Subscribe to the gold package and get unlimited access to Shamra Academy
Register a new userGeneralization of a theorem of Clunie and Hayman
198
0
0.0
(
0
)
Ask ChatGPT about the research
No Arabic abstract
Clunie and Hayman proved that if the spherical derivative of an entire function has order of growth sigma then the function itself has order at most sigma+1. We extend this result to holomorphic curves in projective space of dimension n omitting n hyperplanes in general position.
rate research
Read More
In this paper we first prove a version of $L^{2}$ existence theorem for line bundles equipped a singular Hermitian metrics. Aa an application, we establish a vanishing theorem which generalizes the classical Nadel vanishing theorem.
The purpose of this paper is to present a solution to perhaps the final remaining case in the line of study concerning the generalization of Forellis theorem on the complex analyticity of the functions that are: (1) $mathcal{C}^infty$ smooth at a point, and (2) holomorphic along the complex integral curves generated by a contracting holomorphic vector field with an isolated zero at the same point.
Caratheodory showed that $n$ complex numbers $c_1,...,c_n$ can uniquely be written in the form $c_p=sum_{j=1}^m rho_j {epsilon_j}^p$ with $p=1,...,n$, where the $epsilon_j$s are different unimodular complex numbers, the $rho_j$s are strictly positive numbers and integer $m$ never exceeds $n$. We give the conditions to be obeyed for the former property to hold true if the $rho_j$s are simply required to be real and different from zero. It turns out that the number of the possible choices of the signs of the $rho_j$s are {at most} equal to the number of the different eigenvalues of the Hermitian Toeplitz matrix whose $i,j$-th entry is $c_{j-i}$, where $c_{-p}$ is equal to the complex conjugate of $c_{p}$ and $c_{0}=0$. This generalization is relevant for neutron scattering. Its proof is made possible by a lemma - which is an interesting side result - that establishes a necessary and sufficient condition for the unimodularity of the roots of a polynomial based only on the polynomial coefficients. Keywords: Toeplitz matrix factorization, unimodular roots, neutron scattering, signal theory, inverse problems. PACS: 61.12.Bt, 02.30.Zz, 89.70.+c, 02.10.Yn, 02.50.Ga
Let $n$ be a positive integer. In 1915, Theisinger proved that if $nge 2$, then the $n$-th harmonic sum $sum_{k=1}^nfrac{1}{k}$ is not an integer. Let $a$ and $b$ be positive integers. In 1923, Nagell extended Theisingers theorem by showing that the reciprocal sum $sum_{k=1}^{n}frac{1}{a+(k-1)b}$ is not an integer if $nge 2$. In 1946, ErdH{o}s and Niven proved a theorem of a similar nature that states that there is only a finite number of integers $n$ for which one or more of the elementary symmetric functions of $1,1/2, ..., 1/n$ is an integer. In this paper, we present a generalization of Nagells theorem. In fact, we show that for arbitrary $n$ positive integers $s_1, ..., s_n$ (not necessarily distinct and not necessarily monotonic), the following reciprocal power sum $$sumlimits_{k=1}^{n}frac{1}{(a+(k-1)b)^{s_{k}}}$$ is never an integer if $nge 2$. The proof of our result is analytic and $p$-adic in character.
This article contains a complete proof of Gabrielovs rank Theorem, a fundamental result in the study of analytic map germs. Inspired by the works of Gabrielov and Tougeron, we develop formal-geometric techniques which clarify the difficult parts of the original proof. These techniques are of independent interest, and we illustrate this by adding a new (very short) proof of the Abhyankar-Jung Theorem. We include, furthermore, new extensions of the rank Theorem (concerning the Zariski main Theorem and elimination theory) to commutative algebra.
Log in to be able to interact and post comments
comments
Fetching comments
Sign in to be able to follow your search criteria
