This paper studies the H∞ control of infinitedimensional systems whose transfer matrices are expressible as a series connection of a rational transfer matrix and a scalar (possibly irrational) inner function. This class of systems is adequate for describing many control problems in practice, when weighting functions are rational and plants have at most finitely many unstable modes. We show that...

We consider the nonlinear eigenvalue problem: M(λ)x = 0, where M(λ) is a large parameter-dependent matrix. In several applications, M(λ) has a structure where the higher-order terms of its Taylor expansion have a particular low-rank structure. We propose a new Arnoldi based algorithm that can exploit this structure. More precisely, the proposed algorithm is equivalent to Arnoldi’s method applie...

Lyapunov equation. We analyze the approximation properties of solutions of abstract Lyapunov equations in the setting of a scale of Hilbert spaces associated to an unbounded diagonalizable operator which satisfies the Kato’s square root theorem. We call an (unbounded) operator A diagonalizable if there exists a bounded operator Q, with a bounded inverse, such that the (unbounded) operator Q−1AQ...

The problem of classifying the automorphisms of a free group of infinite countable rank is investigated. Quite a reasonable generating set for the group AutF∞ is described. Some new subgroups of this group and structural results for them are presented. The main result says that the group of all automorphisms is generated (modulo the IA-automorphisms) by strings and lower triangular automorphisms.

Let E be an elliptic curve over a finitely generated infinite field K. Let K denote a separable closure of K, σ an element of the Galois group GK = Gal(K/K), and K(σ) the invariant subfield of K. We prove that if the characteristic of K is not 2 and σ belongs to a suitable open subgroup of GK , then E(K(σ)) has infinite rank. In [2], G. Frey and M. Jarden considered the rank of abelian varietie...

Defect theorem is one of the fundamental results on words, cf [Lo]. Intuitively it states that if n words satisfy a nontrivial relation, then these words can be expressed as products of at most n−1 words. Actually, as discussed in [CK], for example, there does not exist just one defect theorem but several ones depending on restrictions put on the required n−1 words. It is also well-known that t...

Given a set of nonnegative real numbers Λ= {λi}i=0, a Λ-polynomial (or Müntz polynomial) is a function of the form p(x)=ni=0 aizi (n∈N). We denote byΠ(Λ) the space of Λ-polynomials and byΠZ(Λ) := {p(x)=ni=0 aizi ∈Π(λ) : ai ∈ Z for all i≥ 0} the set of integral Λ-polynomials. Clearly, the sets ΠZ(Λ) are subgroups of infinite rank of Z[x] wheneverΛ⊂N, #Λ=∞ (by infinite rank, wemean that the real ...

Today’s focus on sustainability within industry presents a modeling challenge that may be dealt with using dynamic programming over an infinite time horizon. However, the curse of dimensionality often results in a large number of states in these models. These large-scale models require numerically stable solution methods. The best method for infinite-horizon dynamic programming depends on both ...

This paper surveys the methods that have been used to attack conjecture, still open, an abelian variety over a characteristic $0$ field with finitely generated Galois group is always of infinite rank.

Suppose that W is an infinite Coxeter group of finite rank n, and suppose that W has a finite parabolic subgroup WJ of rank n 1. Suppose also that the Coxeter diagram of W has no edges with infinite labels. Then any automorphism of W that preserves reflections lies in the subgroup of AutðWÞ generated by the inner automorphisms and the automorphisms induced by symmetries of the Coxeter graph. If...