George R. Barnes

Multichannel Signal Processing Based on Logic Averaging

The multichannel signal processor described provides an excellent capability for the detection of any kind of signal, on-line, with a SNR of 0.1 dB at a wide frequency range. This very efficient and inexpensive approach to improving the SNR by the processor wiUl reduce the need for other signal processing techniques. The processors reaction on an input signal and noise has been analyzed by a mathematical model. This effect is IMustrated by an in vivo noninvasive detection on a beat-by-beat basis of electrical activity of the His¿Purkinje system of the heart, and the detection of a pulse-like signal from a generator.

A mathematical model for interpersonal relationships in social networks

This paper uses the mathematical structure of algebraic semigroups to model interpersonal relationships. The basis for the model is the periodic surveys administered to each member of a social network. The model uses individual preference to uncover the underlying structure of the team. The model may possibly be used to study the stability of the network.

Ranks of Semigroups Generated by Order-Preserving Transformations with a Fixed Partition Type

Abstract The rank of a finite semigroup is the minimal size of its generating set. Let T n be the semigroup of all total transformations of the set {1, 2,…, n}, and let 𝒪 n be the subsemigroup of T n of all the order-preserving transformations whose images consist of at most n − 1 elements. We study the subsemigroups S 𝒪(τ) of 𝒪 n generated by the order-preserving transformations whose kernels are partitions of X n of a given partition type τ. We characterize idempotent-generated semigroups S 𝒪(τ), and show that S 𝒪(τ) is idempotent-generated precisely when it is regular, and that occurs if and only if the weight r of τ is n − 1 or 1. For arbitrary partition type τ of weight r we determine the Greens relations on S 𝒪(τ), and we show that its rank equals to .

Construction of well separated interaction semigroups to refine the modeling of social networks

Abstract The techniques developed in this paper suggest an objective standard for the process that refines the generation of interaction semigroups associated with social networks. A constructed interaction semigroup S is assessed with respect to measurements used in the original construction. This information, summarized in the pair (m1, m2)(S) associated with the semigroup S, is a comparison of the closeness of the elements of the same class as compared to the closeness of the elements from distinct classes of S. It is desirable to have more separation between the elements from distinct classes and more “closeness” between the elements within a class. We use this idea to produce a separated interaction semigroup with more separation between distinct classes.

Algebraic structure of the interaction semigroup as related to the homogeneity of network

This paper addresses the development of a semigroup model of social networks. Data matrices which represent the perceived relationships between members of a social network are used to construct a (possibly infinite) data semigroup of derived relations defined by (real) matrix multiplication. This complex structure is analyzed by forming interaction semigroups. These semigroups are homomorphic images of the data semigroup. The corresponding congruences are generated by identifying products of finite order which are highly positively correlated. Several methods of generating the interaction semigroups are examined and are shown to generate nonhomomorphic semigroups. For each congruence, an associated triple of numbers can be defined which may serve as an indicator of the validity and/or a measure of the stability of the semigroup model. A series of hypothetical examples is developed to study how the algebraic properties of interaction semigroups reflect and uncover properties of associated networks. Specifi...

Computer Assisted Instruction (CAI) for using statistical tables

In this paper the authors look at the use of Computer Assisted Instruction in the teaching of the use of statistical tables.

Weighted interaction semigroups

Current semigroup models of social systems assume that the system is uniform, that is all the units and all the connections in the system have the same influence. Obviously most social structures are not uniform; some units and connections between some units are more influential and important than some others. This paper initiates a development of the semigroup model of social structures that reflects this non‐uniformity by using weighted inner products in generation of interaction semigroups.