Zero assignment of matrix pencils by additive structured transformations

Matrix pencil models are natural descriptions of linear networks and systems. Changing the values of elements of networks, that is redesigning them, implies changes in the zero structure of the associated pencil and this is achieved by structured additive transformations. The paper examines the problem of zero assignment of regular matrix pencils by a special type of structured additive transformations. For a certain family of network redesign problems the additive perturbations may be described as diagonal perturbations and such modifications are considered here. This problem has certain common features with the pole assignment of linear systemsby structured static compensators and thus the new powerful methodology of global linearization [J. Leventides, N. Karcanias, Sufficient conditions for arbitrary Pole assignment by constant decentralised output feedback, Mathematics of Control for Signals and Systems 8 (1995) 222–240; J. Leventides, N. Karcanias, Global asymptotic linearisation of the pole placement map: A closed form solution for the constant output feedback problem, Automatica 31 (1995) 1303–1309] can be used. For regular pencils with infinite zeros, families of structured degenerate additive transformations are defined and parameterized and this lead to the derivation of conditions for zero structure assignment, as well as methodology for computing such solutions. The case of regular pencils with no infinite zeros is also considered and conditions of zero assignment are developed. The results here provide the means for studying problems of linear network redesign by modification of the non-dynamic elements

Implementation of the Basel Accords I and II in Greek Banking System: The Application of Standardized Approach

This paper focuses on the analysis of the main implications of Basel I and Basel II, based on risk sensitiveness due to credit risk, in Greek Banking System and assesses their effect per portfolio and per Bank in order to evaluate capital charges and to measure credit risk exposure

A comment on ``Pareto improving taxes''

In an article appeared in the Journal of Mathematical Economics, J. Geanakoplos and H. Polemarchakis, [Geanakoplos J. and Polemarchakis H.M.: "Pareto improving taxes", Journal of Mathematical Economics 44 (2008), 682-696], prove on page 685 the following theorem: "Theorem. For almost all economies with separable externalities and L>I, every competitive equilibrium is constrained Pareto suboptimal, that is, for each competitive equilibrium, there exists an anonymous tax package t and a competitive t-equilibrium allocation which Pareto dominates it." It is the purpose of this comment to show that restrictions must be applied on the limiting cases for the theorem to hold. Proposition 1.3, below, gives a counter-positive result and the ensuing Corollary shows that the Theorem in [Geanakoplos & Polemarchakis 2008][p. 685] does not hold for I=2 and subsequently the example given in Section 6, page 693, of Geanakoplos & Polemarchakis (2008)} appears to be incorrect

A comment on ``Pareto improving taxes''

In an article appeared in the Journal of Mathematical Economics, J. Geanakoplos and H. Polemarchakis, [Geanakoplos J. and Polemarchakis H.M.: "Pareto improving taxes", Journal of Mathematical Economics 44 (2008), 682-696], prove on page 685 the following theorem: "Theorem. For almost all economies with separable externalities and L>I, every competitive equilibrium is constrained Pareto suboptimal, that is, for each competitive equilibrium, there exists an anonymous tax package t and a competitive t-equilibrium allocation which Pareto dominates it." It is the purpose of this comment to show that restrictions must be applied on the limiting cases for the theorem to hold. Proposition 1.3, below, gives a counter-positive result and the ensuing Corollary shows that the Theorem in [Geanakoplos & Polemarchakis 2008][p. 685] does not hold for I=2 and subsequently the example given in Section 6, page 693, of Geanakoplos & Polemarchakis (2008)} appears to be incorrect

Distance Optimization and the Extremal Variety of the Grassmann Variety

The approximation of a multivector by a decomposable one is a distance-optimization problem between the multivector and the Grassmann variety of lines in a projective space. When the multivector diverges from the Grassmann variety, then the approximate solution sought is the worst possible. In this paper, it is shown that the worst solution of this problem is achieved, when the eigenvalues of the matrix representation of a related two-vector are all equal. Then, all these pathological points form a projective variety. We derive the equation describing this projective variety, as well as its maximum distance from the corresponding Grassmann variety. Several geometric and algebraic properties of this extremal variety are examined, providing a new aspect for the Grassmann varieties and the respective projective spaces

Approximate decomposability in and the canonical decomposition of 3-vectors

Given a (Figure presented.)3-vector (Formula presented.) the least distance problem from the Grassmann variety (Formula presented.) is considered. The solution of this problem is related to a decomposition of (Formula presented.) into a sum of at most five decomposable orthogonal 3-vectors in (Formula presented.). This decomposition implies a certain canonical structure for the Grassmann matrix which is a special matrix related to the decomposability properties of (Formula presented.). This special structure implies the reduction of the problem to a considerably lower dimension tensor space ⊗3R2 where the reduced least distance problem can be solved efficiently

A direct system for the 3x+1 dynamics

We extend the Collatz map T to a new map T whose trajectories converge to 1. This map gives rise to a fully binary tree which contains shifted copies of the Collatz tree. The new structure allows the simultaneous study of all important features of the conjecture, such as Collatz sequences, transition of parity vectors and the double indexed sequence of signs (-1)Tk(m). Furthermore, the binary tree leads to a direct system that can be used to study the convergence properties of the sequence of signs and it also extends the Collatz conjecture to new directions which are not directly related to the problem. Copyright (C) 2021 The Authors

The Impact of the Basel Accord on Greek Banks: A Stress Test Study

In this paper, we study the impact of extreme events on the loan portfolios of the Greek banking system. These portfolios are grouped into three separate groups based on the size of the bank to which they belong, in particular, large, medium, and small size. A series of extreme scenarios was performed and the increase in capital requirements was calculated for each scenario based on the standardized and internal ratings approach of the Basel II accord. The results obtained show an increase of credit risk during the crisis periods, and the differentiation of risk depending on the size of the banking organization as well as the added capital that will be needed in order to hedge that risk. The execution of the scenarios aims at studying the effects which may be brought about on the capital of the three representative banks by the appearance of adverse events