In this paper, we outline the rudiments of the representation theory of semisimple Lie algebras. We build the necessary theory in order to analyze the representations of sl2, which includes proving that representations of semisimple Lie algebras are completely reducible and preserve the Jordan decomposition. We only assume the reader has a working knowledge of linear algebra and a little famili...

The resolvent (λI − A)−1 of a matrix A is naturally an analytic function of λ ∈ C, and the eigenvalues are isolated singularities. We compute the Laurent expansion of the resolvent about the eigenvalues of A. Using the Laurent expansion, we prove the Jordan decomposition theorem, prove the Cayley-Hamilton theorem, and determine the minimal polynomial of A. The proofs do not make use of determin...

We associate a covering relation to the usual order relation defined in the set of all idempotent endomorphisms projections of a finite-dimensional vector space. A characterization is given of it. This characterization makes this order an order verifying the Jordan-Dedekind chain condition. We give also a property for certain finite families of this order. More precisely, the family of parts in...

Let A belong to an automorphism group, Lie algebra or Jordan algebra of a scalar product. When A is factored, to what extent do the factors inherit structure from A? We answer this question for the principal matrix square root, the matrix sign decomposition, and the polar decomposition. For general A, we give a simple derivation and characterization of a particular generalized polar decompositi...

Suppose that Φ is a reduced irreducible root system, R is an associative commutative ring with unity, G(Φ, R) is the corresponding adjoint Chevalley group, and E(Φ, R) is its elementary subgroup (see Section 5). There are a lot of results (see, e.g., [Wat80], [Pet82], [GMi83], [HO’M89], [Abe93], [Che00], [Bun07], and references therein*) asserting that, under some conditions, all automorphisms ...

The Cartan subalgebra of the quantum loop algebra Uq(Lsl2) is generated by a family of mutually commuting operators, responsible for the l-weight decomposition of finite dimensional Uq(Lsl2)-modules. The natural Jordan filtration induced by these operators is generically non-trivial on l-weight spaces of dimension greater than one. We derive, for every standard module of Uq(Lsl2), the dimension...

Abstract. In this paper we investigate theoretically an approximation technique for avoiding the crowding phenomenon in numerical conformal mapping. The method applies to conformal maps from rectangles to "long quadrilaterals," i.e., Jordan domains bounded by two parallel straight lines and two Jordan arcs, where the two arcs are far apart. We require that these maps take the four corners of th...

The main aim of this study was to examine the effect of education on economic growth in Jordan over the period 1976-2007. Co-integration analysis was adoptedwith five variables: Real Gross National Product (RGNP), Capital (K), Labor (L), Expenditure on Education (EDU), and Technology (T). Two unit root tests (DickeyFuller Test and Philips-Perron Test) have been employed to test the integration ...

We consider QM with non-Hermitian quasi-diagonalizable Hamiltonians, i.e. the Hamiltonians having a number of Jordan cells in particular biorthogonal bases. The ”self-orthogonality” phenomenon is clarified in terms of a correct spectral decomposition and it is shown that ”self-orthogonal” states never jeopardize resolution of identity and thereby quantum averages of observables. The example of ...

Classically, any absolute continuous real function is of bounded variation and hence can always be expressed as a difference of two increasing continuous functions (socalled Jordan decomposition). The effective version of this result is not true. In this paper we give a sufficient and necessary condition for computable real functions which can be expressed as two computable increasing functions...