Open Convex Cone Research Articles

Riesz measures and Wishart laws associated to quadratic maps

We introduce a natural definition of Riesz measures and Wishart laws associated to an $\Omega$-positive (virtual) quadratic map, where $\Omega \subset \real^n$ is a regular open convex cone. We give a general formula for moments of the Wishart laws. Moreover, if the quadratic map has an equivariance property under the action of a linear group acting on the cone $\Omega$ transitively, then the associated Riesz measure and Wishart law are described explicitly by making use of theory of relatively invariant distributions on homogeneous cones.

Todd's maximum-volume ellipsoid problem on symmetric cones

Let V be a Euclidean Jordan algebra and let Ω be the associated symmetric cone, a self-dual homogeneous open convex cone, which is a symmetric space of noncompact type under G(Ω) (the linear automorphism group)-invariant Riemannian metric. We show that the radius of the largest ball centered at a ∈ Ω inscribed in Ω coincides with its minimum eigenvalue and then provide a proof of the problem of finding a point x ∈ Ω to maximize the product of the radii of the largest balls centered at a, b ∈ Ω and inscribed in Ω of the tangent space Tx (Ω) and its dual space Tx–1 (Ω), respectively. We obtain an explicit formula for the maxima; it is precisely the minimal eigenvalue of P (a1/2)b where P denotes the quadratic representation of V. This provides an affirmative answer to a question of Todd on the maxima.

Non-overshooting stabilisation via state and output feedback

The concept of ‘strong stability’ of linear time-invariant (LTI) systems has been introduced in a recent paper (Karcanias, N., Halikias, G., and Papageorgiou, A. (2010), ‘Strong Stability of Internal System Descriptions’, International Journal of Control, 83, 182–205). This is a stronger notion of stability compared to alternative definitions (e.g. stability in the sense of Lyapunov, asymptotic stability), which allows the analysis and design of control systems with non-overshooting response in the state-space for arbitrary initial conditions. This article reviews the notion of ‘strong stability’ (Karcanias et al. 2010) and introduces the problem of non-overshooting stabilisation. It is shown that non-overshooting stabilisation under dynamic and static output feedback are, in a certain sense, equivalent problems. Thus, we turn our attention to static non-overshooting stabilisation problems under state-feedback, output injection and output feedback. After developing a number of preliminary results, we give a geometric interpretation to the problem in terms of the intersection of an affine hyperplane and the interior of an open convex cone. A solution to the problem is finally obtained via linear matrix inequalities, along with the complete parameterisation of the optimal solution set.

On Riesz and Wishart distributions associated with decomposable undirected graphs

Classical Wishart distributions on the open convex cone of positive definite matrices and their fundamental features are extended to generalized Riesz and Wishart distributions associated with decomposable undirected graphs using the basic theory of exponential families. The families of these distributions are parameterized by their expectations/natural parameter and multivariate shape parameter and have a non-trivial overlap with the generalized Wishart distributions defined in Andersson and Wojnar (2004) [4,8]. This work also extends the Wishart distributions of type I in Letac and Massam (2007) [7] and, more importantly, presents an alternative point of view on the latter paper.

Holomorphic extension theorem for tempered ultrahyperfunctions

In this article we are concerned with the space of tempered ultrahyperfunctions corresponding to a proper open convex cone. A holomorphic extension theorem (the version of the celebrated Edge-of-the-Wedge Theorem) will be given for this setting. As application, a version is also given of the principle of determination of an analytic function by its values on a non-empty open real set. The article finishes with the generalization of holomorphic extension theorem à la Martineau.

Tube domain and an orbit of a complex triangular group

Let w be a complex symmetric matrix of order r, and Δ1(w), . . . , Δr(w) the principal minors of w. If w belongs to the Siegel right half-space, then it is known that Re (Δk(w)/Δk-1(w)) > 0 for k = 1, . . . , r. In this paper we study this property in three directions. First we show that this holds for general symmetric right half-spaces. Second we present a series of non-symmetric right half-spaces with this property. We note that case-by-case verifications up to dimension 10 tell us that there is only one such irreducible non-symmetric tube domain. The proof of the property reduces to two lemmas. One is entirely generalized to non-symmetric cases as we prove in this paper. This is the third direction. As a byproduct of our study, we show that the basic relative invariants associated to a homogeneous regular open convex cone Ω studied earlier by the first author are characterized as the irreducible factors of the determinant of right multiplication operators in the complexification of the clan associated to Ω.

Invariant metrics, contractions and nonlinear matrix equations

In this paper we consider the semigroup generated by the self-maps on the open convex cone of positive definite matrices of translations, congruence transformations and matrix inversion that includes symplectic Hamiltonians and show that every member of the semigroup contracts any invariant metric distance inherited from a symmetric gauge function. This extends the results of Bougerol for the Riemannian metric and of Liverani–Wojtkowski for the Thompson part metric. A uniform upper bound of the Lipschitz contraction constant for a member of the semigroup is given in terms of the minimum eigenvalues of its determining matrices. We apply this result to a variety of nonlinear equations including Stein and Riccati equations for uniqueness and existence of positive definite solutions and find a new convergence analysis of iterative algorithms for the positive definite solution depending only on the least contraction coefficient for the invariant metric from the spectral norm.

The Primal-Dual Second-Order Cone Approximations Algorithm for Symmetric Cone Programming

Given any open convex cone K, a logarithmically homogeneous, self-concordant barrier for K, and any positive real number r < 1, we associate, with each direction x e K, a second-order cone Kr(x) containing K. We show that K is the interior of the intersection of the second-order cones Kr(x), as x ranges over all directions in K. Using these second-order cones as approximations to cones of symmetric, positive definite matrices, we develop a new polynomial-time primal-dual interior-point algorithm for semidefinite programming. The algorithm is extended to symmetric cone programming via the relation between symmetric cones and Euclidean Jordan algebras.

A Birkhoff Contraction Formula with Applications to Riccati Equations

In this paper we show that the symplectic Hamiltonian operators on a Hilbert space give rise to linear fractional transformations on the open convex cone of positive definite operators that contract a natural invariant Finsler metric, the Thompson or part metric, on the convex cone. More precisely, the constants of contraction for the Hamiltonian operators satisfy the classical Birkhoff formula: the Lipschitz constant for the corresponding linear fractional transformations on the cone of positive definite operators is equal to the hyperbolic tangent of one fourth the diameter of the image. By means of the close connections between Hamilitonian operators and Riccati equations, this result and the associated machinery are applied to obtain convergence results for discrete algebraic Riccati equations and Riccati differential equations.

Solving symmetric matrix word equations via symmetric space machinery

It has recently been proved by Hillar and Johnson [C.J. Hillar, C.R. Johnson, Symmetric word equations in two positive definite letters, Proc. Amer. Math. Soc. 132 (2004) 945–953] that every symmetric word equation in positive definite matrices has a positive definite solution (existence theorem). In this paper we partially solve the uniqueness conjecture via the symmetric space and non-positive curvature machinery existing in the open convex cone of positive definite matrices. Unique positive definite solutions are obtained in terms of geometric and weighted means and as fixed points of explicit strict contractions on the cone of positive definite matrices.

The gradient maps associated to certain non-homogeneous cones

The gradient map associated to a regular open convex cone gives a diffeomorphism from the cone onto its dual cone. If the cone is homogeneous, the inverse of the map is known to be equal to the gradient map associated to the dual cone. However, we show that this is no longer true for a general case by presenting a simple counterexample.

99/03445 Energy planning in urban historical centres. A methodological approach with a case study

Cost and the input demand correspondence are formally defined and the fundamental implications for them of the price-taking cost minimization behavioral postulate are established, including the Law of Demand for conditional demands. These implications are mostly reinterpretations of the implications of the price-taking profit maximization behavioral postulate. Cost-minimizing behavior is shown to be part of profit-maximizing behavior and is useful even when price-taking output selection to maximize profit may not be an appropriate behavioral postulate. The envelope relationship between the cost function and cost objective is explained, leading to the relationship between cost and input demand known as Shephard’s Lemma, and the symmetry and negative semidefiniteness of the matrix of demand price slopes. Additional properties arising from additional features of production sets are explored, including the elasticity identities and reciprocity. Comparative statics of the demand choices are derived for the important special case of a firm that produces one output, including the shadow price interpretation of the Lagrange multiplier as marginal cost and the concepts of complementary, substitute and normal inputs. Sufficiency of the fundamental properties of the cost function is formally proven, thereby establishing that no additional properties of cost are implied by the price-taking cost minimization behavioral postulate. There is always a nonempty, closed, free disposal, and convex cost-relevant input requirement correspondence underlying any function possessing the homogeneity and concavity properties of cost. Similarly, a homogeneous of degree zero vector-valued function defined on an open convex cone in the price space, with a symmetric negative semidefinite Jacobian, is an input demand function for a price-taking firm that minimizes cost over the implied cost-relevant input requirement sets. The chapter also shows how to derive cost-relevant production and profit functions, and how to derive a profit-relevant cost function from a given profit function.

The cauchy and poisson kernels as ultradifferentiable functions

Let C be an open convex cone in such that does not contain any entire straight line. We previously have shown that the Cauchy and Poisson kernel functions corresponding to the tube are elements of the ultradifferentiable function spaces , where * is either (Mp ) or {Mp }. In this paper we prove that the same is true for the values 1 ≤.s < 2 for the Cauchy kernel and for the values 1 ≤ s < 2 for the Poisson kernel. This allows us to extend results for the Cauchy and Poisson integrals of ultradistributions in , previously obtained for 2 ≤ s ≤ ∞ to 1 ≤ s ≤ 2 and 1 ≤ s < 2, respectively, now. Thus the study of the Cauchy integral of ultradistributions and of distributions corresponding to arbitrary tubes is now shown to hold for all the values of s as in Tillmann's [17] original work on the representation of distributions in as boundary values of functions analytic in tubes where the C μ, are the 2 2 n-rants in

Cauchy and poisson intergral of ultredistributions

Let C be an open convex cone in such that [Cbar] does not contain any entire straight line and let be the corresponding rube domain. Generalized Cauchy and Poisson integrals of ultradistributions of Beurling type and of Roumieu type both of which generalize the Schwartz distributions are defined and studied for appropriate values of s The Caucthy integral is shown to be holomorphic in T to satisfy a growth property, and to obtain an ultradistribution boundary value. Which leads to a holomorphic decomposition dresult for the ultradistributions. The Poission integral is shown to have an ultradistribution boundary value. The analysis of this paper lays the foundation and is necessary for future research concerning a study of holomorphic functions in tubes which are characterixed by either pointwise or norm growths, their boundary values, their recovery in terms of generalized integrals, and other associated properties.

Positivity of Stable Densities

We settle two conjectures of Taylor about the positivity of the densities $p(t,x)$ of a drift-free, nondegenerate, stable process on $d$-dimensional Euclidean space ${R^d}$ starting at the origin. If $0 < \alpha < 1$ and $p(1,0) = 0$, we show that $x:\;p(t,x) > 0$ for some $t > 0$ is an open convex cone with vertex 0 and that $p(t,x) > 0$ for all $t > 0$ for each $x$ in this cone. If $\alpha = 1$ we show that $p(t,x) > 0$ for all $t > 0$ and all $x \in {R^d}$.

Positivity of stable densities

We settle two conjectures of Taylor about the positivity of the densities p ( t , x ) p(t,x) of a drift-free, nondegenerate, stable process on d d -dimensional Euclidean space R d {R^d} starting at the origin. If 0 &gt; α &gt; 1 0 &gt; \alpha &gt; 1 and p ( 1 , 0 ) = 0 p(1,0) = 0 , we show that x : p ( t , x ) &gt; 0 x:\;p(t,x) &gt; 0 for some t &gt; 0 t &gt; 0 is an open convex cone with vertex 0 and that p ( t , x ) &gt; 0 p(t,x) &gt; 0 for all t &gt; 0 t &gt; 0 for each x x in this cone. If α = 1 \alpha = 1 we show that p ( t , x ) &gt; 0 p(t,x) &gt; 0 for all t &gt; 0 t &gt; 0 and all x ∈ R d x \in {R^d} .

Boundary values of generalizations of Hρ functions in tubes

In previous work we have defined and studied holomorphic functions in tubes which generalize the Hardy Hρ (TB) spaces; the base B of the tube TB is a proper open subset of If B=C, an open convex cone in and 1 < ρ ⩽ 2, we have proved that our new functions have distributional boundary values in the strong topology of p as . In this paper we continue to study these new functions and extend the existence of boundary values to the cases 2 < ρ < ∞ We prove that our functions obtain boundary values in the strong topology of P as , where C is a polygonal cone or a regular cone, for the cases 2 < ρ < ∞.

Holomorphic Functions in Tubes Which Have Distributional Boundary Values and Which Are $H^p $ Functions

Let C be an open convex cone in ${\bf R}^n $, n-dimensional real space, such that $\bar C$ does not contain any entire straight line. We consider holomorphic functions in tubes $T^C = {\bf R}^n + iC \subset {\bf C}^n $, n-dimensional complex space, which are known to have $\mathfrak{S}'$ distributional boundary values. We prove that if the $\mathfrak{S}'$ boundary value of such a holomorphic function is an $L^p $ function, $1 \leq p \leq \infty $, then the holomorphic function is in the Hardy space $H^p (T^C )$, $1 \leq p \leq \infty $, corresponding to the tube $T^C $ and can be recovered as the Poisson integral of the distributional boundary value. A similar result is proved wih respect to the holomorphic functions in tubes $T^C $ which are known to have boundary values in the distribution spaces $\mathfrak{Z}',K'_r ,(\mathfrak{S}^\alpha )'$, and $(W^\Omega )'$ and which do not necessarily have $S'$ distributional boundary values. In all cases we prove converse theorems. Our basic results are motivated ...

A differential geometric characterization of homogeneous self-dual cones

defines a tensor fieldon £?,we denote this tensor fieldby the same letter F. An open convex set Q in Vn is called a cone with vertex o if o+/l(x―o)qQ for all a;eX?and ^>0. An open convex cone Q with vertex o is said to be a selfdual cone if Vn admits an inner product such that (i) (x ―o,y ―0>>O for all x,y£Q; (ii) if # F is a vector such that O ―o,y― ^0 for all y£Q then a;ei2, wh^rp O is thp.rinsnrp of .0 in Vn

Root Systems and Cartan Matrices

This paper is concerned with two things. The first is a (primarily) geometric axiomatic description for the systems of real roots of Lie algebras arising from (generalized) Cartan matrices. The description is base free and is a natural extension of the well-known axiomatic description of finite root systems. The primary component of our description is an open convex cone which, following Looijenga [3], we call the Tits cone. In fact it was Looijenga's paper that led to this axiomatic formulation. Unlike his construction, the dimension of the Tits cone is not tightly connected to the dimension of the Cartan matrix which it eventually yields. This leads us to the second part of the paper which concerns the construction of Cartan matrices of low row rank. We can show that if we have an l × l Cartan matrix of row rank n, then we can model an axiomatic description of it with a cone of dimension n + 1.