Properties Of Linear Spaces Research Articles

The Theory of Knowledge Fields: A Thermodynamics Approach

The emergence of knowledge economy and knowledge management revealed the need for reconsidering the concept of knowledge in a larger framework than that created by philosophers from ancient times. While the epistemology as a theory of knowledge and justification considers knowledge as a justified true belief, experts in knowledge management consider knowledge as a strategic resource. The new economic interpretation of knowledge as a strategic resource and a key contributor to achieving a competitive advantage generated a search of new metaphors to supply the attributes needed in constructing the new framework of understanding and operating with a working concept of knowledge in management. The most widespread knowledge metaphors are based on analogies with stocks, flows, and stock-and-flows. These metaphors induce, beyond some useful attributes, the Newtonian mechanics paradigm which is limited by the properties of linear spaces and reversible processes. The purpose of this paper is to show how we can enrich the theory of knowledge by introducing the concepts of knowledge fields and knowledge dynamics based on metaphorical thinking and the thermodynamics principles. The focus of our research is the energy metaphor which considers energy as a source semantic field. The main outcome of the present research is that knowledge can be considered as a field, which is manifesting in different forms like energy. This thermodynamics framework opens new directions for research in knowledge management, decision-making and leadership.

Some comments on the properties of unitary transformations on linear spaces having an indefinite metric and the connection with the theory of spin

After a brief review of the history of the discovery of the spin, some fundamental properties of linear spaces having an indefinite metric are being discussed. The study starts with an elementary survey of the theory of matrices and their stability problem. It is emphasized that—by a similarity transformation—all matrices may be brought to classical canonical form characterized by the diagonal elements called eigenvalues, their multiplicities, their Jordan blocks, and their Segre characteristics. In connection with the reduced Cayley-Hamilton equation, the existence of the product projection operators and their main properties is briefly discussed. Particular attention is paid to the concept of a basis for the linear space and the associated metric matrix, which is self-adjoint and may be brought to diagonal form with the eigenvalues ± 1 by a unitary transformation, which reveals the indices of inertia, p and q. The Minkowski space having p = 3 and q = 1 is used as an example. After this introduction, some properties of linear operators defined on an indefinite space are discussed, and it is pointed out that self-adjoint operators and unitary operators may now have a rather peculiar and unexpected behavior, and the special Lorentz transformations are used as an example. It is then shown that these features are of essential importance in studying rotations as special cases of unitary transformations defined on an indefinite space. The rotations are here defined by means of their reduced Cayley-Hamilton equation, and their properties are studied by means of the associated product projection operators, which are idempotent, mutually exclusive, and form a resolution of the identity. In a previous article, it was shown that, in a positive definite space, there is a close connection between the requirement that all rotations around an external axis form a group and the existence of an anticommutator algebra leading to the concept of spinors. The rotations are expressed in the exponential form U=exp(iO), where O is a self-adjoint operator which is independent of any choice basis, coordinate system, etc., and which is, hence, a true invariant. It is shown that this approach may now be extended also to indefinite spaces and may lead to operators O which are both rotationally and relativistically invariant. In this connection, the full Lorentz transformations are given a particularly simple form. The article may be considered as a simple exercise in linear algebra, in which the mathematical connection between the concept of rotations and the existence of spinors is strongly emphasized. © John Wiley & Sons, Inc.