Table of contents

An electrical network with the structure of a random tree is considered: starting from a root vertex, in one iteration each leaf (a vertex with zero or one adjacent edges) of the tree is extended by either a single edge with probability p or two edges with probability 1 − p. With each edge having a resistance equal to 1Ω, the total resistance R_n between the root vertex and a busbar connecting all the vertices at the nth level is considered. A dynamical system is presented which approximates R_n, it is shown that the mean value 〈R_n〉 for this system approaches (1 + p)/(1 − p) as n → ∞, the distribution of R_n at large n is also examined. Additionally, a random sequence construction akin to a random Fibonacci sequence is used to approximate R_n; this sequence is shown to be related to the Legendre polynomials and its mean is shown to converge with |〈R_n〉 − (1 + p)/(1 − p)| ∼ n^−1/2.

DNA-binding proteins recognize their cognate sites on the template DNA more efficiently when the thermally driven flipping of their DNA-binding domains between the fast- and slow-moving conformations is coupled to the search dynamics. We show that there exists an optimum barrier height (∼k_BT ln2) that separates these fast- and slow-moving states of DNA-binding domains, at which the efficiency associated with the thermodynamic coupling of thermally driven flipping and the overall search dynamics is the maximum. Furthermore, the dynamics of DNA-binding domains resembles that of typical downhill folding proteins at their midpoint denaturation temperatures. We further show that the average one-dimensional scanning lengths of slow- and fast-moving states of DNA-binding domains of LacI repressor protein are tuned to minimize the overall search time that is required to locate its cognate sites on DNA.

Key aspects of the cluster distribution in the case of directed, compact percolation near a damp wall are derived as functions of the bulk occupation probability p and the wall occupation probability p_w. The mean length of finite clusters and mean number of contacts with the wall are derived exactly, and we find that both results involve elliptic integrals and further multiple sum functions of two variables. Despite the added complication of these multiple sum functions, our analysis shows that the critical behaviour is similar to the dry wall case where these functions do not appear. We derive the critical amplitudes as a function of p_w.

At the end of his paper (Oprocha 2006 J. Phys. A: Math. Gen.39 14 559–65), Oprocha said that it is difficult and interesting to know the exact value of the principal measure of an annihilation operator $\widehat{a}$ of the unforced quantum harmonic oscillator. Motivated by this, in this paper, we prove that this exact value is equal to 1. Besides, we point out that $\widehat{a}$ exhibits distributional -chaos for any 0 < < 2.

We introduce a new infinite class of superintegrable quantum systems in the plane. Their Hamiltonians involve reflection operators. The associated Schrödinger equations admit the separation of variables in polar coordinates and are exactly solvable. The angular part of the wavefunction is expressed in terms of little −1 Jacobi polynomials. The spectra exhibit 'accidental' degeneracies. The superintegrability of the model is proved using the recurrence relation approach. The (higher order) constants of motion are constructed and the structure equations of the symmetry algebra are obtained.

We use symbol correspondence and quantum normal form theory to develop a more general method for finding uniform asymptotic approximations. We then apply this method to derive a result we announced in an earlier paper, namely the uniform approximation of the 6j-symbol in terms of the rotation matrices. The derivation is based on the Stratonovich–Weyl symbol correspondence between matrix operators and functions on a spherical phase space. The resulting approximation depends on a canonical, or area-preserving, map between two pairs of intersecting level sets on the spherical phase space.

In the paper of Sheftel and Malykh (2009 J. Phys. A: Math. Theor.42 395202) on the classification of second-order PDEs with four independent variables that possess partner symmetries, an asymmetric heavenly equation appears as one of the canonical equations admitting partner symmetries. Here, for the asymmetric heavenly equation formulated in a two-component form, we present the Lax pair of Olver–Ibragimov–Shabat type and obtain its multi-Hamiltonian structure. Therefore, by Magri's theorem, it is a completely integrable bi-Hamiltonian system in four dimensions.

When is a q-series modular? This is an interesting open question in mathematics that has deep connections to conformal field theory. In this paper, we define a particular r-fold q-hypergeometric series f_{A, B, C}, with data given by a matrix A, a vector B and a scalar C, all rational, and ask when f_{A, B, C} is modular. In the past much work has been done to predict which values of A give rise to modular f_{A, B, C}; however, there is no straightforward method for calculating corresponding values of B. We approach this problem from the point of view of conformal field theory, by considering (2n + 3, 2)-minimal models, and coset models of the form $\widehat{su}(2)_k /\widehat{u}(1)$. By calculating the characters of these models and comparing them to the functions f_{A, B, C}, we succeed in computing appropriate B-values in many cases.

Soliton perturbation theory is used to obtain analytical solutions describing solitary wave tails or shelves, due to numerical discretization error, for soliton solutions of the nonlinear Schrödinger equation. Two important implicit numerical schemes for the nonlinear Schrödinger equation, with second-order temporal and spatial discretization errors, are considered. These are the Crank–Nicolson scheme and a scheme, due to Taha [1], based on the inverse scattering transform. The first-order correction for the solitary wave tail, or shelf, is in integral form and an explicit expression is found for large time. The shelf decays slowly, at a rate of $t^{-{1\over 2}}$, which is characteristic of the nonlinear Schrödinger equation. Singularity theory, usually used for combustion problems, is applied to the explicit large-time expression for the solitary wave tail. Analytical results are then obtained, such as the parameter regions in which qualitatively different types of solitary wave tails occur, the location of zeros and the location and amplitude of peaks. It is found that three different types of tail occur for the Crank–Nicolson and Taha schemes and that the Taha scheme exhibits some unusual symmetry properties, as the tails for left and right moving solitary waves are different. Optimal choices of the discretization parameters for the numerical schemes are also found, which minimize the amplitude of the solitary wave tail. The analytical solutions are compared with numerical simulations, and an excellent comparison is found.

In this study, Green's function of the two-dimensional radiative transfer equation is derived for an infinitely extended anisotropically scattering medium, which is illuminated by a unidirectional source distribution. In the steady-state domain, the final results, which are based on eigenvalues and eigenvectors, are given analytically apart from the eigenvalues. For the time-dependent case an additional numerical inverse Fourier transform is required. The obtained solutions were successfully validated with another exact analytical solution in the time domain for isotropically scattering and with the Monte Carlo method for anisotropically scattering media.

We consider the full Dicke spin-boson model composed by a single bosonic mode and an ensemble of N identical two-level atoms with different couplings for the resonant and anti-resonant interaction terms, and incorporate a dipole–dipole interaction between the atoms. Assuming that the system is in thermal equilibrium with a reservoir at temperature β⁻¹, we compute the free energy in the thermodynamic limit N → ∞ in the saddle-point approximation to the path integral and determine the critical temperature for the super-radiant phase transition. In the zero temperature limit, we recover the critical coupling of the quantum phase transition, presented in the literature.

By means of simple quantum-mechanical models we show that under certain conditions the main assumptions of the connected moments expansion (CMX) are no longer valid. In particular, we consider two-level systems: the harmonic oscillator and the pure quartic oscillator. Although derived from such simple models, we think that the results of this investigation may be of utility in future applications of the approach to realistic problems. We show that a straightforward analysis of the CMX exponential parameters may provide a clear indication of the success of the approach.

It is pointed out that the 60 complex rays in four dimensions associated with a system of two qubits yield over 10⁹ critical parity proofs of the Kochen–Specker theorem. The geometrical properties of the rays are described, an overview of the parity proofs contained in them is given and examples of some of the proofs are exhibited.

Some properties of the quantum discord based on the geometric measure advanced by Dakic et al (2010 Phys. Rev. Lett.105 190502) are discussed here by recourse to a systematic survey of the two-qubit state-space, with emphasis on Werner and MEM states. We explore the dependence of quantum discord on the degree of mixedness of the bipartite states, and also its connection with non-locality as measured by the maximum violation of a Bell inequality within the CHSH scenario. Some considerations regarding the XY model are also made.

We study a binary mixture of Bose–Einstein condensates, confined in a generic potential, in the Thomas–Fermi approximation. We search for the zero-temperature ground state of the system, both in the case of fixed numbers of particles and fixed chemical potentials. For generic potentials, we analyze the transition from mixed to separated ground-state configurations as the inter-species interaction increases. We derive a simple formula that enables one to determine the location of the domain walls. Finally, we find criteria for the energetic stability of separated configurations, depending on the number and the position of the domain walls separating the two species.

We show that some recently derived sum rules for the hydrogen atom are incorrect because the authors did not take into account the continuous part of the spectrum in the sum over intermediate states.

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