Table of contents

We show that aeroplane boarding can be asymptotically modelled by two-dimensional Lorentzian geometry. Boarding time is given by the maximal proper time among curves in the model. Discrepancies between the model and simulation results are closely related to random matrix theory. The models can be used to explain why some commonly practised airline boarding policies are ineffective and even detrimental.

Two neurons coupled by unreliable synapses are modelled by leaky integrate-and-fire neurons and stochastic on–off synapses. The dynamics is mapped to an iterated function system. Numerical calculations yield a multifractal distribution of interspike intervals. The covering, information and correlation dimensions are calculated as a function of synaptic strength and transmission probability.

Recently semiclassical approximations have been successfully applied to study the effect of a superconducting lead on the density of states and conductance in ballistic billiards. However, the summation over classical trajectories involved in such theories was carried out using the intuitive picture of Andreev reflection rather than semiclassical reasoning. We propose a method to calculate the semiclassical sums which allows us to go beyond the diagonal approximation in these problems. In particular, we address the question of whether the off-diagonal corrections could explain the gap in the density of states of a chaotic Andreev billiard.

We study a two-dimensional electron system in the presence of both Rashba and Dresselhaus spin–orbit interactions in a perpendicular magnetic field. Defining two suitable boson operators and using the unitary transformations we are able to obtain the exact Landau levels in the range of all the parameters. When the strengths of the Rashba and Dresselhaus spin–orbit interactions are equal, a new analytical solution for the vanishing Zeeman energy is found, where the orbital and spin wavefunctions of the electron are separated. It is also shown that in this case the Zeeman and spin–orbit splittings are independent of the Landau level index n. Due to the Zeeman energy, new crossing between the eigenstates |n, k, s = 1, σ⟩ and |n + 1, k, s' = −1, σ'⟩ is produced at a certain magnetic field for larger Rashba spin–orbit coupling. This degeneracy leads to a resonant spin Hall conductance if it happens at the Fermi level.

A three-dimensional quasiperiodic lattice, with overlapping unit cells and periodic in one direction, is constructed using grid and projection methods pioneered by de Bruijn. Each unit cell consists of 26 points, of which 22 are the vertices of a convex polytope , and 4 are interior points also shared with other neighbouring unit cells. Using Kronecker's theorem the frequencies of all possible types of overlapping are found.

We consider a gas of classical particles in having q distinct colours, interacting via a mean-field Potts potential and subject to an external field; a colour-independent molecular interaction of a mean-field type is also admitted. In contrast to the usual lattice Potts model, the Potts gas exhibits the specific volume v as a parameter, in addition to colour-ordering. The v-dependence of the Potts gas is studied in detail, and the complete phase diagram is derived. It turns out that, for q ⩾ 3, the transition to colour-ordering implies a jump of density. For q = 2, this transition is continuous but may become discontinuous under the influence of a suitable molecular interaction.

The present work is an endeavour to determine analytically features of the stationary measure of a non-integrable zero-range process, and to investigate the possible existence of phase transitions for such a nonequilibrium model. The rates defining the model do not satisfy the constraints necessary for the stationary measure to be a product measure. Even in the absence of a drive, detailed balance with respect to this measure is violated. Analytical and numerical investigations on the complete graph demonstrate the existence of a first-order phase transition between a fluid phase and a condensed phase, where a single site has macroscopic occupation. The transition is sudden from an imbalanced fluid where both species have densities larger than the critical density, to a critical neutral fluid and an imbalanced condensate.

In this paper, we study an exact solution of the asymmetric simple exclusion process on a periodic lattice of finite sites with two typical updates, i.e., random and parallel. Then, we find that the explicit formulae for the partition function and the average velocity are expressed by the Gauss hypergeometric function. In order to obtain these results, we effectively exploit the recursion formula for the partition function for the zero-range process. The zero-range process corresponds to the asymmetric simple exclusion process if one chooses the relevant hop rates of particles, and the recursion gives the partition function, in principle, for any finite system size. Moreover, we reveal the asymptotic behaviour of the average velocity in the thermodynamic limit, expanding the formula as a series in system size.

A veritable zoo of different knots is seen in the ensemble of looped polymer chains, whether created computationally or observed in vitro. At short loop lengths, the spectrum of knots is dominated by the trivial knot (unknot). The fractional abundance of this topological state in the ensemble of all conformations of the loop of N segments follows a decaying exponential form, ∼exp(−N/N₀), where N₀ marks the crossover from a mostly unknotted (i.e., topologically simple) to a mostly knotted (i.e., topologically complex) ensemble. In the present work, we use computational simulation to look closer into the variation of N₀ for a variety of polymer models. Among models examined, N₀ is smallest (about 240) for the model with all segments of the same length, it is somewhat larger (305) for Gaussian-distributed segments, and can be very large (up to many thousands) when the segment length distribution has a fat power-law tail.

Using the Wang–Landau Monte Carlo method, we study the antiferromagnetic (AF) three-state Potts model with a staggered polarization field on the square lattice. We obtain two phase transitions: one belongs to the ferromagnetic three-state Potts universality class and the other to the Ising universality class. The phase diagram obtained is quantitatively consistent with the transfer matrix calculation. The Ising transition in the large nearest-neighbour interaction limit has been made clear by the detailed analysis of the energy density of states.

We derive the steady state properties of a general directed 'sandpile' model in one dimension. Using a central limit theorem for dependent random variables we find the precise conditions for the model to belong to the universality class of the totally asymmetric Oslo model, thereby identifying a large universality class of directed sandpiles. We map the avalanche size to the area under a Brownian curve with an absorbing boundary at the origin, motivating us to solve this Brownian curve problem. Thus, we are able to determine the moment generating function for the avalanche-size probability in this universality class, explicitly calculating amplitudes of the leading order terms.

In this paper, we numerically study the probability P_ac of the occurrence of car accidents in the Nagel–Schreckenberg (NS) model with a speed limit zone. Numerical results show that the probability for car accidents to occur P_ac is determined by the maximum speed v'_max of the speed limit zone, but is independent of the length L_v of the speed limit zone in the deterministic NS model. However in the nondeterministic NS model, the probability of the occurrence of car accidents P_ac is determined not only by the maximum speed v'_max, but also the length L_v. The probability P_ac increases accordingly with the increase of the maximum speed of the speed limit zone, but decreases with the increase of the length of the speed limit zone, in the low-density region. However in the case of v'_max = 1, the probability P_ac increases with the increase of the length in the low-density region, but decreases in the interval between the low-density and high-density regions. The speed limit zone also causes an inhomogeneous distribution of car accidents over the whole road. Theoretical analyses give an agreement with numerical results in the nondeterministic NS model with v'_max = 1 and v_max = 5.

This paper presents a computer-assisted verification of hyperchaos in the well-known Saito hysteresis chaos generator (SHCG) by virtue of topological horseshoe theory. By means of interval analysis we find two disjoint compact subsets in a carefully chosen 3D cross section that can guarantee the existence of a topological horseshoe for the corresponding third-return Poincaré map. Numerical studies show that the Poincaré map expands in two directions. It justifiably indicates that there exists hyperchaos in the SHCG.

We analyse the structure of the exact, dark and bright soliton solutions of the driven nonlinear Schrödinger equation. A wide class of solutions, phase locked with the source, is identified for distinct parameter ranges. These contain periodic as well as localized solutions, which can be singular implying extreme increase in intensity. Conditions for obtaining non-propagating solutions are also found. A special case, where the scale of the soliton emerges as a free parameter, is obtained. We also study the highly restrictive structure of the localized solutions, when the phase and amplitude get coupled. Numerical solutions are obtained for this case, which reveals presence of periodic solutions. Stability analysis is also carried out through the Crank–Nicolson method.

We present a self-dual Drinfeld double structure underlying the A_n series of simple Lie algebras. Such double is constructed through a central extension t_n of gl(n), and is obtained by pairing two disjoint solvable subalgebras coming from positive and negative roots. The Cartan–Weyl basis of gl(n) is shown to be completely determined by the compatibility conditions in the double. A natural Lie bialgebra structure on gl(n) is obtained that offers a new perspective for the construction of its quantum deformations.

We consider a general framework for integrable hierarchies in Lax form and derive certain universal equations from which 'functional representations' of particular hierarchies (such as KP, discrete KP, mKP, AKNS), i.e. formulations in terms of functional equations, are systematically and quite easily obtained. The formalism genuinely applies to hierarchies where the dependent variables live in a noncommutative (typically matrix) algebra. The obtained functional representations can be understood as 'noncommutative' analogues of 'Fay identities' for the KP hierarchy.

We study some geometrical aspects of two-dimensional orientable surfaces arising from the study of sigma models. To this aim we employ an identification of with the Lie algebra by means of which we construct a generalized Weierstrass formula for immersion of such surfaces. The structural elements of the surface like its moving frame, the Gauss–Weingarten and the Gauss–Codazzi–Ricci equations are expressed in terms of the solution of the model defining it. Further, the first and second fundamental forms, the Gaussian curvature, the mean curvature vector, the Willmore functional and the topological charge of surfaces are expressed in terms of this solution. We present detailed implementation of these results for surfaces immersed in and Lie algebras.

An antilinear operator in complex vector spaces is an important operator in the study of modern quantum theory, quantum and semiclassical optics, quantum electronics and quantum chemistry. Consimilarity of complex matrices arises as a result of studying an antilinear operator referred to different bases in complex vector spaces, and the theory of consimilarity of complex matrices plays an important role in the study of quantum theory. This paper, by means of a real representation of a complex matrix, studies the relation between consimilarity and similarity of complex matrices, sets up an algebraic bridge between consimilarity and similarity and turns the theory of consimilarity into that of ordinary similarity. This paper also gives some applications of consimilarity of complex matrices.

We establish differential conditions to be fulfilled by the slope functions α(x, y, z), β(x, y, z) corresponding to a given two-parametric family of orbits (given in the form f(x, y, z) = c₁, g(x, y, z) = c₂) traced in space by a material point so that these families can result in the presence of an axisymmetric potential V = V(x² + y², z). All possible cases for the 'given' orbits are studied and some pairs of families and potentials are found.

We present stationary expressions for the on-shell time delay operator and for its trace in quantum-mechanical potential scattering. The trace of this operator is shown to represent the time delay for a monoenergetic and well-collimated beam of initial states. For exponentially decaying potentials the total scattering cross section and the trace of the on-shell time delay operator have analytic extensions to complex energies, with poles at most at the usual resonance poles (poles of the analytically continued S-matrix) in the lower half plane and at their mirror images with respect to the real axis. At real energies close to such a pole both the scattering cross section and the time delay are approximately of Breit–Wigner type, the sign of the time delay being related to that of the virial. These results corroborate the idea that a resonance should show up through a peak in the scattering cross section and a large (though positive) time delay.

The Bohm–Gadella theory, sometimes referred to as the time asymmetric quantum theory of scattering and decay, is based on the Hardy axiom. The Hardy axiom asserts that the solutions of the Lippmann–Schwinger equation are functionals over spaces of Hardy functions. The preparation–registration arrow of time provides the physical justification for the Hardy axiom. In this paper, it is shown that the Hardy axiom is incorrect, because the solutions of the Lippmann–Schwinger equation do not act on spaces of Hardy functions. It is also shown that the derivation of the preparation–registration arrow of time is flawed. Thus, Hardy functions neither appear when we solve the Lippmann–Schwinger equation nor should they appear. It is also shown that the Bohm–Gadella theory does not rest on the same physical principles as quantum mechanics, and that it does not solve any problem that quantum mechanics cannot solve. The Bohm–Gadella theory must therefore be abandoned.

We provide time-evolution operators, gauge transformations and a perturbative treatment for non-Hermitian Hamiltonian systems, which are explicitly time dependent. We determine various new equivalence pairs for Hermitian and non-Hermitian Hamiltonians, which are therefore pseudo-Hermitian and in addition in some cases also invariant under PT-symmetry. In particular, for the harmonic oscillator perturbed by a cubic non-Hermitian term, we evaluate explicitly various transition amplitudes, for the situation when these systems are exposed to a monochromatic linearly polarized electric field.

We give a simple spectral condition in terms of the ordered eigenvalues of the state of a bipartite quantum system which is sufficient for separability.

Supersymmetrical intertwining relations of second order in the derivatives are investigated for the case of supercharges with deformed hyperbolic metric g_ik = diag(1, − a²). Several classes of particular solutions of these relations are found. The corresponding Hamiltonians do not allow the conventional separation of variables, but they commute with symmetry operators of fourth order in momenta. For some of these models the specific SUSY procedure of separation of variables is applied.

We study how imperfect quantum gates affect the quantum random-walk search algorithm. We find that systematic errors in phase inversions result in the reduction of the maximum probability of the marked state and lower the algorithm efficiency with an increasing degree of inaccuracy. The size of the database should be limited due to the inevitable errors. Finally, we compare the phase noise caused by such errors in the random-walk search algorithm with that in the Grover search algorithm.

We analyse quantum games with correlated noise through a generalized quantization scheme. Four different combinations on the basis of entanglement of initial quantum state and the measurement basis are analysed. It is shown that the quantum player only enjoys an advantage over the classical player when both the initial quantum state and the measurement basis are in entangled form. Furthermore, it is shown that for maximum correlation the effects of decoherence diminish and it behaves as a noiseless game.

The scattering properties of a three-parameter family of point dipole-like interactions constructed from a sequence of barrier-well rectangles are studied in the zero-range limit. Besides the real (unrenormalized) δ'-interaction, the derivative of Dirac's delta function, a whole family of point dipole interactions with a renormalized coupling constant are analysed. Depending on the parameter values, all these interactions being self-adjoint extensions of the one-dimensional Schrödinger operator are shown to be divided into four types: (i) interactions will full transparency, (ii) non-transparent interactions, (iii) partially transparent interactions acting effectively as a δ-interaction and (iv) interactions with partial transparency at discrete resonant values of the coupling constant.

Based on mirror symmetry, we discuss geometric engineering of N = 1 ADE quiver models from F-theory compactifications on elliptic K3 surfaces fibred over certain four-dimensional base spaces. The latter are constructed as intersecting 4-cycles according to ADE Dynkin diagrams, thereby mimicking the construction of Calabi–Yau threefolds used in geometric engineering in type II superstring theory. Matter is incorporated by considering D7-branes wrapping these 4-cycles. Using a geometric procedure referred to as folding, we discuss how the corresponding physics can be converted into a scenario with D5-branes wrapping 2-cycles of ALE spaces.

The use of master actions to prove duality at quantum level becomes cumbersome if one of the dual fields interacts nonlinearly with other fields. This is the case of the theory considered here consisting of U(1) scalar fields coupled to a self-dual field through a linear and a quadratic term in the self-dual field. Integrating perturbatively over the scalar fields and deriving effective actions for the self-dual and the gauge field we are able to consistently neglect awkward extra terms generated via master action and establish quantum duality up to cubic terms in the coupling constant. The duality holds for the partition function and some correlation functions. The absence of ghosts imposes restrictions on the coupling with the scalar fields.

All indecomposable finite-dimensional representations of the homogeneous Galilei group which when restricted to the rotation subgroup are decomposed to spin-0, -1/2 and -1 representations are constructed and classified. These representations are also obtained via contractions of the corresponding representations of the Lorentz group. Finally, the obtained representations are used to derive a general Pauli anomalous interaction term as well as to deduce wave equations which describe Darwin and spin–orbit couplings of a Galilei particle interacting with an external electric field.

The hallmark of a good book of problems is that it allows you to become acquainted with an unfamiliar topic quickly and efficiently. The Quantum Mechanics Solver fits this description admirably. The book contains 27 problems based mainly on recent experimental developments, including neutrino oscillations, tests of Bell's inequality, Bose--Einstein condensates, and laser cooling and trapping of atoms, to name a few.

Unlike many collections, in which problems are designed around a particular mathematical method, here each problem is devoted to a small group of phenomena or experiments. Most problems contain experimental data from the literature, and readers are asked to estimate parameters from the data, or compare theory to experiment, or both. Standard techniques (e.g., degenerate perturbation theory, addition of angular momentum, asymptotics of special functions) are introduced only as they are needed. The style is closer to a non-specialist seminar rather than an undergraduate lecture. The physical models are kept simple; the emphasis is on cultivating conceptual and qualitative understanding (although in many of the problems, the simple models fit the data quite well). Some less familiar theoretical techniques are introduced, e.g. a variational method for lower (not upper) bounds on ground-state energies for many-body systems with two-body interactions, which is then used to derive a surprisingly accurate relation between baryon and meson masses.

The exposition is succinct but clear; the solutions can be read as worked examples if you don't want to do the problems yourself. Many problems have additional discussion on limitations and extensions of the theory, or further applications outside physics (e.g., the accuracy of GPS positioning in connection with atomic clocks; proton and ion tumor therapies in connection with the Bethe--Bloch formula for charged particles in solids).

The problems use mainly non-relativistic quantum mechanics and are organised into three sections: Elementary Particles, Nuclei and Atoms; Quantum Entanglement and Measurement; and Complex Systems. The coverage is not comprehensive; there is little on scattering theory, for example, and some areas of recent interest, such as topological aspects of quantum mechanics and semiclassics, are not included. The problems are based on examination questions given at the École Polytechnique in the last 15 years. The book is accessible to undergraduates, but working physicists should find it a delight.

This book is a new volume of a series designed to introduce the curious reader to anything from ancient Egypt and Indian philosophy to conceptual art and cosmology. Very handy in pocket size, Chaos promises an introduction to fundamental concepts of nonlinear science by using mathematics that is `no more complicated than X=2.'

Anyone who ever tried to give a popular science account of research knows that this is a more challenging task than writing an ordinary research article. Lenny Smith brilliantly succeeds to explain in words, in pictures and by using intuitive models the essence of mathematical dynamical systems theory and time series analysis as it applies to the modern world. In a more technical part he introduces the basic terms of nonlinear theory by means of simple mappings. He masterly embeds this analysis into the social, historical and cultural context by using numerous examples, from poems and paintings over chess and rabbits to Olbers' paradox, card games and `phynance'.

Fundamental problems of the modelling of nonlinear systems like the weather, sun spots or golf balls falling through an array of nails are discussed from the point of view of mathematics, physics and statistics by touching upon philosophical issues. At variance with Laplace's demon, Smith's 21st century demon makes `real world' observations only with limited precision. This poses a severe problem to predictions derived from complex chaotic models, where small variations of initial conditions typically yield totally different outcomes. As Smith argues, this difficulty has direct implications on decision-making in everyday modern life. However, it also asks for an inherently probabilistic theory, which somewhat reminds us of what we are used to in the microworld.

There is little to criticise in this nice little book except that some figures are of poor quality thus not really reflecting the beauty of fractals and other wonderful objects in this field. I feel that occasionally the book is also getting a bit too intricate for the complete layman, and experts may not agree on all details of the more conceptual discussions.

Altogether I thoroughly enjoyed reading this book. It was a happy companion while travelling and a nice bedtime literature. It is furthermore an excellent reminder of the `big picture' underlying nonlinear science as it applies to the real world. I will gladly recommend this book as background literature for students in my introductory course on dynamical systems. However, the book will be of interest to anyone who is looking for a very short account on fundamental problems and principles in modern nonlinear science.