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## Polynomial normal form and finitely smooth equivalence of autonomous systems with one zero eigenvalue

In a neighborhood of a singular point, we consider autonomous systems of ordinary differential equations for which one eigenvalue of the matrix of the linear part is zero and the remaining eigenvalues do not belong to the imaginary axis. We study the reducibility of such systems to normal form. We prove that the problem of finitely smooth equivalence can be solved for such systems by using finite segments of the Taylor series of their right-hand sides.

In this paper, in a neighborhood of a singular point, we consider autonomous systems of ordinary differential equations such that the matrix of their linear part has one zero eigenvalue, while the other eigenvalues lie outside the imaginary axis. We prove that the problem of finitely smooth equivalence can be solved for such systems by using finite segments of the Taylor series of their right-hand sides.

In a neighborhood of a singular point, we consider autonomous systems of ordinary differential equations such that the matrix of their linear part has two purely imaginary eigenvalues, while the other eigenvalues lie outside the imaginary axis. We study the reducibility of such systems to pseudonormal form. We prove that the problem of finitely smooth equivalence can be solved for such systems by using finite segments of the Taylor series of their right-hand sides.

The Autonomous Agents and MultiAgent Systems (AAMAS) conference series brings together researchers from around the world to share the latest advances in the field. It is the premier forum for research in the theory and practice of autonomous agents and multi-agent systems. AAMAS 2002, the first of the series, was held in Bologna, followed by Melbourne (2003), New York (2004), Utrecht (2005), Hakodate (2006), Honolulu (2007), Estoril (2008), Budapest (2009), Toronto (2010), Taipei (2011), Valencia (2012), Saint Paul (2013), Paris (2014), and Istanbul (2015). This volume constitutes the proceedings of AAMAS 2016, the fifteenth conference in the series, held in Singapore in May 2016.

In line with previous editions, AAMAS 2016 attracted submissions for a general track and five special tracks: Innovative Applications, Robotics, Embodied Virtual Agents and Human-Agent Interaction, Blue Sky Ideas track, and the JAAMAS presentation track. The special tracks were chaired by leading researchers in their corresponding fields: Onn Shehory and Noa Agmon chaired the Innovative Applications track, Francesco Amigoni and Roderich Gross the Robotics track, Tim Bickmore and Hannes Vilhjálmsson the Embodied Virtual Agents and Human-Agent Interaction track, and Frank Dignum the Blue Sky Ideas track. As a new initiative, the chairs of AAMAS 2016 also solicited articles published in the Journal of Autonomous Agents and Multiagent Systems for the JAAMAS Presentation Track. Only papers that have appeared in the Journal of Autonomous Agents and Multi-agent Systems (JAAMAS) in the 12 months period preceding the AAMAS notification date were eligible. This new track was chaired by Peter Stone.

Jointly with the PC chairs the special track chairs were responsible for appointing the Programme Committee (PC) members and the Senior Programme Committee members (SPC) for their tracks, and they made acceptance/rejection recommendations for their tracks in consultation with Programme Chairs based on input provided by the track PC, SPC, and Area Experts. This year the PC chairs introduced the new role of Area Experts, i.e., SPC members with additional responsibilities, to assist with selecting SPC members for specific research areas, identifying appropriate keywords, and assisting in potential issues during discussion phase. This new role was a success and increased the quality of our SPC and PC, and also the reviewing process in general.

Full paper submissions (8 pages plus bibliographic references) and Blue Sky Ideas paper submissions (4 pages plus references) were solicited for AAMAS 2016. Some of the full paper submissions were accepted as extended abstracts (2 pages). The papers were selected by means of a thorough review and discussion process, which included an opportunity for authors to respond to reviewer comments during a rebuttal phase. All SPC members, Area Experts, and Track Chairs followed and contributed to the technical discussions on the papers they were overseeing. The JAAMAS presentation Track submissions published as extended abstracts were handled by the track chair.

Overall, out of 550 submissions, 137 (25%) were accepted as full papers and 143 (26%) were accepted as extended abstracts. Additionally, all 16 JAAMAS track submissions were accepted.

Full papers were presented orally in 20 minute slots; all extended abstracts and, optionally, full papers were presented as posters during the conference.

Out of the 550 submissions, 351 (64%) had a student as the primary author, 82 of these were accepted as full papers (23%), and a further 90 (26%) were accepted as extended abstracts.

The proceedings also contain 17 Demonstration papers, 13 Doctoral Consortium papers, as well as abstracts of the invited talks and details of some of the awards given.

Let k be a field of characteristic zero, let G be a connected reductive algebraic group over k and let g be its Lie algebra. Let k(G), respectively, k(g), be the field of k- rational functions on G, respectively, g. The conjugation action of G on itself induces the adjoint action of G on g. We investigate the question whether or not the field extensions k(G)/k(G)^G and k(g)/k(g)^G are purely transcendental. We show that the answer is the same for k(G)/k(G)^G and k(g)/k(g)^G, and reduce the problem to the case where G is simple. For simple groups we show that the answer is positive if G is split of type A_n or C_n, and negative for groups of other types, except possibly G_2. A key ingredient in the proof of the negative result is a recent formula for the unramified Brauer group of a homogeneous space with connected stabilizers. As a byproduct of our investigation we give an affirmative answer to a question of Grothendieck about the existence of a rational section of the categorical quotient morphism for the conjugating action of G on itself.

Let G be a connected semisimple algebraic group over an algebraically closed field k. In 1965 Steinberg proved that if G is simply connected, then in G there exists a closed irreducible cross-section of the set of closures of regular conjugacy classes. We prove that in arbitrary G such a cross-section exists if and only if the universal covering isogeny Ĝ → G is bijective; this answers Grothendieck's question cited in the epigraph. In particular, for char k = 0, the converse to Steinberg's theorem holds. The existence of a cross-section in G implies, at least for char k = 0, that the algebra k[G]G of class functions on G is generated by rk G elements. We describe, for arbitrary G, a minimal generating set of k[G]G and that of the representation ring of G and answer two Grothendieck's questions on constructing generating sets of k[G]G. We prove the existence of a rational (i.e., local) section of the quotient morphism for arbitrary G and the existence of a rational cross-section in G (for char k = 0, this has been proved earlier); this answers the other question cited in the epigraph. We also prove that the existence of a rational section is equivalent to the existence of a rational W-equivariant map T- - - >G/T where T is a maximal torus of G and W the Weyl group.