I’M SORRY (information is out-of-date… I’m now updating it slowly).

**TITLE**: Multiplicity results for Schrödinger-Poisson systems

**LOCATION**: Department of Information Engineering, Computer Science and Mathematics – University of L’Aquila [url:01]

**ABSTRACT**: We discuss the core of some results about the multiplicity of solutions of Schrödinger-Poisson systems of the following type

for some classes of nonlinearities f (e.g. indefinite nonlinearity or involving the critical exponent). These systems can be seen as nonlocal equations where variational methods and critical point theory may be applied. A direct method will be shown which do not involve Palais-Smale condition or the Ekeland variational principle. Special phenomena of these systems, compared with semilinear elliptic equations, will be point out. We emphasize open questions about the extension of these systems, a numerical analysis approach based on a recent reproducing kernel Hilbert and Tikhonov regularization technique, and discuss their possible applications to superconductors and nanoscience (computational chemistry).

Joint work with L. Huang and J. Chen.

Based on some papers as: (a) Nonlin. Anal. Real World Appl. 28(2016), 1-19 (to appear); (b) Libertas Math. 34:2(2014), 91-104; (c) J. Math. Anal. Appl. 408(2013), 55-69; (d) J. Differential Equations 255(2013), 2463-2483.

Today, we published the NIM Game in the Google Play store [url:01].

**About the standard NIM game**

Tradicionally, the NIM game is a board game for two players with quite simple rules. Starting with any number of counters distributed in any number of piles, two players take turns to remove any number of counters from a single pile. The winner is the player who takes the last counter. In figure,* a stone “Nim” probably originating from Chinese Jian Shi Zi in the 15th century.*

Matematically, NIM is a subject of study in Combinatorial Game Theory, for which there is a ‘winning strategy’ and all the moves can be analysed. See for example:

- NIM-like Games [url:02]
- Combinatorial Game Theory [url:03]
- The Eletronic Journal of Combinatorics – Combinatorial Games [url:04]
- Integers: Electronic Journal of Combinatorial Number Theory [url:05]
- Journal of Combinatorial Theory [url:06]

**Our NIM Game for Android**

The game was developed by the group Geometrix. Besides the standard features, we added the following:

- Three variations to the standard rules: Misére, NIM21 and Fibonacci;
- Each game can have more than 2 players (e.g. against computer players and/or other humans) in the local multiplayer mode or in the online multiplayer mode;
- A ‘power-up’ is available which allows a player to break the adversary winning strategy (just once), based on a harmonic oscillator equation (i.e. a second order ordinary differential equation). Graphically, the power-up is implemented as a gauge with damping, oscillating in the screen for ten seconds, where the frequency and amplitude are function of the difficulty level, the number of counters in the board, etc. The player may add or remove counters with it.

**Our aim in developing this game**

Besides the fun and mathematical interest of the NIM game by itself, we intend to use this game as a tool to research how an anonymous player deal with a combinatorial problem. In particular, we are interested in the research issues behind the understanding of building and classifying user’s learning profiles by using techniques from deep learning, linear optimization, algebra and graph theory. For such reason, we collect basic statistics about players performance and their way of playing. If you are a player, we acknowledge your valuable collaboration to this project.

(will be updated)

I was member of the evaluation committee of the PhD of *Sandrina Santos* with the thesis “**Nonlinear elliptic equations with nonsmooth potential – variational and topological methods**“, on 7 December 2009, at the Department of Mathematics of the University of Aveiro. The other members of the committee were Jean Mawhin, Luis Sanchez, Nikolaos Papageorgiou, and Vasile Staicu.

I gave the talk “**Linear and nonlinear elliptic equations with critical exponents, Hardy-Sobolev terms and combined singular terms**” at the regular seminar of the Group Functional Analysis and Applications of UIMA.

Abstract

I was member of the evaluation committee of the PhD of *Sofia Pinheiro* with the thesis “**Besov spaces and the Laplacian on fractal h-sets**“, on 28 July 2009, at the Department of Mathematics of the University of Aveiro.