Mogens Esrom Larsen

General equilibrium dynamics of multi-sector growth models

This paper analyzes Walrasian general equilibrium systems and calculates the static and dynamic solutions for competitive market equilibria. The Walrasian framework encompasses the basic multi-sector growth (MSG) models with neoclassical production technologies inN sectors (industries). The endogenous behavior of all the relative prices are analyzed in detail, as are sectorial allocations of the primary factors, labor and capital. Dynamic systems of Walrasian multi-sector economies and the family of solutions (time paths) for steady-state and persistent growth per capita are parametrically characterized. The technology parameters of the capital good industry are decisive for obtaining long-run per capita growth in closed (global) economies. Brief comments are offered on the MSG literature, together with apects on the studies of industrial (structural) evolution and economic history.

Growth and long-run stability

This paper analyses the implications of persistent growth upon the stability properties of dynamic models. Besides the traditional concept of asymptotic stability, new stability criteria-strong/weak absolute, strong/weak relative, strong/weak logarithmic stability-are introduced, and global stability conditions for satisfying these criteria are stated for general first-order autonomous differential equations. The conflict between rapidity of growth and the degree of stability is demonstrated. Economic applications of the stability theorems are illustrated within the growth models of Harrod and Solow.

The Eternal Triangle?A History of a Counting Problem

Mogens Esrom Larsen was born in 1942. He graduated from the University of Copenhagen in 1965 where he remained, except for a recess at M.I.T during 196970, and is now associate professor of mathematics. His interests cover most of mathematics from several complex variables, to differen? tial equations, to finite groups, e.g., Rubiks cubes, and from pre-Euclidean history of mathematics to the teaching and applications of modern mathematics, including numerical analysis. In addition, he writes a monthly problem column in a popular Scandinavian science journal where this triangle problem was presented. Fond of games, he also founded the Copenhagen Go Club in 1972.

Combinatorial Identities: A Generalization of Dougall’s Identity

In this paper, I will discuss combinatorial identities as a tool for individuals working with combinatorial problems. I will also present a generalization of Dougall’s (1907) identity. In the notations of this paper, the general combinatorial identity established is the following: If p = b + c + d + e − n + 1 is a non-negative integer then we have:

Coupled linear differential equations with real coefficients

\begin{gathered} \sum\limits_{{k = 0}}^{n} {\left( {\begin{array}{*{20}{c}} n \\ k \\ \end{array} } \right){{{\left[ {n + 2a} \right]}}_{k}}{{{\left[ {b + a} \right]}}_{k}}} {{\left[ {c + a} \right]}_{k}}{{\left[ {d + a} \right]}_{k}} \hfill \\ \times {{\left[ {n - 2a} \right]}_{{n - k}}}{{\left[ {b - a} \right]}_{{n - k}}}{{\left[ {c - a} \right]}_{{n - k}}}{{\left[ {d - a} \right]}_{{n - k}}}{{\left[ {e - a} \right]}_{{n - k}}}\left( {n + 2a - 2k} \right) \hfill \\ = {{\left( { - 1} \right)}^{n}}{{\left[ {n + 2a} \right]}_{{2n - 1}}}{{\left[ {b + d - p} \right]}_{{n - p}}}{{\left[ {b + e - p} \right]}_{{n - p}}}{{\left[ {d + e - p} \right]}_{{n - p}}} \hfill \\ \times \sum\limits_{{j = 0}}^{p} {\left( {\begin{array}{*{20}{c}} p \\ j \\ \end{array} } \right){{{\left[ n \right]}}_{j}}{{{\left[ {c + a} \right]}}_{j}}{{{\left[ {c - a} \right]}}_{j}}{{{\left[ {b + e - n} \right]}}_{{p - j}}}{{{\left[ {d + e - n} \right]}}_{{p - j}}}{{{\left[ {d + e - n} \right]}}_{{p - j}}}} \hfill \\ \end{gathered}

On the topology of complex projective manifolds

where [x] k denotes the descending factorial. Dougall’s identity, which is usually written in terms of a hypergeometric series, corresponds to the case p = 0.

On the Homotopy Groups of Complex Projective Algebraic Manifolds.

In describing many phenomena, we are naturally inclined to order our observations by their occurrence in time. As a scalar variable t, time is often treated as affording an instance of the abstract concept of a continuum (infinite divisibility), which accordingly is represented by the set of real numbers, t ∈ R. Where the relevant state variables may also take values on a continuous domain, and when their rates of change, at any time, only depend on their present (current) state (i.e., the past only affects the future through the present state, irrespective of how the present state has been attained), then the theory of ordinary differential equations and the powerful tools of infinitesimal calculus are available for determining the subsequent history (solutions) of the state variables in continuous time.

RUBIK'S REVENGE: THE GROUP THEORETICAL SOLUTION

Introduction. In a first course the case of two coupled linear differential equations tends to fall between two stools. The teachers unrequited love for eigenvalues drives him into the complex domain, a maze in which he seldom finds the simple, real solutions of the original problem. And even if the complex numbers can be avoided he has difficulty returning through the coordinate transforms. It would seem that if the students had an adequate basis in algebra, everything would be easy. However, on the one hand, it is too much to include all that algebra. On the other hand, that particular subject is not something that can be used now and explained later. Hence, it is tempting to look for a simple, direct solution, which works in the real domain and only requires straightforward ideas.