It is well known that the group of all nonsingular lower block -triangular pxp matrices acts transitively on the cone p* of all positive definite pxp matrices. This result has been applied to obtain several major reSUlts in mUltivariate statistical distribution theory and decision theory. Here a converse is established: if a matrix group acts transitively on P*, then its group algebra must be (...

Interval Pairwise Comparison Matrices have been widely used to account for uncertain statements concerning the preferences of decision makers. Several approaches have been proposed in the literature, such as multiplicative and fuzzy interval matrices. In this paper, we propose a general unified approach to Interval Pairwise Comparison Matrices, based on Abelian linearly ordered groups. In this ...

In this paper we extend one of the main tools used in veri cation of discrete systems namely Binary Decision Diagrams BDD to treat probabilistic transition systems We show how probabilistic vectors and matrices can be represented canonically and succinctly using proba bilistic trees and graphs and how simulation of large scale probabilistic systems can be performed We consider this work as an i...

The weighted geometric mean ordering vector and logarithmic least squares are proposed to aggregate information of decision makers to generate weight vector from uncertainty comparison matrices in uncertain group AHP model. Especially, when the judgment matrices are incomplete, the latter can be developed to derive priorities under the condition of minimal deviation. Some concepts in uncertaint...

In this document we provide a couple of exercises that help understand the basic concepts of game theory, including pure and mixed strategies, zero sum and non-zero sum games, and saddle-point and Nash solutions. The definition for concepts and the ideas for these exercises were taken in part from [1, 2]. Before we start, let us introduce some notation. We are going to assume that we have only ...

The paper considers calculation of autocorrelation functions on finite dyadic groups over decision diagrams. The methods exploit recursive structure of both autocorrelation matrices and decision diagrams. First, it is discussed calculation of the autocorrelation through the Wiener-Khinchin theorem implemented over decision diagrams. Then, it is proposed a method for calculation of separate auto...

In this paper we extend one of the main tools used in verii-cation of discrete systems, namely Binary Decision Diagrams (BDD), to treat probabilistic transition systems. We show how probabilistic vectors and matrices can be represented canonically and succinctly using proba-bilistic trees and graphs, and how simulation of large-scale probabilistic systems can be performed. We consider this work...

Algebraic structures sometimes need to be typed. For example, matrices over a ring form a ring, but the product is a only a partial operation: dimensions have to agree. Therefore, an easy way to look at matrices algebraically is to consider “typed rings”. We prove some “untyping” theorems: in some algebras (semirings, Kleene algebras, residuated monoids), types can be reconstructed from valid u...

let $ m , n in mathbb{n}$, $d$ be a division ring, and $m_{m times n}(d)$ denote the bimodule of all $m times n$ matrices with entries from $d$. first, we characterize one-sided submodules of $m_{m times n}(d)$ in terms of left row reduced echelon or right column reduced echelon matrices with entries from $d$. next, we introduce the notion of a nest module of matrices with entries from $d$. we ...

Few studies have examined the impact of age on reactivity to concurrent think-aloud (TA) verbal reports. An initial study with 30 younger and 31 older adults revealed that thinking aloud improves older adult performance on a short form of the Raven's Matrices (Bors & Stokes, 1998, Educational and Psychological Measurement, 58, p. 382) but did not affect other tasks. In the replication experimen...