On the functional equation F(A(z))=G(B(z)),F(A(z))=G(B(z)), where A,BA,B are polynomialand and F,G$ are continuous functions

In this note we describe solutions of the equation: $F(A(z))=G(B(z)),$ where $A,B$ are polynomials and $F,G$ are continuous functions on the Riemann sphere.Comment: 6 pages, revise

On the Mahler measure of Jones polynomials under twisting

We show that the Mahler measures of the Jones polynomial and of the colored Jones polynomials converge under twisting for any link. Moreover, almost all of the roots of these polynomials approach the unit circle under twisting. In terms of Mahler measure convergence, the Jones polynomial behaves like hyperbolic volume under Dehn surgery. For pretzel links P(a_1,...,a_n), we show that the Mahler measure of the Jones polynomial converges if all a_i tend to infinity, and approaches infinity for a_i = constant if n tend to infinity, just as hyperbolic volume. We also show that after sufficiently many twists, the coefficient vector of the Jones polynomial and of any colored Jones polynomial decomposes into fixed blocks according to the number of strands twisted.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol5/agt-5-1.abs.htm

Multi-variable Polynomial Solutions to Pell's Equation and Fundamental Units in Real Quadratic Fields

For each positive integer $n$ it is shown how to construct a finite collection of multivariable polynomials $\{F_{i}:=F_{i}(t,X_{1},..., X_{\lfloor \frac{n+1}{2} \rfloor})\}$ such that each positive integer whose squareroot has a continued fraction expansion with period $n+1$ lies in the range of exactly one of these polynomials. Moreover, each of these polynomials satisfy a polynomial Pell's equation $C_{i}^{2} -F_{i}H_{i}^{2} = (-1)^{n-1}$ (where $C_{i}$ and $H_{i}$ are polynomials in the variables $t,X_{1},..., X_{\lfloor \frac{n+1}{2} \rfloor}$) and the fundamental solution can be written down. Likewise, if all the $X_{i}$'s and $t$ are non-negative then the continued fraction expansion of $\sqrt{F_{i}}$ can be written down. Furthermore, the congruence class modulo 4 of $F_{i}$ depends in a simple way on the variables $t,X_{1},..., X_{\lfloor \frac{n+1}{2} \rfloor}$ so that the fundamental unit can be written down for a large class of real quadratic fields. Along the way a complete solution is given to the problem of determining for which symmetric strings of positive integers $a_{1},..., a_{n}$ do there exist positive integers $D$ and $a_{0}$ such that $\sqrt{D} = [ a_{0};\bar{a_{1}, >..., a_{n},2a_{0}}]$.Comment: 13 page

Composite lacunary polynomials and the proof of a conjecture of Schinzel

Let $g(x)$ be a fixed non-constant complex polynomial. It was conjectured by Schinzel that if $g(h(x))$ has boundedly many terms, then h(x)\in \C[x] must also have boundedly many terms. Solving an older conjecture raised by R\'enyi and by Erd\"os, Schinzel had proved this in the special cases $g(x)=x^d$; however that method does not extend to the general case. Here we prove the full Schinzel's conjecture (actually in sharper form) by a completely different method. Simultaneously we establish an "algorithmic" parametric description of the general decomposition $f(x)=g(h(x))$, where $f$ is a polynomial with a given number of terms and $g,h$ are arbitrary polynomials. As a corollary, this implies for instance that a polynomial with $l$ terms and given coefficients is non-trivially decomposable if and only if the degree-vector lies in the union of certain finitely many subgroups of $\Z^l$.Comment: 9 page

Identification of γ-ray emission from 3C 345 and NRAO 512

For more than 15 years, since the days of the Energetic Gamma-Ray Experiment Telescope (EGRET) on board the Compton Gamma-Ray Observatory (CGRO; 1991−2000), it has remained an open question why the prominent blazar 3C 345 was not reliably detected at γ-ray energies ≥ 20 MeV. Recently a bright γ-ray source (0FGL J1641.4+3939/1FGL J1642.5+3947), potentially associated with 3C 345, was detected by the Large Area Telescope (LAT) on Fermi. Multiwavelength observations from radio bands to X-rays (mainly GASP-WEBT and Swift) of possible counterparts (3C 345, NRAO 512, B3 1640 + 396) were combined with 20 months of Fermi-LAT monitoring data (August 2008 − April 2010) to associate and identify the dominating γ-ray emitting counterpart of 1FGL J1642.5+3947. The source 3C 345 is identified as the main contributor for this γ-ray emitting region. However, after November 2009 (15 months), a significant excess of photons from the nearby quasar NRAO 512 started to contribute and thereafter was detected with increasing γ-ray activity, possibly adding flux to 1FGL J1642.5+3947. For the same time period and during the summer of 2010, an increase of radio, optical and X-ray activity of NRAO 512 was observed. No γ-ray emission from B3   1640 + 396 was detected

Uniparental Disomy and Genomic Imprinting in Humans

Uniparental disomy (UPD), the inheritance of both homologues from one chromosome from the same parent, was first proposed in 1980 by Erik Engel [1] to be a potential cause of congenital developmental defects in hymans. First hints from the premolecular era towards its existence came from instances where a pericentric inversion was present on one homologue in a parent and on both in one offspring [2] and where there was transmission of an interhomologous Robertsonian translocation (of chromosome 22) from a healthy mother to healthy offspring [3-4]. In mice, UPD was experimentally produced by crossing two mice lines with different Robertsonian translocations both involving the same chromosome [for 2 review see ref. 5]. Through this approach, it was possible to define imprinted regions, chromosomes and chromosomal segments for which either maternal or paternal or both types of uniparental disomy led to phenotypic abnormalities. The latter are explained by genomic imprinting, the differential silencing of a gene or genes from one of the parents (the mother or the father) during any stage of embryogenesis or later in life. If, for example, the maternal homologue of a given gene is imprinted (and hence only the paternal allele is active), maternal UPD would lead to loss of the active allele and thus might cause consequences due to loss of functio

Triples of Positive Integers with the same Sum and the same Product

It is proved that for every k there exist k triples of positive integers with the same sum and the same product

Sew Zambia! Home Economics in Primary & Secondary Schools of the Eastern & Southern Provinces

Introduction: Home Economics (HE) is within the main curriculum for primary school students in Zambia, yet it remains unclear what is included. Methods: Survey research conducted between May-June 2022 on 6-7th-grade students within 9 Zambian schools. Classroom observations were made, and teachers were interviewed in primary and secondary public schools. Results: Most know how to cook and sew. Zambian women cook more than men and learn at an older age. Both males and females are trained to sew learning at a much younger age. Discussion: HE remains critical to learning life skills in Zambia and the curriculum reflects skills related to rural living: 54.6% of Zambians work in agriculture while only 2% work in agriculture in the US (FAO, 2023)