Advances on the Simplification of Sine–Cosine Equations

AbstractIn this paper we contribute several results to the approach initiated by Hommel and Kovács (well documented with applications in a recent book by Kovács (1993)) on the symbolic simplification of sine–cosine polynomials that arise, for instance, as determining equations for joint values in robotics inverse kinematic problems. We present, taking into consideration for the first time sine–cosine polyomials, fast algorithms for the functional decomposition and factorization problems, reducing the solving of suchs–cequations to a sequence of lower degree ones. Moreover, we show that triangularization of a given sine–cosine equation provides a conceptual understanding of the conditions that yield extraneous roots in the half-angle tangent substitution (and therefore that imply a reduction of the degree in the determining equation of a givens–csystem)

Some issues on the automatic computation of plane envelopes in interactive environments

This paper addresses some concerns, and describes some proposals, on the elusive concept of envelope of an algebraic family of varieties, and on its automatic computation. We describe how to use the recently developed Gröbner Cover algorithm to study envelopes of families of algebraic curves, and we give a protocol toward its implementation in dynamic geometry environments. The proposal is illustrated through some examples. A beta version of GeoGebra is used to highlight new envelope abilities in interactive environments, and limitations of our approach are discussed, since the computations are performed in an algebraically closed field

Factoring analytic multivariate polynomials and non-standard Cauchy–Riemann conditions

Motivated by previous work on the simplification of parametrizations of curves, in this paper we generalize the well-known notion of analytic polynomial (a bivariate polynomial P (x , y ), with complex coefficients, which arises by substituting z → x + iy on a univariate polynomial View the MathML source, i.e. p (z ) → p (x + iy ) = P (x , y )) to other finite field extensions, beyond the classical case of View the MathML source. In this general setting we obtain different properties on the factorization, gcd's and resultants of analytic polynomials, which seem to be new even in the context of Complex Analysis. Moreover, we extend the well-known Cauchy–Riemann conditions (for harmonic conjugates) to this algebraic framework, proving that the new conditions also characterize the components of generalized analytic polynomials

: A PLATFORM FOR SHARING DYNAMIC GEOMETRY RESOURCES ALL OVER EUROPE

International audienceThe poster presents the I2Geo platform developed within the Intergeo project whose ambition was to develop a Pan-European math teacher community enabled to share resources and practices in using dynamic geometry (DG). We present the main tools and services i2geo offers to stimulate the use of DG in mathematics classes

Professional development in mathematics education: evaluation of a MOOC on outdoor mathematics

In this paper, we examine the impact of a massive open online course (MOOC) in thecontext of outdoor mathematics on the participating teachers’ professional development. We firstlyintroduce the theoretical background on outdoor mathematics, focusing on math trails with the digitaltool MathCityMap and professional development to be accomplished using MOOCs. By taking intoaccount the MOOC “Task Design for Math Trails”, with 93 finalists, we analyze the learning progressof 19 selected case studies from different nations and learning levels by taking into account theiranswers in a pre- and post-questionnaire and their posts on a specific communication message board,with a special focus on the MOOC’s topics’ task design for outdoor mathematics and the digital toolMathCityMap. The analysis is performed using different quantitative and qualitative approaches.The results show that the teachers studied have benefited from professional development, which isevident in the expansion/evolution of their knowledge from a content, pedagogical, and technologicalperspective. Finally, we formulate consequences for professional development in STEM education,and conclude the paper with limitations to be drawn and a perspective for further research.C41E-A4D9-3C8D | Elisabete Ferraz da CunhaN/

Divergent Effects of Metformin on an Inflammatory Model of Parkinson’s Disease

The oral antidiabetic drug metformin is known to exhibit anti-inflammatory properties through activation of AMP kinase, thus protecting various brain tissues as cortical neurons, for example. However, the effect of metformin on the substantia nigra (SN), the main structure affected in Parkinson’s disease (PD), has not yet been studied in depth. Inflammation is a key feature of PD and it may play a central role in the neurodegeneration that takes place in this disorder. The aim of this work was to determine the effect of metformin on the microglial activation of the SN of rats using the animal model of PD based on the injection of the pro-inflammogen lipopolysaccharide (LPS). In vivo and in vitro experiments were conducted to study the activation of microglia at both the cellular and molecular levels. Our results indicate that metformin overall inhibits microglia activation measured by OX-6 (MHCII marker), IKKβ (pro-inflammatory marker) and arginase (anti-inflammatory marker) immunoreactivity. In addition, qPCR experiments reveal that metformin treatment minimizes the expression levels of several pro- and anti-inflammatory cytokines. Mechanistically, the drug decreases the phosphorylated forms of mitogen-activated protein kinases (MAPKs) as well as ROS generation through the inhibition of the NADPH oxidase enzyme. However, metformin treatment fails to protect the dopaminergic neurons of SN in response to intranigral LPS. These findings suggest that metformin could have both beneficial and harmful pharmacological effects and raise the question about the potential use of metformin for the prevention and treatment of PD.España MINECO SAF2015-64171-

The Ubiquitin Proteasome System in Neuromuscular Disorders: Moving Beyond Movement

Neuromuscular disorders (NMDs) affect 1 in 3000 people worldwide. There are more than 150 different types of NMDs, where the common feature is the loss of muscle strength. These disorders are classified according to their neuroanatomical location, as motor neuron diseases, peripheral nerve diseases, neuromuscular junction diseases, and muscle diseases. Over the years, numerous studies have pointed to protein homeostasis as a crucial factor in the development of these fatal diseases. The ubiquitin-proteasome system (UPS) plays a fundamental role in maintaining protein homeostasis, being involved in protein degradation, among other cellular functions. Through a cascade of enzymatic reactions, proteins are ubiquitinated, tagged, and translocated to the proteasome to be degraded. Within the ubiquitin system, we can find three main groups of enzymes: E1 (ubiquitin-activating enzymes), E2 (ubiquitin-conjugating enzymes), and E3 (ubiquitin-protein ligases). Only the ubiquitinated proteins with specific chain linkages (such as K48) will be degraded by the UPS. In this review, we describe the relevance of this system in NMDs, summarizing the UPS proteins that have been involved in pathological conditions and neuromuscular disorders, such as Spinal Muscular Atrophy (SMA), Charcot-Marie-Tooth disease (CMT), or Duchenne Muscular Dystrophy (DMD), among others. A better knowledge of the processes involved in the maintenance of proteostasis may pave the way for future progress in neuromuscular disorder studies and treatments.Ministerio de Economía y Competitividad RTI2018-098645-B-10