1,022 research outputs found
Diophantine equations with Euler polynomials
In this paper we determine possible decompositions of Euler polynomials
, i.e. possible ways of writing Euler polynomials as a functional
composition of polynomials of lower degree. Using this result together with the
well-known criterion of Bilu and Tichy, we prove that the Diophantine equation
with of
degree at least and , has only finitely many integers solutions
unless polynomial can be decomposed in ways that we list explicitly.Comment: to appear in Acta Arithmetic
On equal values of power sums of arithmetic progressions
In this paper we consider the Diophantine equation \begin{align*}b^k
+\left(a+b\right)^k &+ \cdots + \left(a\left(x-1\right) + b\right)^k=\\ &=d^l +
\left(c+d\right)^l + \cdots + \left(c\left(y-1\right) + d\right)^l,
\end{align*} where are given integers. We prove that, under some
reasonable assumptions, this equation has only finitely many integer solutions.Comment: This version differs slightly from the published version in its
expositio
On conjectures and problems of Ruzsa concerning difference graphs of S-units
Given a finite nonempty set of primes S, we build a graph with
vertex set by connecting x and y if the prime divisors of both the
numerator and denominator of x-y are from S. In this paper we resolve two
conjectures posed by Ruzsa concerning the possible sizes of induced
nondegenerate cycles of , and also a problem of Ruzsa concerning
the existence of subgraphs of which are not induced subgraphs.Comment: 15 page
Synthesis and magnetic properties of NiFe_{2-x}Al_{x}O_{4} nanoparticles
Nanocrystalline Al-doped nickel ferrite powders have been synthesized by
sol-gel auto-ignition method and the effect of non-magnetic aluminum content on
the structural and magnetic properties has been studied. The X-ray diffraction
(XRD) revealed that the powders obtained are single phase with inverse spinel
structure. The calculated grain sizes from XRD data have been verified using
transmission electron microscopy (TEM). TEM photographs show that the powders
consist of nanometer-sized grains. It was observed that the characteristic
grain size decreases from 29 to 6 nm as the non-magnetic Al content increases,
which was attributed to the influence of non-magnetic Al concentration on the
grain size. Magnetic hysteresis loops were measured at room temperature with a
maximum applied magnetic field of 1T. As aluminum content increases, the
measured magnetic hysteresis curves become more and more narrow and the
saturation magnetization and remanent magnetization both decreased. The
reduction of agnetization compared to bulk is a consequence of spin
non-collinearity. Further reduction of magnetization with increase of aluminum
content is caused by non-magnetic Al^{3+} ions and weakened interaction between
sublattices. This, as well as the decrease in hysteresis was understood in
terms of the decrease in particle size.Comment: 24 pages, 6 figure
Tumour Cell Heterogeneity.
The population of cells that make up a cancer are manifestly heterogeneous at the genetic, epigenetic, and phenotypic levels. In this mini-review, we summarise the extent of intra-tumour heterogeneity (ITH) across human malignancies, review the mechanisms that are responsible for generating and maintaining ITH, and discuss the ramifications and opportunities that ITH presents for cancer prognostication and treatment
30 years of collaboration
We highlight some of the most important cornerstones of the long standing and very fruitful collaboration of the Austrian Diophantine Number Theory research group and the Number Theory and Cryptography School of Debrecen. However, we do not plan to be complete in any sense but give some interesting data and selected results that we find particularly nice. At the end we focus on two topics in more details, namely a problem that origins from a conjecture of Rényi and Erdős (on the number of terms of the square of a polynomial) and another one that origins from a question of Zelinsky (on the unit sum number problem). This paper evolved from a plenary invited talk that the authors gaveat the Joint Austrian-Hungarian Mathematical Conference 2015, August 25-27, 2015 in Győr (Hungary)
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