. Each irreducible spine, I, thus defines an equivalence class consisting of all graphs {G_i} such that G_i.ω = I. We then show that reduction by ω preserves many key properties. Specifically it preserves shortest path lengths in I, and (under rather general conditions) preserves the location of distance and betweeness centers. In essence, every member graph of the equivalence class will have much the same path structure. Next, an algorithm is presented which, given any irreducible spine I, will generate (randomly) a graph Gk in the equivalence class defined by I, that is, we must have G_k.ω = I. Since I.e.ω = I, e = ω^-1. It is of some importance in network analysis to be able to quantify the notion of network similarity; to be able to say that a network G1 is closer to the network G2 than to G3. A popular method of comparing similarity between two networks G1 and G2 is to calculate the Spearman coefficient based on corresponding node degrees. We do this for 5 random graphs in the equivalence class of our running example. Finally, we show that any irreducible spine consists of a system of cycles. We then note that every spine, I, is topologically equivalent to a cubic graph. Thus individual cubic graphs generate a second, and coarser, equivalence relation on the family of all undirected graphs, which we call its genus.

Recent research has demonstrated the transfer of induced mood functions through equivalence relations by means of a musical mood-induction procedure. The research described in this article replicated and extended such work, primarily with the inclusion of a baseline and two types of reversal procedures. First, 16 adult participants were trained and tested for the formation of two three-member equivalence classes (A1-B1-C1 and A2-B2-C2). A musical mood-induction procedure was used to induce happy and sad mood states in the presence of the B stimuli, and a transfer of mood functions test was administered to determine whether the corresponding mood functions had transferred via equivalence to the C stimuli. Participants were then exposed to either Reversal 1, in which only the functions attached to the B stimuli were reversed, or to Reversal 2, in which the structure of the equivalence classes was altered. At all stages throughout the experiment, participants were asked to rate their current mood state by using mood and incentive rating scales. The findings of the current study replicated the results of previous research in demonstrating a clear transfer of mood functions via equivalence relations. This outcome is compared with the existing literature about mood induction and the transfer of mood functions.

In a recent paper by H. KARZEL and R. STANIK, Abh. Math. Sem. Univ. Hamburg, a characterization of euclidean metric planes (the metric is defined there by a separable quadratic field extension L ⊃ K of the base field K) by means of a congruence relation on segments is given. Here we develop a theory of congruence of angles for these planes to get an axiomatic foundation, which also works when L ⊃ K is inseparable. The main result is of the following type: Let be an affine plane together with an equivalence relation $$\sphericalangle $$ onG x G, whereG denotes the set of lines — any equivalence class is then what we call an angle — such that some simple properties (W1) to (W3) of angles and one further condition (V) on equality of certain angles in quadrangles hold, then is pappian and $$\sphericalangle $$ results from a quadratic field extension L ⊃ K.

Hepaxpert-I is a medical expert system for interpretive analysis of hepatitis A and B serology findings. The knowledge base of HEPAXPERT-I contains 13 rules for hepatitis A and 106 rules for hepatitis B serology. After entering an input pattern of test results, a pattern matching algorithm based on indexing is employed as efficient look-up procedure for the interpretive text. Formally, a partition of the sets of possible serological finding patterns—64 for hepatitis A and 4,096 for hepatitis B serology—induced by an equivalence relation, divides the elements of the sets into disjoint subsets, the equivalence classes. Each equivalence class together with its associated interpretive text is represented as one rule that assigns to every member of the equivalence class one interpretive text. Since September 1, 1989, Hepaxpert-I has been routinely used at the hepatitis serology laboratory of the 2nd Department for Gastroenterology and Hepatology of the University of Vienna Medical School (Vienna General Hospital). By February 28, 1990, 3,408 interpretive reports had been automatically generated and sent out. Beforehand, the system had been validated with 24,242 serological test results. The acceptance of the generated reports by the physicians requiring the findings is extremely high.

An examination of the translation invariance of V0 under dyadic rationals is presented, generating a new equivalence relation on the collection of wavelets. The equivalence classes under this relation are completely characterized in terms of the support of the Fourier transform of the wavelet. Using operator interpolation, it is shown that several equivalence classes are non-empty.

<p>This paper demonstrates the pontential of verification based on state spaces reduced by equivalence relations. The basic observation is that quite often, some states of a system are similar, i.e., they induce similar behaviours. Similarity can be formally expressed by defining equivalence relations on the set of states and the set of actions of a system under consideration. A state space can be constructed, in which the nodes correspond to equivalence classes of states and the arcs correspond to equivalence classes of actions. Such a state space is often much smaller than the ordinary full state space, but does allow derivation of many verification results.</p><p>Other researches have taken advantage of the symmetries of systems, which induce a certain kind of equivalence. The contribution fo this paper is to show that a more general notion of equivalence is useful. As example, a communication protocol modelled in the formalism of Coloured Petri Nets is verified. Aided by a computer tool supporting state spaces with equivalence classes, significant reduction of state spaces are exhibited.</p><p><strong>Topics</strong>. State space reduction methods, equivalence vs. symmetry, High-level Petri Nets, communication protocols.</p>

Reversible computation is one of the intensively developing research areas nowadays in the field of computer science and engineering. Although the field of reversible circuit synthesis has been studied during the last 15 years few attempts have been published to propose classification of reversible functions. In an earlier paper we considered whether all component functions of a reversible Boolean function either can have the same property in the sense of classical logic synthesis or can belong to the same equivalent class under some equivalence relations. This problem has a direct relationship to studying different aspects of classification of reversible functions. In this paper we present reversible ternary functions of any number of variables with all component functions having the same property or belonging to the same equivalence class.

Extension of the category Og and a vanishing theorem for the Ext functor for Kac-Moody algebras

Many sophisticated schemes are springing up recently with better characteristics, such as higher capacity and better security. However, if we tune the size of the secret message progressively, most methods do not provide the progressive quality characteristic which means that the relationship between the quality of stego image and the size of the secret message could be represented by a smooth curve without any jump points. This paper designs a novel hiding strategy based on an equivalence relation, which not only provides the progressive quality characteristic but also enhances remarkably the quality of stego image without sacrificing the security and capacity compared with original steganography schemes. In the proposed strategy, all hiding units can be partitioned into equivalence classes according to a constructed equivalence relation based on the capacity of hiding units. Following that, the hiding procedure is performed in predefined order in equivalence classes as the traditional steganography scheme. Because of considering the relation between the length of message and capacity, the hiding method using proposed hiding strategy outperforms the original approaches when embedding same message. Experimental results indicate that the proposed strategy gains up to 4.0 dB over existing hiding schemes.

In previous papers, we have developed the theoretical background for the design of deadlock-free adaptive routing algorithms for store-and-forward and wormhole networks. Some definitions and theorems have been proposed, developing conditions to verify that an adaptive algorithm is deadlock-free, even when there are cyclic dependencies between channels. Also, two design methodologies have been proposed. In this paper, we propose a partial order between channels as well as an equivalence relation. This relation splits the set of channels into equivalence classes. Then, we extend our previous theory by considering equivalence classes (channel classes) instead of channels. This extension drastically simplifies the verification of deadlock freedom for adaptive routing algorithms with cyclic dependencies between channels. Finally, we present an example.

The paper presents a novel, rough set-based clustering method that enables the evaluation of classification knowledge simplicity during the clustering procedure. The method iteratively refines equivalence relations so that they become a more simple set of relations that give adequate coarse classification to the objects. At each step of the iteration, the importance of the equivalence relation is evaluated on the basis of the newly introduced measure, indiscernibility degree. An indiscernibility degree is defined as a ratio of equivalence relations that classify the two objects into the same equivalence class. If an equivalence relation has the ability to discern two objects that have a high indiscernibility degree, a very fine classification is performed and then modified to regard them as indiscernible objects. The refinement is repeated, decreasing the threshold level of indiscernibility degree, and finally simple clusters can be obtained. Experimental results on the artificial data shows that iterative refinement of equivalence relation leads to successful generation of coarse clusters that can be represented by simple knowledge.

For p prime and for a field F containing a root of unity of order p, we generalize Marshall's equivalence relation on orderings to arbitrary subgroups of F× of index p. The equivalence classes then correspond to free pro-p factors of the maximal pro-p Galois group of F. We generalize to this setting results of Jacob on the maximal pro-2 Galois group of a Pythagorean field.