42 research outputs found
Discontinuity of the Lempert function of the spectral ball
We give some further criteria for continuity or discontinuity of the Lempert
funtion of the spectral ball , with respect to one or both of its
arguments, in terms of cyclicity the matrices involved.Comment: 12 pages; the converse to theorem 1.3, mistakenly claimed in v1 & 2,
excised from v3, is now proved in some case
Disks extremal with respect to interpolation constants
We define a function µ from the set of sequences in the unit ball to R*+ by taking the greatest lower bound of the reciprocal of the interpolating constant of the sequences of the disk which get mapped to the given sequence by a holomorphic mapping from the disk to the ball. Its properties are studied in the spirit of the work of Amar and Thomas
Restricted triangulation on circulant graphs
The restricted triangulation existence problem on a given graph decides whether there exists a triangulation on the graph’s vertex set that is restricted with respect to its edge set. Let G = C ( n , S ) be a circulant graph on n vertices with jump value set S . We consider the restricted triangulation existence problem for G . We determine necessary and sufficient conditions on S for which G admitting a restricted triangulation. We characterize a set of jump values S ( n ) that has the smallest cardinality with C ( n , S ( n )) admits a restricted triangulation. We present the measure of non-triangulability of K n − G for a given G
Pluricomplex Green and Lempert functions for equally weighted poles
For a domain in , the pluricomplex Green function with
poles is defined as .
When there is only one pole, or two poles in the unit ball, it turns out to be
equal to the Lempert function defined from analytic disks into by . It is known
that we always have . In the more general case where we
allow weighted poles, there is a counterexample to equality due to Carlehed and
Wiegerinck, with equal to the bidisk.
Here we exhibit a counterexample using only four distinct equally weighted
poles in the bidisk. In order to do so, we first define a more general notion
of Lempert function "with multiplicities", analogous to the generalized Green
functions of Lelong and Rashkovskii, then we show how in some examples this can
be realized as a limit of regular Lempert functions when the poles tend to each
other. Finally, from an example where in the case of
multiple poles, we deduce that distinct (but close enough) equally weighted
poles will provide an example of the same inequality. Open questions are
pointed out about the limits of Green and Lempert functions when poles tend to
each other.Comment: 25 page
Khi nào thì chúng ta tin người lạ?
Khi chúng ta còn trẻ, lời khuyên phổ biến từ cha mẹ thường là "Đừng tin người lạ". Và ở những tình huống khác, họ cũng hay động viên sự tự tin của chúng ta bằng cách nhắc nhở: "Hãy tin vào bản thân mình". Tuy nhiên, khi đối diện với người nào đó có vẻ ngoài giống chúng ta ngay từ cái nhìn đầu tiên, liệu chúng ta có tin tưởng họ không
Green functions of the spectral ball and symmetrized polydisk
The Green function of the spectral ball is constant over the isospectral
varieties, is never less than the pullback of its counterpart on the
symmetrized polydisk, and is equal to it in the generic case where the pole is
a cyclic (non-derogatory) matrix. When the pole is derogatory, the inequality
is always strict, and the difference between the two functions depends on the
order of nilpotence of the strictly upper triangular blocks that appear in the
Jordan decomposition of the pole. In particular, the Green function of the
spectral ball is not symmetric in its arguments. Additionally, some estimates
are given for invariant functions in the symmetrized polydisc, e.g.
(infinitesimal versions of) the Carath\'eodory distance and the Green function,
that show that they are distinct in dimension greater or equal to .Comment: 12 page