Using level-2 fuzzy sets to combine uncertainty and imprecision in fuzzy regions

In many applications, spatial data need to be considered but are prone to uncertainty or imprecision. A fuzzy region - a fuzzy set over a two dimensional domain - allows the representation of such imperfect spatial data. In the original model, points of the fuzzy region where treated independently, making it impossible to model regions where groups of points should be considered as one basic element or subregion. A first extension overcame this, but required points within a group to have the same membership grade. In this contribution, we will extend this further, allowing a fuzzy region to contain subregions in which not all points have the same membership grades. The concept can be used as an underlying model in spatial applications, e.g. websites showing maps and requiring representation of imprecise features or websites with routing functions needing to handle concepts as walking distance or closeby

Fuzzy regions: adding subregions and the impact on surface and distance calculation

In the concept of fuzzy regions we introduced before, a region was considered to be a fuzzy set of points, each having its own membership grade. While this allows the modelling of regions in which points only partly belong to the region, it has the downside that all the points are considered independently, which is too loose a restriction for some situations. The model is not able to support the fact that some points may be linked together. In this contribution, we propose an extension to the model, so that points can be made related to one another. It will permit the user to, for instance, specify points or even (sub)regions within the fuzzy region that are linked together: they all belong to the region to the same extent at the same time. By letting the user specify such subregions, the accuracy Of the model can be increased: the model can match the real situation better; while at the same time decreasing the fuzziness: if points are known to be related, there is no need to consider them independently. As an example, the use of such a fuzzy region to represent a lake with a variable water level can be considered: as the water level rises, a set of points will become flooded; it is interesting to represent this set of points as a. subset of the region, as these points are somewhat related (the same can be done for different water levels). The impact of this extension to the model on both surface area calculation an distance measurement are considered, and new appropriate definitions are introduced

Topological nature of spinons and holons: Elementary excitations from matrix product states with conserved symmetries

We develop variational matrix product state (MPS) methods with symmetries to determine dispersion relations of one dimensional quantum lattices as a function of momentum and preset quantum number. We test our methods on the XXZ spin chain, the Hubbard model and a non-integrable extended Hubbard model, and determine the excitation spectra with a precision similar to the one of the ground state. The formulation in terms of quantum numbers makes the topological nature of spinons and holons very explicit. In addition, the method also enables an easy and efficient direct calculation of the necessary magnetic field or chemical potential required for a certain ground state magnetization or particle density.Comment: 13 pages, 4 pages appendix, 8 figure

Nonlocal resources in the presence of Superselection Rules

Superselection rules severely alter the possible operations that can be implemented on a distributed quantum system. Whereas the restriction to local operations imposed by a bipartite setting gives rise to the notion of entanglement as a nonlocal resource, the superselection rule associated with particle number conservation leads to a new resource, the \emph{superselection induced variance} of local particle number. We show that, in the case of pure quantum states, one can quantify the nonlocal properties by only two additive measures, and that all states with the same measures can be asymptotically interconverted into each other by local operations and classical communication. Furthermore we discuss how superselection rules affect the concepts of majorization, teleportation and mixed state entanglement.Comment: 4 page

Minimally Entangled Typical Thermal State Algorithms

We discuss a method based on sampling minimally entangled typical thermal states (METTS) that can simulate finite temperature quantum systems with a computational cost comparable to ground state DMRG. Detailed implementations of each step of the method are presented, along with efficient algorithms for working with matrix product states and matrix product operators. We furthermore explore how properties of METTS can reveal characteristic order and excitations of systems and discuss why METTS form an efficient basis for sampling. Finally, we explore the extent to which the average entanglement of a METTS ensemble is minimal.Comment: 18 pages, 14 figure

Fermionic concurrence in the extended Hubbard dimer

In this paper, we introduce and study the fermionic concurrence in a two-site extended Hubbard model. Its behaviors both at the ground state and finite temperatures as function of Coulomb interaction $U$ (on-site) and $V$ (nearest-neighbor) are obtained analytically and numerically. We also investigate the change of the concurrence under a nonuniform field, including local potential and magnetic field, and find that the concurrence can be modulated by these fields.Comment: 5 pages, 7 figure

Cross-linked cationic diblock copolymer worms are superflocculants for micrometer-sized silica particles

A series of linear cationic diblock copolymer nanoparticles are prepared by polymerization-induced self-assembly (PISA) via reversible addition–fragmentation chain transfer (RAFT) aqueous dispersion polymerization of 2-hydroxypropyl methacrylate (HPMA) using a binary mixture of non-ionic and cationic macromolecular RAFT agents, namely poly(ethylene oxide) (PEO113, Mn = 4400 g mol−1; Mw/Mn = 1.08) and poly([2-(methacryloyloxy)ethyl]trimethylammonium chloride) (PQDMA125, Mn = 31 800 g mol−1, Mw/Mn = 1.19). A detailed phase diagram was constructed to determine the maximum amount of PQDMA125 stabilizer block that could be incorporated while still allowing access to a pure worm copolymer morphology. Aqueous electrophoresis studies indicated that zeta potentials of +35 mV could be achieved for such cationic worms over a wide pH range. Core cross-linked worms were prepared via statistical copolymerization of glycidyl methacrylate (GlyMA) with HPMA using a slightly modified PISA formulation, followed by reacting the epoxy groups of the GlyMA residues located within the worm cores with 3-aminopropyl triethoxysilane (APTES), and concomitant hydrolysis/condensation of the pendent silanol groups with the secondary alcohol on the HPMA residues. TEM and DLS studies confirmed that such core cross-linked cationic worms remained colloidally stable when challenged with either excess methanol or a cationic surfactant. These cross-linked cationic worms are shown to be much more effective bridging flocculants for 1.0 μm silica particles at pH 9 than the corresponding linear cationic worms (and also various commercial high molecular weight water-soluble polymers.). Laser diffraction studies indicated silica aggregates of around 25–28 μm diameter when using the former worms but only 3–5 μm diameter when employing the latter worms. Moreover, SEM studies confirmed that the cross-linked worms remained intact after their adsorption onto the silica particles, whereas the much more delicate linear worms underwent fragmentation under the same conditions. Similar results were obtained with 4 μm silica particles

Entanglement and bifurcations in Jahn-Teller models

We compare and contrast the entanglement in the ground state of two Jahn-Teller models. The $E\otimes\beta$ system models the coupling of a two-level electronic system, or qubit, to a single oscillator mode, while the $E\otimes\epsilon$ models the qubit coupled to two independent, degenerate oscillator modes. In the absence of a transverse magnetic field applied to the qubit, both systems exhibit a degenerate ground state. Whereas there always exists a completely separable ground state in the $E\otimes\beta$ system, the ground states of the $E\otimes\epsilon$ model always exhibit entanglement. For the $E\otimes\beta$ case we aim to clarify results from previous work, alluding to a link between the ground state entanglement characteristics and a bifurcation of a fixed point in the classical analogue. In the $E\otimes\epsilon$ case we make use of an ansatz for the ground state. We compare this ansatz to exact numerical calculations and use it to investigate how the entanglement is shared between the three system degrees of freedom.Comment: 11 pages, 9 figures, comments welcome; 2 references adde

Stimulus-responsive non-ionic diblock copolymers: protonation of a tertiary amine end-group induces vesicle-to-worm or vesicle-to-sphere transitions

A well-defined poly(glycerol monomethacrylate) (PGMA) macromolecular chain transfer agent (macroCTA) with a mean degree of polymerisation (DP) of 43 was prepared by reversible addition–fragmentation chain transfer (RAFT) polymerisation using a morpholine-functionalised trithiocarbonate-based chain transfer agent (MPETTC). Chain extension of this macro-CTA by RAFT aqueous dispersion polymerisation of 2-hydroxypropyl methacrylate (HPMA) at pH 7.0–7.5 produced a series of four MPETTC-PGMA43- PHPMAy vesicles (where y = 190, 200, 220 or 230). Protonation of the morpholine end-group increases the hydrophilic character of the PGMA stabiliser block, which leads to a reduction in the packing parameter for the diblock copolymer chains. However, such pH-responsive behaviour critically depends on the value of y. For y = 190 or 200, lowering the solution pH to pH 3 induces a vesicle-to-worm transition at 20 °C according to dynamic light scattering, aqueous electrophoresis, transmission electron microscopy and turbidimetry studies. This order–order transition is suppressed in the presence of added electrolyte, which screens the cationic end-groups. In addition, no change in copolymer morphology was observed on lowering the solution temperature at neutral pH, regardless of the y value. The diblock copolymer nano-objects obtained at pH 3 were also cooled to 4 °C to examine their dual stimulusresponsive behaviour to both pH and temperature triggers. In all four cases, a change in morphology from either worms or vesicles to afford spheres (or spheres plus relatively short worms) was observed. Temperature-dependent oscillatory rheology experiments performed on cationic worms at pH 3 indicated a worm-to-sphere transition on cooling from 20 °C to 4 °C, which leads to reversible degelation. In summary, spheres, worms or vesicles can be obtained for MPETTC-PGMA-PHPMA diblock copolymers on first lowering the solution pH to pH 3, followed by cooling from 20 °C to 4 °C

Violation of area-law scaling for the entanglement entropy in spin 1/2 chains

Entanglement entropy obeys area law scaling for typical physical quantum systems. This may naively be argued to follow from locality of interactions. We show that this is not the case by constructing an explicit simple spin chain Hamiltonian with nearest neighbor interactions that presents an entanglement volume scaling law. This non-translational model is contrived to have couplings that force the accumulation of singlet bonds across the half chain. Our result is complementary to the known relation between non-translational invariant, nearest neighbor interacting Hamiltonians and QMA complete problems.Comment: 9 pages, 4 figure