46 research outputs found
Hurwitz rational functions
A generalization of Hurwitz stable polynomials to real rational functions is
considered. We establishe an analogue of the Hurwitz stability criterion for
rational functions and introduce a new type of determinants that can be treated
as a generalization of the Hurwitz determinants.Comment: 10 page
On the spectra of Schwarz matrices with certain sign patterns
The direct and inverse spectral problems are solved for a wide subclass of
the class of Schwarz matrices. A connection between the Schwarz matrices and
the so-called generalized Hurwitz polynomials is found. The known results due
to H. Wall and O. Holtz are briefly reviewed and obtained as particular cases.Comment: 15 page
A generalized Hermite-Biehler theorem
The classical Hermite-Biehler theorem describes possible zero sets of complex
linear combinations of two real polynomials whose zeros strictly interlace. We
provide the full characterization of zero sets for the case when this
interlacing is broken at exactly one location.
Using this we solve the direct and inverse spectral problem for rank-one
multiplicative perturbations of finite Hermitian matrices. We also treat
certain rank two additive perturbations of finite Jacobi matrices
Maximal univalent disks of real rational functions and Hermite-Biehler polynomials
The well-known Hermite-Biehler theorem claims that a univariate monic
polynomial s of degree k has all roots in the open upper half-plane if and only
if s=p+iq where p and q are real polynomials of degree k and k-1 resp. with all
real, simple and interlacing roots, and q has a negative leading coefficient.
Considering roots of p as cyclically ordered on RP^1 we show that the open disk
in CP^1 having a pair of consecutive roots of p as its diameter is the maximal
univalent disk for the function R=\frac{q}{p}. This solves a special case of
the so-called Hermite-Biehler problem.Comment: 10 pages, 4 figure
Integral Laplacian graphs with a unique double Laplacian eigenvalue, II
The set
, is called Laplacian realizable if there exists a simple
connected graph whose Laplacian spectrum is . In this
case, the graph is said to realize . In this paper, we
completely describe graphs realizing the sets with
and determine the structure of these graphs.Comment: 14 page