Ground state of low-dimensional dipolar gases: Linear and zigzag chains

We study the ground-state phase diagram of ultracold dipolar gases in highly anisotropic traps. Starting from a one-dimensional geometry, by ramping down the transverse confinement along one direction, the gas reaches various planar distributions of dipoles. At large linear densities, when the dipolar gas exhibits a crystal-like phase, critical values of the transverse frequency exist below which the configuration exhibits transverse patterns. These critical values are found by means of a classical theory, and are in full agreement with classical Monte Carlo simulations. The study of the quantum system is performed numerically with Monte Carlo techniques and shows that the quantum fluctuations smoothen the transition and make it completely disappear in a gas phase. These predictions could be experimentally tested and would allow one to reveal the effect of zero-point motion on self-organized mesoscopic structures of matter waves, such as the transverse pattern of the zigzag chain.

Mixing operators on spaces with weak topology

We prove that a continuous linear operator T on a topological vector space X with weak topology is mixing if and only if the dual operator T' has no finite dimensional invariant subspaces. This result implies the characterization of hypercyclic operators on the space $\omega$ due to Herzog and Lemmert and implies the result of Bayart and Matheron, who proved that for any hypercyclic operator T on $\omega$, $T\oplus T$ is also hypercyclic.

Zero-One Designs Produce Small Hard SAT Instances

Some basics of combinatorial block design are combined with certain constraint satisfaction problems of interest to the satisfiability community. The paper shows how such combinations lead to satisfiability problems, and shows empirically that these are some of the smallest very hard satisfiability problems ever constructed. Partially balanced (0,1) designs (PB01Ds) are introduced as an extension of balanced incomplete block designs (BIBDs) and (0,1) designs. Also, (0,1) difference sets are introduced as an extension of certain cyclical difference sets. Constructions based on (0,1) difference sets enable generation of PB01Ds over a much wider range of parameters than is possible for BIBDs. Building upon previous work of Spence, it is shown how PB01Ds lead to small, very hard, unsatisfiable formulas. A new three-dimensional form of combinatorial block design is introduced, and leads to small, very hard, satisfiable formulas. The methods are validated on solvers that performed well in the SAT 2009 and earlier competitions.

Investigation of tanpura string vibrations using a two-dimensional time-domain model incorporating coupling and bridge friction

Tanpura string vibrations have been investigated previously using numerical models based on energy conserving schemes derived from a Hamiltonian description in one-dimensional form. Such time-domain models have the property that, for the lossless case, the numerical Hamiltonian (representing total energy of the system) can be proven to be constant from one time step
to the next, irrespective of any of the system parameters; in practice the Hamiltonian can be shown to be conserved within machine precision. Models of this kind can reproduce a jvari effect, which results from the bridge-string interaction. However the one-dimensional formulation has recently been shown to fail to replicate the jvaris strong dependence on the thread placement. As a first step towards simulations which accurately emulate this sensitivity to the thread placement, a twodimensional model is proposed, incorporating coupling of controllable level between the two string polarisations at the string termination opposite from the barrier. In addition, a friction force acting when the string slides across the bridge in horizontal direction is introduced, thus effecting a further damping mechanism. In this preliminary study, the string is terminated at the position of the thread. As in the one-dimensional model, an implicit scheme has to be used to solve the system, employing Newton's method to calculate the updated positions and momentums of each string segment. The two-dimensional model is proven to be energy conserving when the loss parameters are set to zero, irrespective of the coupling constant. Both frequency-dependent and independent losses are then added to the string, so that the model can be compared to analogous instruments. The influence of coupling and the bridge friction are investigated.

Violation of Bell inequalities in larger Hilbert spaces: robustness and challenges

We explore the challenges posed by the violation of Bell-like inequalities by d-dimensional systems exposed to imperfect state-preparation and measurement settings. We address, in particular, the limit of high-dimensional systems, naturally arising when exploring the quantum-to-classical transition. We show that, although suitable Bell inequalities can be violated, in principle, for any dimension of given subsystems, it is in practice increasingly challenging to detect such violations, even if the system is prepared in a maximally entangled state. We characterize the effects of random perturbations on the state or on the measurement settings, also quantifying the efforts needed to certify the possible violations in case of complete ignorance on the system state at hand.

Three dimensional Sklyanin algebras and Gröbner bases

We consider Sklyanin algebras $S$ with 3 generators, which are quadratic algebras over a field $\K$ with $3$ generators $x,y,z$ given by $3$ relations $pxy+qyx+rzz=0$, $pyz+qzy+rxx=0$ and $pzx+qxz+ryy=0$, where $p,q,r\in\K$. this class of algebras has enjoyed much attention. In particular, using tools from algebraic geometry, Feigin, Odesskii \cite{odf}, and Artin, Tate and Van Den Bergh, showed that if at least two of the parameters $p$, $q$ and $r$ are non-zero and at least two of three numbers $p^3$, $q^3$ and $r^3$ are distinct, then $S$ is Artin--Schelter regular. More specifically, $S$ is Koszul and has the same Hilbert series as the algebra of commutative polynomials in 3 indeterminates (PHS). It has became commonly accepted that it is impossible to achieve the same objective by purely algebraic and combinatorial means like the Groebner basis technique. The main purpose of this paper is to trace the combinatorial meaning of the properties of Sklyanin algebras, such as Koszulity, PBW, PHS, Calabi-Yau, and to give a new constructive proof of the above facts due to Artin, Tate and Van Den Bergh. Further, we study a wider class of Sklyanin algebras, namely
the situation when all parameters of relations could be different. We call them generalized Sklyanin algebras. We classify up to isomorphism all generalized Sklyanin algebras with the same Hilbert series as commutative polynomials on
3 variables. We show that generalized Sklyanin algebras in general position have a Golod–Shafarevich Hilbert series (with exception of the case of field with two elements).