Kollektív viselkedés két dimenzióban = Collective behaviour in two-dimensions

A 2D rendszerek kollektiv viselkedésének formáit és törvényszerűségeit tanulmányoztuk úgy kvantummechanikai, mint klasszikus sokrészecskés esetekben. a) Kvantummechanikai rendszerekre módszereket dolgoztunk ki egzakt alapállapotok levezetésére még nem-integrálható esetekre is. Ezeket 2D-ben alkalmazva fém-szigetelő átmenetet mutattunk ki rendezetlen és kölcsönható rendszerekben, stripe és sakktábla fázisokat vezettünk le illetve normálfázisú nem-Fermi folyadék és szigetelő fázisokat kaptunk. b) Kerámiák esetében nemegyensúlyi körülmények között végzett mérésekkel nőveltük a mérések lokális felbontóképességét és a szupravezető Tc feletti tartományban különböző korrelációs hossz és koherencia élettartammal rendelkező tartományt mutattunk ki. c) Elemeztük granulált anyagok erőláncait, dipolusok aggregációs és kristályosodási folyamatait és mágneses zaj létrejöttét törésben. d) Rendezetlen mágnesek esetében javaslatot tettünk egy optimalizációs algoritmusra. Ezen hiszterézises optimalizáció működőképességét spinüveg modelleken és az utazó ügynök problémáján demonstráltuk. e) Epitaxiális felületnővekedés esetében egy térfogati diffúziót, spinodális dekompoziciót és a felületnövekedést magába foglaló modell segitségével megvizsgáltuk az epitaxiális növekedés során fellépő önszervező spontán kompozició modulációkat. Három különböző növekedési módust kaptunk eredményül, amelyek mindegyikében egydimenziós laterális kompozició modulációk alakulnak ki. | We have studied the possibilities and principles of the 2D collective behavior for many body systems holding both quantum mechanical, or classical properties. a) For the quantum case we elaborated procedures which allow the deduction of exact ground states even in non-integrable cases. These used in 2D led to metal-insulator transition for disordered and interacting systems, stripe and checkerboards, normal phase non-Fermi liquids and insulators. b) For ceramics, by measurements effectuated under non-equilibrium conditions we have increased the local resolution of measurements and we have found above the superconducting Tc several regions described by different correlation lengths and coherence lifetimes. c) We studied the force chains in granular media, aggregation and crystallization in dipolar monolayers and magnetic noise during fracture. d) For the case of random magnet systems we introduced an optimization algorithm. We demonstrated the performances of this hysteretic optimization method on spin glass models and on the traveling salesman problem. e) For the surface epitaxial growth case we constructed a model including bulk diffusion, spinodal decomposition, and surface growth to study the self-organized superlattice formation during the process under consideration. We found three growth regimes in which one dimensional lateral composition modulations occur

Modular elliptic curves over real abelian fields and the generalized Fermat equation x2ℓ+y2m=zpx^{2\ell}+y^{2m}=z^p

Using a combination of several powerful modularity theorems and class field theory we derive a new modularity theorem for semistable elliptic curves over certain real abelian fields. We deduce that if $K$ is a real abelian field of conductor $n<100$, with $5 \nmid n$ and $n \ne 29$, $87$, $89$, then every semistable elliptic curve $E$ over $K$ is modular. Let $\ell$, $m$, $p$ be prime, with $\ell$, $m \ge 5$ and $p \ge 3$.To a putative non-trivial primitive solution of the generalized Fermat $x^{2\ell}+y^{2m}=z^p$ we associate a Frey elliptic curve defined over $\mathbb{Q}(\zeta_p)^+$, and study its mod $\ell$ representation with the help of level lowering and our modularity result. We deduce the non-existence of non-trivial primitive solutions if $p \le 11$, or if $p=13$ and $\ell$, $m \ne 7$.Comment: Introduction rewritten to emphasise the new modularity theorem. Paper revised in the light of referees' comment

How to obtain lattices from (f,σ,δ)-codes via a generalization of Construction A

We show how cyclic (f,σ,δ)-codes over finite rings canonically induce a Z-lattice in RN by using certain quotients of orders in nonassociative division algebras defined using the skew polynomial f. This construction generalizes the one using certain σ-constacyclic codes by Ducoat and Oggier, which used quotients of orders in non-commutative associative division algebras defined by f, and can be viewed as a generalization of the classical Construction A for lattices from linear codes. It has the potential to be applied to coset coding, in particular to wire-tap coding. Previous results by Ducoat and Oggier are obtained as special cases

Credal Networks under Epistemic Irrelevance

A credal network under epistemic irrelevance is a generalised type of Bayesian network that relaxes its two main building blocks. On the one hand, the local probabilities are allowed to be partially specified. On the other hand, the assessments of independence do not have to hold exactly. Conceptually, these two features turn credal networks under epistemic irrelevance into a powerful alternative to Bayesian networks, offering a more flexible approach to graph-based multivariate uncertainty modelling. However, in practice, they have long been perceived as very hard to work with, both theoretically and computationally. The aim of this paper is to demonstrate that this perception is no longer justified. We provide a general introduction to credal networks under epistemic irrelevance, give an overview of the state of the art, and present several new theoretical results. Most importantly, we explain how these results can be combined to allow for the design of recursive inference methods. We provide numerous concrete examples of how this can be achieved, and use these to demonstrate that computing with credal networks under epistemic irrelevance is most definitely feasible, and in some cases even highly efficient. We also discuss several philosophical aspects, including the lack of symmetry, how to deal with probability zero, the interpretation of lower expectations, the axiomatic status of graphoid properties, and the difference between updating and conditioning