The output feedback pole assignment problem is a classical problem in linear systems theory. In this paper we calculate the number of complex dynamic compensators of order q assigning a given set of poles for a q-nondegenerate m-input, p-output system of McMillan degree n = q(m+p? 1) + mp. As a corollary it follows that when this number is odd, the generic system can be arbitrarily pole assigne...

It is proved that the range of a Sylvester map defined by two matrices of sizes p× p and q × q, respectively, plus matrices whose ranks are bounded above, cover all p× q matrices. The best possible upper bound on the ranks is found in many cases. An application is made to a minimal rank problem that is motivated by the theory of minimal factorizations of rational matrix functions.

The third, fifth and sixth Painlevé equations are studied by means of the weighted projective spaces CP (p, q, r, s) with suitable weights (p, q, r, s) determined by the Newton polyhedrons of the equations. Singular normal forms of the equations, symplectic atlases of the spaces of initial conditions, Riccati solutions and Boutroux’s coordinates are systematically studied in a unified way with ...

Let G be a elementary abelian p-group of order q = p. Let V be a faithful indecomposable modular representation of G with dimension 2. We consider representations of the form S(S(V )). In particular we prove that S(S(V )) is projective whenever d+m ≥ q, and either d < p and m < q or m < p and d < q. More generally we show that if d+m ≥ q and d < q, m < q, then each summand of S(S(V )) is induce...

The two-dimensional systems of first order nonlinear differential equations (S1) x? = p(t)y?, y? q(t)x? and (S2) + p(t)y? 0, 0 are analyzed using the theory rapid variation. This approach allows us to prove that all strongly increasing solutions system (and, respectively, decreasing ) rapidly varying functions under assumption p q varying. Also, asymptotic equivalence relations for these given.

This is a report for my presentation at the upcoming meeting on Berechenbarkeitstheorie (“Computability Theory”), Oberwolfach, January 21–27, 2001. We use 2 to denote the space of infinite sequences of 0’s and 1’s. For X, Y ∈ 2, X ≤T Y means that X is Turing reducible to Y . For P,Q ⊆ 2 we say that P is Muchnik reducible to Q, abbreviated P ≤w Q, if for all Y ∈ Q there exists X ∈ P such that X ...

For a compact Riemann surface X of genus g > 1, Hom(π1(X),PU(p, q))/PU(p, q) is the moduli space of flat PU(p, q)-connections on X. There are two integer invariants, dP , dQ, associated with each σ ∈ Hom(π1(X),PU(p, q))/ PU(p, q). These invariants are related to the Toledo invariant τ by τ = 2P −pdQ p+q . This paper shows, via the theory of Higgs bundles, that if q = 1, then −2(g − 1) ≤ τ ≤ 2(g...

We consider summations over digamma and polygamma functions, often with summands of the form (±1)nψ(n + p/q)/nr and (±1)nψ(m)(n + p/q)/nr, where m, p, q, and r are positive integers. We develop novel general integral representations and present explicit examples. Special cases of the sums reduce to known linear Euler sums. The sums of interest find application in quantum field theory, including...

It was discovered some years ago that there exist non-integer real numbers q > 1 for which only one sequence (ci) of integers 0 ≤ ci < q satisfies the equality P ∞ i=1 ciq −i = 1. The set of such “univoque numbers” has a rich topological structure, and its study revealed a number of unexpected connections with measure theory, fractals, ergodic theory and Diophantine approximation. In this paper...