Characterizing Triviality of the Exponent Lattice of A Polynomial through Galois and Galois-Like Groups

The problem of computing \emph{the exponent lattice} which consists of all the multiplicative relations between the roots of a univariate polynomial has drawn much attention in the field of computer algebra. As is known, almost all irreducible polynomials with integer coefficients have only trivial exponent lattices. However, the algorithms in the literature have difficulty in proving such triviality for a generic polynomial. In this paper, the relations between the Galois group (respectively, \emph{the Galois-like groups}) and the triviality of the exponent lattice of a polynomial are investigated. The \bbbq\emph{-trivial} pairs, which are at the heart of the relations between the Galois group and the triviality of the exponent lattice of a polynomial, are characterized. An effective algorithm is developed to recognize these pairs. Based on this, a new algorithm is designed to prove the triviality of the exponent lattice of a generic irreducible polynomial, which considerably improves a state-of-the-art algorithm of the same type when the polynomial degree becomes larger. In addition, the concept of the Galois-like groups of a polynomial is introduced. Some properties of the Galois-like groups are proved and, more importantly, a sufficient and necessary condition is given for a polynomial (which is not necessarily irreducible) to have trivial exponent lattice.Comment: 19 pages,2 figure

A parthenogenetic quasi-program causes teratoma-like tumors during aging in wild-type C. elegans

A long-standing belief is that aging (senescence) is the result of stochastic damage accumulation. Alternatively, senescent pathology may also result from late-life, wild-type gene action (i.e., antagonistic pleiotropy, as argued by Williams) leading to non-adaptive run-on of developmental programs (or quasi-programs) (as suggested more recently by Blagosklonny). In this study, we use existing and new data to show how uterine tumors, a prominent form of senescent pathology in the nematode Caenorhabditis elegans, likely result from quasi-programs. Such tumors develop from unfertilized oocytes which enter the uterus and become hypertrophic and replete with endoreduplicated chromatin masses. Tumor formation begins with ovulation of unfertilized oocytes immediately after exhaustion of sperm stocks. We show that the timing of this transition between program and quasi-program (i.e., the onset of senescence), and the onset of tumor formation, depends upon the timing of sperm depletion. We identify homology between uterine tumors and mammalian ovarian teratomas, which both develop from oocytes that fail to mature after meiosis I. In teratomas, futile activation of developmental programs leads to the formation of differentiated structures within the tumor. We report that older uterine tumors express markers of later embryogenesis, consistent with teratoma-like activation of developmental programs. We also present evidence of coupling of distal gonad atrophy to oocyte hypertrophy. This study shows how the Williams Blagosklonny model can provide a mechanistic explanation of this component of C. elegans aging. It also suggests etiological similarity between teratoma and some forms of senescent pathology, insofar as both are caused by quasi-programs

Time to Upgrade: A New OpenSPIM Guide to Build and Operate Advanced OpenSPIM Configurations.

OpenSPIM is an Open Access platform for Selective Plane Illumination Microscopy (SPIM) and allows hundreds of laboratories around the world to generate and process light-sheet data in a cost-effective way due to open-source hardware and software. While setting up a basic OpenSPIM configuration can be achieved expeditiously, correctly assembling and operating more complex OpenSPIM configurations can be challenging for routine standard OpenSPIM users. Detailed instructions on how to equip an OpenSPIM with two illumination sides and two detection axes (X-OpenSPIM) are provided, and a solution is also provided on how the temperature can be controlled in the sample chamber. Additionally, it is demonstrated how to operate it by implementing an ArduinoUNO microcontroller and introducing μOpenSPIM, a new software plugin for OpenSPIM, to facilitate image acquisition. The new software works on any OpenSPIM configuration comes with drift correction functionality, on-the-fly image processing, and gives users more options in the way time-lapse movies are initially set up and saved. Step-by-step guides are also provided within the Supporting Information and on the website on how to align the lasers, configure the hardware, and acquire images using μOpenSPIM. With this, current OpenSPIM users are empowered in various ways, and newcomers striving to use more advanced OpenSPIM systems are helped