12,302 research outputs found
A Stability Version of the Gauss-Lucas Theorem and Applications
Let be a polynomial. The Gauss-Lucas
theorem states that its critical points, , are contained in the
convex hull of its roots. We prove a stability version whose simplest form is
as follows: suppose has roots where are inside the unit disk,
\max_{1 \leq i \leq n}{|a_i|} \leq 1, \quad \mbox{and $m$ are outside} \quad
\min_{n+1 \leq i \leq n+m}{ |a_i|} \geq d > 1 + \frac{2 m}{n}, then has
roots inside the unit disk and roots at distance at least from the origin and the involved constants are sharp. We also
discuss a pairing result: in the setting above, for sufficiently large each
of the roots has a critical point at distance .Comment: to appear in Journal of the Australian Mathematical Societ
On the generalized Riemann hypothesis II
Some of my previous publications were incomplete in the sense that non
trivial zeros belonging to a particular type of fundamental domain have been
inadvertently ignored. Due to this fact, I was brought to believe that
computations done by some authors in order to show counterexamples to RH were
affected of approximation errors. In this paper I illustrate graphically the
correctness of those computations and I fill the gaps in my publications.Comment: 12 pages, 2 figure
An elementary introduction to quantum graphs
We describe some basic tools in the spectral theory of Schr\"odinger operator
on metric graphs (also known as "quantum graph") by studying in detail some
basic examples. The exposition is kept as elementary and accessible as
possible. In the later sections we apply these tools to prove some results on
the count of zeros of the eigenfunctions of quantum graphs.Comment: 31 pages, 17 figure
On the distribution of zeros of the derivative of Selberg's zeta function associated to finite volume Riemann surfaces
W. Luo has investigated the distribution of zeros of the derivative of the
Selberg zeta function associated to compact hyperbolic Riemann surfaces. In
essence, the main results in Luo's article involve the following three points:
Finiteness for the number of zeros in the half plane to the left of the
critical line; an asymptotic expansion for the counting function measuring the
vertical distribution of zeros; and an asymptotic expansion for the counting
function measuring the horizontal distance of zeros from the critical line. In
the present article, we study the more complicated setting of distribution of
zeros of the derivative of the Selberg zeta function associated to a
non-compact, finite volume hyperbolic Riemann surface. There are numerous
difficulties which exist in the non-compact case that are not present in the
compact setting, beginning with the fact that in the non-compact case the
Selberg zeta function does not satisfy the analogue of the Riemann hypothesis.
To be more specific, we actually study the zeros of the derivative of ZH, where
Z is the Selberg zeta function and H is the Dirichlet series component of the
scattering matrix, both associated to an arbitrary finite-volume hyperbolic
Riemann surface. Our main results address finiteness of zeros in the half plane
to the left of the critical line, an asymptotic count for the vertical
distribution of zeros, and an asymptotic count for the horizontal distance of
zeros
Factorization of Linear Quantum Systems with Delayed Feedback
We consider the transfer functions describing the input-output relation for a
class of linear open quantum systems involving feedback with nonzero time
delays. We show how such transfer functions can be factorized into a product of
terms which are transfer functions of canonical physically realizable
components. We prove under certain conditions that this product converges, and
can be approximated on compact sets. Thus our factorization can be interpreted
as a (possibly infinite) cascade. Our result extends past work where linear
open quantum systems with a state-space realization have been shown to have a
pure cascade realization [Nurdin, H. I., Grivopoulos, S., & Petersen, I. R.
(2016). The transfer function of generic linear quantum stochastic systems has
a pure cascade realization. Automatica, 69, 324-333.]. The functions we
consider are inherently non-Markovian, which is why in our case the resulting
product may have infinitely many terms.Comment: 32 pages, 4 figure
Complex Zeros of Eigenfunctions of 1D Schr\"odinger Operators
In this article we study the semi-classical distribution of complex zeros of
the eigenfunctions of the 1D Schr\"odinger operators for the class of
polynomial potentials of even degree, when an energy level E is fixed.Comment: 17 pages, 5 figure
Plancherel-Rotach formulae for average characteristic polynomials of products of Ginibre random matrices and the Fuss-Catalan distribution
Formulae of Plancherel-Rotach type are established for the average
characteristic polynomials of certain Hermitian products of rectangular Ginibre
random matrices on the region of zeros. These polynomials form a general class
of multiple orthogonal hypergeometric polynomials generalizing the classical
Laguerre polynomials. The proofs are based on a multivariate version of the
complex method of saddle points. After suitable rescaling the asymptotic zero
distributions for the polynomials are studied and shown to coincide with the
Fuss-Catalan distributions. Moreover, introducing appropriate coordinates,
elementary and explicit characterizations are derived for the densities as well
as for the distribution functions of the Fuss-Catalan distributions of general
order.Comment: 18 page
Asymptotics for characteristic polynomials of Wishart type products of complex Gaussian and truncated unitary random matrices
Based on the multivariate saddle point method we study the asymptotic
behavior of the characteristic polynomials associated to Wishart type random
matrices that are formed as products consisting of independent standard complex
Gaussian and a truncated Haar distributed unitary random matrix. These
polynomials form a general class of hypergeometric functions of type . We describe the oscillatory behavior on the asymptotic interval of zeros
by means of formulae of Plancherel-Rotach type and subsequently use it to
obtain the limiting distribution of the suitably rescaled zeros. Moreover, we
show that the asymptotic zero distribution lies in the class of Raney
distributions and by introducing appropriate coordinates elementary and
explicit characterizations are derived for the densities as well as for the
distribution functions
Grassmann convexity and multiplicative Sturm theory, revisited
In this paper we settle a special case of the Grassmann convexity conjecture
formulated earlier by B.and M.Shapiro. We present a conjectural formula for the
maximal total number of real zeros of the consecutive Wronskians of an
arbitrary fundamental solution to a disconjugate linear ordinary differential
equation with real time. We show that this formula gives the lower bound for
the required total number of real zeros for equations of an arbitrary order
and, using our results on the Grassmann convexity, we prove that the
aforementioned formula is correct for equations of orders and .Comment: 23 pages, 12 figures, exposition improve
Harmonic maps between three-spheres
It is shown that smooth maps contain two countable
families of harmonic representatives in the homotopy classes of degree zero and
one.Comment: 17 pages, 2 figures available on reques
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