The rich literature concerning “asymptotic behavior of Hankel determinants” concerns the behavior, as the order n tends to ∞, of Hankel determinants whose entries are numbers, e.g., with a combinatorial interest or arising as values of special classes of functions. Such determinants are numbers depending on n, playing roles in number theory, combinatorics, random matrices and the like;and mathematicians in the involved fields have been interested in their asymptotic behaviors as n goes to ∞, as previously mentioned, with no single exception to the author’s knowledge. The study carried on in the present paper treats an altogether different situation as suggested by the specification in the title “as the variable tends to +∞”. We deal with those types of Hankel determinants (purposely called Hankelians) which are special cases of Wronskians and, continuing our work on the asymptotics of Wronskians, we study the asymptotic behaviors of n-order Hankelians, whose entries involve either regularly- or rapidly-varying functions, when the variable tends to +∞. As in the study of Wronskians, the treatment of this case also needs the whole apparatus of the theory of higher-order types of asymptotic variation, but the most demanding results are not automatic corollaries of the general theory. In fact, in the study of generic Wronskians (study motivated by applications to asymptotic expansions), the entries were required to belong to one of the classes of “higher-order regular or rapid variation”;on the contrary, in the case of Hankelians, we are confronted with functions whose logarithms are either “regularly- or rapidly-varying functions”, roughly classifiable as “ultrarapidly-varying functions”, and the study requires both special devices and a number of preliminary lemmas about products and linear combinations of functions in the mentioned classes.
Cyclopia is a rare congenital brain malformation frequently associated with facial anomalies. It is characterized by the presence of a single eye with varying degrees of doubling of the intrinsic ocular structures located in the middle of the face. It is the most severe facial expression of holoprosencephaly. Its aetiology is still poorly understood, but several factors could play a role in its occurrence, including certain viruses contracted during pregnancy. Obstetrical ultrasound has made antenatal diagnosis and the search for associated malformations possible. This diagnosis must be made antenatally because the prognosis is poor, hence the decision to terminate the pregnancy. We report a case of cyclopia associated with ambiguity of the external genitalia, discovered intraoperatively in a patient with poor prenatal follow-up, in whom a coronavirus infection (COVID-19) had been diagnosed in early pregnancy.
Post-disaster aid is widely regarded as important in helping local recovery and development.This paper examines the effectiveness of post-disaster aid on exports,which are a driving factor of economic development.It reports a natural experiment in China–the case of post-disaster aid following the Wenchuan earthquake in 2008–to examine how donors'experiences affected the exports of manufacturing firms in disaster-stricken counties.The export experience of the donor was important.Aid coming from donors with more export experience was more beneficial to the exports of firms in recipient counties than aid from less experienced donors.“Learning from the donor”is a potential channel through which this effect occurred.That is,firms in recipient counties learned from donors'export experience by exporting more products similar to those of donors,exporting more to the destination countries of donors,and participating in the donors'supply-chain networks by exporting more of the donors'exports.Such“learning from the donor”effects show that knowledge spillover can occur between spatially distant parties,which complements the literature.
The eccentricity matrix of a graph is obtained from the distance matrix by keeping the entries that are largest in their row or column,and replacing the remaining entries by zero.This matrix can be interpreted as an opposite to the adjacency matrix,which is on the contrary obtained from the distance matrix by keeping only the entries equal to 1.In the paper,we determine graphs having the second largest eigenvalue of eccentricity matrix less than 1.
