Table of contents

We propose a new correlator in one-dimensional quantum spin chains, the s-emptiness formation probability (s-EFP). This is a generalization of the emptiness formation probability (EFP), which is the probability that the first n spins of the chain are all aligned downwards. In the s-EFP we let the spins in question be separated by s sites. The usual EFP corresponds to the special case when s = 1. Taking s > 1 allows us to quantify non-local correlations. We express the s-EFP for the anisotropic XY model in a transverse magnetic field, a system with both critical and non-critical regimes, in terms of a Toeplitz determinant. For the isotropic XY model we find that the magnetic field induces an interesting length scale.

The concept of self-Fourier functions, functions that equal their Fourier transform, is considered using differential operators. The goal is to analyse these functions and determine their properties without evaluating any Fourier, or any other type, transform integrals. Certain known results are generalized and the theory is extended to include integral and fractional-differential operators. Moreover, it is shown that the problem of defining these functions, in its original formulation, is equivalent to this method and in doing so, the concept of a Fourier eigenoperator is introduced. We also describe a procedure for applying this approach to general cyclic transforms.

In this letter, the generalized nonlinear Schrödinger (GNLS) equation, phase-locked a source κ e^{i[χ(ξ)−ωt]}: , is investigated. Firstly, we reduce this equation to a second-order non-homogeneous nonlinear ordinary differential equation via a plane wave transformation and some constraint conditions. Then we use some fractional transformations to study exact solutions in obtaining the GNLS equation. As a consequence, many types of exact solutions are deduced such as envelope rational solutions, envelope periodic wave solutions, envelope solitary wave solutions and envelope doubly periodic solutions. Similarly, the corresponding exact solutions can also be obtained for the Hirota-type GNLS equation with a source and their combined equation.

We propose a general classification of nonequilibrium steady states in terms of their stationary probability distribution and the associated probability currents. The stationary probabilities can be represented graph theoretically as directed labelled trees; closing a single loop in such a graph leads to a representation of probability currents. This classification allows us to identify all choices of transition rates, based on a master equation, which generate the same nonequilibrium steady state. We explore the implications of this freedom, e.g., for entropy production.

We review the progress made in dynamic bulk critical behaviour in equilibrium in the last 25 years since the review of Halperin and Hohenberg. We unify the presentation of the theoretical background by restricting ourselves to the field-theoretic renormalization group method. The main results obtained in the different universality classes are presented. This contains the critical dynamics near the gas–liquid transition in pure fluids (model H), the plait point and consolute point in mixtures (model H'), the superfluid transition in ⁴He (model F) and ⁴He–³He mixtures (model F'), the Curie point (model J) and Neel point (model G) in Heisenberg magnets and the superconducting transition. In comparison with experimental results, it became clear that in most cases one has to consider apart from the universal asymptotic critical behaviour also the non-universal effective behaviour. Either because it turned out to be inevitable due to a small dynamical transient exponent inhibiting the system to reach the asymptotics (e.g., at the superfluid transition) or because one is interested in the region further away from the phase transition like in pure fluids and mixtures at their gas–liquid or demixing transition. The calculation of the critical dynamics is adequate in most cases only in two-loop order. We review these results and present the solution to unreasonable features found for some models. Thus, we consider model C where relaxational and diffusive dynamics are coupled and the scaling properties and the limit to a purely relaxational model (model A) have not been understood. In general for models where the order parameter couples to other conserved densities time scale ratios between the kinetic coefficients of the order parameter and the conserved densities play an important role. Their fixed-point values and the approach to the fixed point are changed considerably in two-loop order compared to their values in one-loop order. These considerations are relevant for the explanation of the dynamical critical shape functions of systems such as superfluid helium (model F) and the isotropic antiferromagnet (model G). As far as possible, the comparison of results obtained by the renormalization group theory with numerical simulations has been made.

We show that generalized extreme value statistics—the statistics of the kth largest value among a large set of random variables—can be mapped onto a problem of random sums. This allows us to identify classes of non-identical and (generally) correlated random variables with a sum distributed according to one of the three (k-dependent) asymptotic distributions of extreme value statistics, namely the Gumbel, Fréchet and Weibull distributions. These classes, as well as the limit distributions, are naturally extended to real values of k, thus providing a clear interpretation to the onset of Gumbel distributions with non-integer index k in the statistics of global observables. This is one of the very few known generalizations of the central limit theorem to non-independent random variables. Finally, in the context of a simple physical model, we relate the index k to the ratio of the correlation length to the system size, which remains finite in strongly correlated systems.

Amplitude weighting can improve the accuracy of frequency measurements in signals corrupted by multiplicative speckle noise. When the speckle field constitutes a circular complex Gaussian process, the optimal function of amplitude weighting is provided by the field intensity, corresponding to the intensity-weighted phase derivative statistic. In this paper, we investigate the phase derivative and intensity-weighted phase derivative returned from a two-dimensional random walk, which constitutes a generic scattering model capable of producing both Gaussian and non-Gaussian fluctuations. Analytical results are developed for the correlation properties of the intensity-weighted phase derivative, as well as limiting probability densities of the scattered field. Numerical simulation is used to generate further probability densities and determine optimal weighting criteria from non-Gaussian fields. The results are relevant to frequency retrieval in radiation scattered from random media.

We study the integrable hierarchy underlying topological Landau–Ginzburg models of D-type proposed by Takasaki. Since this integrable hierarchy contains the dBKP hierarchy as a sub-hierarchy, we refer it to the extended dBKP (EdBKP) hierarchy. We give a dressing formulation to the EdBKP hierarchy and investigate additional symmetries associated with the solution space of the hierarchy. We obtain hodograph solutions of its finite-dimensional reductions via Riemann–Hilbert problem (twistor construction) and derive Bäcklund transformations of the (2 + 1)-dimensional dBKP equation from additional flows. Finally, the modified partner of the dBKP hierarchy is also established through a Miura transformation.

We focus on the transformation matrices between the standard Young–Yamanouchi basis of an irreducible representation for the symmetric group S_n and the split basis adapted to the direct product subgroups . We introduce the concept of subduction graph and show that it conveniently describes the combinatorial structure of the equation system arisen from the linear equation method. Thus we can outline an improved algorithm to solve the subduction problem in symmetric groups by a graph searching process. We conclude by observing that the general matrix form for multiplicity separations, resulting from orthonormalization, can be expressed in terms of Sylvester matrices relative to a suitable inner product in the multiplicity space.

The solutions of the Drinfeld equation corresponding to the full set of different carrier subalgebras are explicitly constructed. The obtained Hopf structures are studied. It is demonstrated that the presented twist deformations can be considered as limits of the corresponding quantum analogues (q-twists) defined for the q-quantized algebras.

We study local conservation laws with non-vanishing conserved densities and corresponding boundary conditions for the potential Kadomtsev–Petviashvili equation. We analyse an infinite symmetry group of the equation, and generate a finite number of conserved densities corresponding to infinite symmetries through appropriate boundary conditions.

We consider a system of two coupled (2 + 1)-dimensional nonlinear Schrödinger equations, describing two-component disc-shaped Bose–Einstein condensates. We present three different asymptotic reductions of this system. In particular, we derive the Mel'nikov system, the Yajima–Oikawa system as well as the Davey–Stewartson system (the latter is found as a special case of the Djordjevic–Redekopp system). Conditions for integrability of the reduced systems, their soliton solutions and the asymptotic relevance of such solutions to the original system are also discussed. Numerical results pertaining to the reduction to the Davey–Stewartson system are found to be in good agreement with the analytical predictions.

We study the possible generalized boundary conditions and the corresponding solutions for the quantum mechanical oscillator model on Kähler conifold. We perform it by self-adjoint extension of the initial domain of the effective radial Hamiltonian. Remarkable effect of this generalized boundary condition is that at certain boundary condition the orbital angular momentum degeneracy is restored! We also recover the known spectrum in our formulation, which of course corresponds to some other boundary condition.

We consider spontaneous emission of two two-level atoms interacting with vacuum fluctuations. We study the process of disentanglement in this system and show the possibility of changing disentanglement time by local unitary operations.

Solving the s-wave Dirac equation for the Eckart potential with spin and pseudospin symmetry by using the supersymmetric quantum mechanics approach and function analysis method, we obtain the exact energy equation and corresponding two-component spinor wavefunctions. The restriction conditions of existing bound states are analysed.

We develop a variational method to obtain accurate bounds for the eigenenergies of H = −Δ + V in arbitrary dimensions N > 1, where V(r) is the nonpolynomial oscillator potential . The variational bounds are compared with results previously obtained in the literature. An infinite set of exact solutions is also obtained and used as a source of comparison eigenvalues.

We present a set of necessary conditions for the existence of a biorthonormal basis composed of eigenvectors of non-Hermitian operators. As an illustration, we examine these conditions in the case of normal operators. We also provide a generalization of the conditions which is applicable to non-diagonalizable operators by considering not only eigenvectors but also all root vectors.

In this paper, we investigate the synthesis of ternary reversible circuits in the absence of ancilla bits. We demonstrated that 2-qudit ternary Swap, NOT and 1-controlled-NOT gates are universal for realization of arbitrary ternary n-qudit reversible circuits without ancilla qudits, and all even ternary n-qudit reversible circuits can be constructed by ternary NOT and ternary 1-controlled-NOT gates without ancilla qudits. The realization approach is constructive. This result is significantly different from the binary case.

Stokes parameters form a Minkowskian 4-vector under various optical transformations. As a consequence, the resulting two-by-two density matrix constitutes a representation of the Lorentz group. The associated Poincaré sphere is a geometric representation of the Lorentz group. Since the Lorentz group preserves the determinant of the density matrix, it cannot accommodate the decoherence process through the decaying off-diagonal elements of the density matrix, which yields to an increase in the value of the determinant. It is noted that the O(3, 2) de Sitter group contains two Lorentz subgroups. The change in the determinant in one Lorentz group can be compensated by the other. It is thus possible to describe the decoherence process as a symmetry transformation in the O(3, 2) space. It is shown also that these two coupled Lorentz groups can serve as a concrete example of Feynman's rest of the universe.