388 research outputs found
Asymptotic behavior of the least common multiple of consecutive reducible quadratic progression terms
Let and be two integers with , and let be the
product of two linear polynomials with integer coefficients. In this paper, we
show that , where is a
constant depending only on , and .Comment: 13 page
Uniform lower bound for the least common multiple of a polynomial sequence
Let be a positive integer and be a polynomial with nonnegative
integer coefficients. We prove that except that and and that
with being an integer and , where denotes the
smallest integer which is not less than . This improves and extends the
lower bounds obtained by Nair in 1982, Farhi in 2007 and Oon in 2013.Comment: 6 pages. To appear in Comptes Rendus Mathematiqu
The elementary symmetric functions of a reciprocal polynomial sequence
Erd\"{o}s and Niven proved in 1946 that for any positive integers and
, there are at most finitely many integers for which at least one of the
elementary symmetric functions of are
integers. Recently, Wang and Hong refined this result by showing that if , then none of the elementary symmetric functions of is an integer for any positive integers and . Let be a
polynomial of degree at least and of nonnegative integer coefficients. In
this paper, we show that none of the elementary symmetric functions of is an integer except for with being
an integer and .Comment: 4 pages. To appear in Comptes Rendus Mathematiqu
The distribution of divisors of polynomials
Let be an irreducible polynomial with integer coefficients and degree
at least 2. For , denote by the number of
integers such that has at least one divisor with . We determine the order of magnitude of uniformly for
and , showing that the order is
the same as the order of , the number of positive integers
with a divisor in . Here is an arbitrarily large constant and
is arbitrarily small.Comment: v2. minor edits and correction
Witnessing criticality in non-Hermitian systems via entropic uncertainty relation
Non-Hermitian systems with exceptional points lead to many intriguing
phenomena due to the coalescence of both eigenvalues and corresponding
eigenvectors, in comparison to Hermitian systems where only eigenvalues
degenerate. In this paper, we have investigated entropic uncertainty relation
(EUR) in a non-Hermitian system and revealed a general connection between the
EUR and the exceptional points of non-Hermitian system. Compared to the
unitarity dynamics determined by a Hermitian Hamiltonian, the behaviors of EUR
can be well defined in two different ways depending on whether the system is
located in unbroken phase or broken phase regimes. In unbroken phase regime,
EUR undergoes an oscillatory behavior while in broken phase regime where the
oscillation of EUR breaks down. The exceptional points mark the oscillatory and
non-oscillatory behaviors of the EUR. In the dynamical limit, we have
identified the witness of critical behavior of non-Hermitian systems in terms
of the EUR. Our results reveal that the witness can detect exactly the critical
points of non-Hermitian systems beyond (anti-) PT-symmetric systems. Our
results may have potential applications to witness and detect phase transition
in non-Hermitian systems.Comment: 8 pages,7fugure
Estimation of a partially linear additive model for data from an outcome-dependent sampling design with a continuous outcome
Outcome-dependent sampling (ODS) designs have been well recognized as a cost-effective way to enhance study efficiency in both statistical literature and biomedical and epidemiologic studies. A partially linear additive model (PLAM) is widely applied in real problems because it allows for a flexible specification of the dependence of the response on some covariates in a linear fashion and other covariates in a nonlinear non-parametric fashion. Motivated by an epidemiological study investigating the effect of prenatal polychlorinated biphenyls exposure on children's intelligence quotient (IQ) at age 7 years, we propose a PLAM in this article to investigate a more flexible non-parametric inference on the relationships among the response and covariates under the ODS scheme. We propose the estimation method and establish the asymptotic properties of the proposed estimator. Simulation studies are conducted to show the improved efficiency of the proposed ODS estimator for PLAM compared with that from a traditional simple random sampling design with the same sample size. The data of the above-mentioned study is analyzed to illustrate the proposed method
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