The relation between motion in −1/ r and r 2 potentials, known since Newton, can be demonstrated by the substitution r → r 2 in the classical/quantum radial equations of the Kepler/Hydrogen problems versus the harmonic oscillator. This suggests a duality-type relationship between these systems. However, when both radial and angular components of these systems are included the possibility of a true

We consider several models of state dependent delay differential equations (SDDEs), in which the delay is affected by a small parameter. This is a very singular perturbation since the nature of the equation changes. Under some conditions, we construct formal power series, which solve the SDDEs order by order. These series are quasi-periodic functions of time. This is very similar to the Lindstedt procedure

Sarkar and Kumar recently conjectured (2019 J. Phys. A: Math. Theor. 52 295203) that for a bipartite system of Hilbert dimension mn , the mean values of quantum purity and von Neumann entropy of a subsystem of dimension m ⩽ n over the Bures–Hall measure are given by ##IMG## [http://ej.iop.org/images/1751-8121/53/23/235203/aab8d07ieqn1.gif] {$\frac{2n\left(2n\,+\,m\right)\,-\,{m}^{2}\,+\,1}{2n\left(2mn\

The action principle by Low (1958 Proc. R. Soc. Lond. A 248 282–7) for the classic Vlasov–Maxwell system contains a mix of Eulerian and Lagrangian variables. This renders the Noether analysis of reparametrization symmetries inconvenient, especially since the well-known energy- and momentum-conservation laws for the system are expressed in terms of Eulerian variables only. While an Euler–Poincaré formulation

We discuss modifications in the integral representation of the Riemann zeta-function that lead to generalizations of the Riemann functional equation that preserves the symmetry s → (1 − s ) in the critical strip. By modifying one integral representation of the zeta-function with a cut-off that does exhibit the symmetry x ↦ 1/ x , we obtain a generalized functional equation involving Bessel functions

We show the Godbillon–Vey invariant arises as a ‘restricted Casimir’ invariant for three-dimensional ideal fluids associated to a foliation. We compare to a finite-dimensional system, the rattleback, where analogous phenomena occur.

In this paper, we analyse the phase space structure of the roaming dynamics in a 2 degree of freedom potential energy surface consisting of two identical planar Morse potentials separated by a distance. This potential energy surface was previously studied in Carpenter B K et al (2018 Regul. Chaotic Dyn. 23 60–79), and it has two potential wells surrounded by an unbounded flat region containing no critical

The equilibrium properties of an open harmonic oscillator are considered in three steps: first the creation and destruction operators are generalized for open dynamics and the creation operator is used to construct coherent states. The second step consists of the introduction of the Heisenberg representation where the dynamical decoherence is identified. Finally it is pointed out that the quantum fluctuations

Machine learning algorithms relying on deep neural networks recently allowed a great leap forward in artificial intelligence. Despite the popularity of their applications, the efficiency of these algorithms remains largely unexplained from a theoretical point of view. The mathematical description of learning problems involves very large collections of interacting random variables, difficult to handle

Biological molecular motors are special enzymes that support biological processes such as intracellular transport, vesicle locomotion, RNA translation and many more. Experimental works suggest that the motor proteins interact among each other and moreover they experience a push by other motors during the intracellular transport. To incorporate these dynamics, we consider a variant of open one-dimensional

We recently introduced a class of ##IMG## [http://ej.iop.org/images/1751-8121/53/23/235201/aab8b36ieqn2.gif] {${\mathbb{Z}}_{N}$} graded discrete Lax pairs and studied the associated discrete integrable systems (lattice equations). We discuss differential–difference equations which then we interpret as symmetries of the discrete systems. In particular, we present nonlocal symmetries which are associated

For complex scalar waves, a convenient measure of the local oscillations and (‘faster than Fourier’) superoscillations is the phase gradient vector: the local wavevector, or weak value of the momentum operator. This vanishes for standing waves, described by real functions ψ ( r ); for such waves, an alternative descriptor of oscillations is the local weak value of the square of one of the momentum

We consider intertwining relations of the triangular q -Onsager algebra, and obtain generic triangular boundary K -operators in terms of the Borel subalgebras of U q ( sl 2 ). These K -operators solve the reflection equation.

We study the intrinsic effects of dimensional reduction on the transport equation of a perfectly two-dimensional Landau–Fermi liquid. By employing the orthogonality condition on the 2D analog of the Fourier–Legendre expansion, we find that the equilibrium and non-equilibrium properties of the fermionic system differ from its three-dimensional counterpart, with the latter changing drastically. Specifically

This review describes a link between Lax operators, embedded surfaces and thermodynamic Bethe ansatz equations for integrable quantum field theories. This surprising connection between classical and quantum models is undoubtedly one of the most striking discoveries that emerged from the off-critical generalisation of the ODE/IM correspondence, which initially involved only conformal invariant quantum

Recently Mertens and Moore (2019 Phys . Rev . Lett . 123 230605) showed that site percolation ‘is odd’. By this they mean that on an M × N square lattice the number of distinct site configurations that allow for vertical percolation is odd. We report here an alternative proof, based on recursive use of geometric symmetry, for both free and periodic boundary conditions.

The position x ( t ) of a particle diffusing in a one-dimensional uncorrelated and time dependent random medium is simply Gaussian distributed in the typical direction, i.e. along the ray x = v 0 t , where v 0 is the average drift. However, it has been found that it exhibits at large time sample to sample fluctuations characteristic of the Kardar–Parisi–Zhang (KPZ) universality class when observed

We study the null space degeneracy of open quantum systems with multiple non-abelian, strong symmetries. By decomposing the Hilbert space representation of these symmetries into an irreducible representation involving the direct sum of multiple, commuting, invariant subspaces we derive a tight lower bound for the stationary state degeneracy. We apply these results within the context of open quantum

The CSL model predicts a progressive breakdown of the quantum superposition principle, with a noise randomly driving the state of the system towards a localized one, thus accounting for the emergence of a classical world within a quantum framework. In the original model the noise is supposed to be white, but since white noises do not exist in nature, it becomes relevant to identify some of its spectral

Given a probability vector x with its components sorted in non-increasing order, we consider the closed ball ##IMG## [http://ej.iop.org/images/1751-8121/53/21/215305/aab8674ieqn1.gif] {${\mathcal{B}}_{{\epsilon}}^{p}\left(x\right)$} with p ⩾ 1 formed by the probability vectors whose ℓ p -norm distance to the center x is less than or equal to a radius ϵ . Here, we provide an order-theoretic characterization

The quantum Hall effect is studied in a spherical geometry using the Dirac operator for non-interacting fermions in a background magnetic field, which is supplied by a Wu–Yang magnetic monopole at the center of the sphere. Wave functions are cross-section of a non-trivial U (1) bundle, the zero point energy then vanishes and no perturbations can lower the energy. The Atiyah–Singer index theorem constrains

In this paper, we address the problem of how to represent a classical data distribution in a quantum system. The proposed method is to learn the quantum Hamiltonian, that is such that its ground state approximates the given classical distribution. We review previous work on the quantum Boltzmann machine (QBM) (Kieferová M and Nathan W 2017 Phys. Rev. A 96 062327, Amin M H et al 2018 Phys. Rev. X 8

We compute semi-analytic and numerical estimates for the largest Lyapunov exponent in a many-particle system with long-range interactions, extending previous results for the Hamiltonian mean field model with a cosine potential. Our results evidence a critical exponent associated to a power law decay of the largest Lyapunov exponent close to second-order phase transitions, close to the same value as

Coherent subspaces spanned by a finite number of coherent states are introduced, in a quantum system with Hilbert space that has odd prime dimension d . The set of all coherent subspaces is partitioned into equivalence classes, with d 2 subspaces in each class. The corresponding coherent projectors within an equivalence class, have the ‘closure under displacements property’ and also resolve the identity

We consider a quench in a free-fermion chain by joining two homogeneous half-chains via a defect. The time evolution of the entanglement negativity is studied between adjacent segments surrounding the defect. In case of equal initial fillings, the negativity grows logarithmically in time and essentially equals one-half of the Rényi mutual information with index α = 1/2 in the limit of large segments

Hydrodynamics of the non-relativistic compressible fluid in the curved spacetime is derived using the generalized framework of the stochastic variational method (SVM) for continuum medium. The fluid-stress tensor of the resultant equation becomes asymmetric for the exchange of the indices, different from the standard Euclidean one. Its incompressible limit suggests that the viscous term should be represented

The Whitham approach is a well-studied method to describe non-linear integrable systems. Although approximate in nature, its results may predict rather accurately the time evolution of such systems in many situations given initial conditions. A similarly powerful approach has recently emerged that is applicable to quantum integrable systems, namely the generalized hydrodynamics approach. This paper

The Mittag–Leffler (ML) relaxation function, E α (− t α ) (0 < α ⩽ 1), describes multiscale relaxation processes with a broad range of relaxation rates, where α = 1 corresponds to exponential relaxation. For 0 < α < 1 it decays asymptotically ∼ t − α and is thus asymptotically self-similar, i.e. form invariant under a scale transform t → μt . In the language of asymptotic analysis, such functions are

Two models are first presented, of a one-dimensional discrete-time quantum walk (DTQW) with temporal noise on the internal degree of freedom (i.e., the coin): (i) a model with both a coin-flip and a phase-flip channel, and (ii) a model with random coin unitaries. It is then shown that both these models admit a common limit in the spacetime continuum, namely, a Lindblad equation with Dirac-fermion Hamiltonian

This is the story of a simple observation made by Peter (and independently by Haim Harari) more than half a century ago and whose unexpectedly rich consequences have kept bugging my mind, on and off, till these days.

We show that the unresolved examples of Christol’s conjecture ##IMG## [http://ej.iop.org/images/1751-8121/53/20/205201/aab82dcieqn3.gif] {${\;}_{3}{F}_{2}\left(\left[2/9,5/9,8/9\right],\left[2/3,1\right],x\right)$} and ##IMG## [http://ej.iop.org/images/1751-8121/53/20/205201/aab82dcieqn4.gif] {${\;}_{3}{F}_{2}\left(\left[1/9,4/9,7/9\right],\left[1/3,1\right],x\right)$} , are indeed diagonals of rational

We study bulk-boundary correspondences and related surface phenomenastabilized by the second Chern number in three-dimensional insulators driven in adiabatic cycles. Magnetic fields and disorder effects are incorporated in our analysis using operator algebraic methods. We use the connecting maps between the K -theories of bulk and boundary algebras as engines for the bulk-boundary correspondences.

Remembering the foundational contributions of Peter Freund to supergravity, and especially to the problems of dimensional compactification, reduction is considered with a non-compact space transverse to the lower dimensional theory. The known problem of a continuum of Kaluza–Klein states is avoided here by the occurrence of a mass gap between a single normalizable zero-eigenvalue transverse wavefunction

Projected least squares is an intuitive and numerically cheap technique for quantum state tomography: compute the least-squares estimator and project it onto the space of states. The main result of this paper equips this point estimator with rigorous, non-asymptotic convergence guarantees expressed in terms of the trace distance. The estimator’s sample complexity is comparable to the strongest convergence

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We discuss the attempts made at the end of the sixties and at the beginning of the seventies in order to get a consistent N -point amplitude involving the lightest hadrons, such as the pion, that then brought to string theory containing massless gluons and gravitons but not massless pions. This gives us the opportunity to remember the period in which I first met Peter Freund and also to summarise the

Hydrodynamic type systems in Riemann invariants arise in a whole range of applications in fluid dynamics, Whitham averaging procedure, differential geometry and the theory of Frobenius manifolds. In this paper we discuss parabolic (Jordan block) analogues of diagonalisable systems. Our main observation is that integrable quasilinear systems of Jordan block type are parametrised by solutions of the

We consider a classical and superadiabatic version of an iterative quantum adiabatic algorithm to solve combinatorial optimization problems. This algorithm is deterministic because it is based on purely classical dynamics, that is, it does not rely on any stochastic approach to mimic quantum dynamics. Moreover, we use the exact shortcut to adiabaticity for stationary states of classical spin systems

Recently, machine learning techniques have become popular for analysing physical systems and solving problems occurring in quantum computing. In this paper we focus on using such techniques for finding the sequence of physical operations implementing the given quantum logical operation. In this context we analyse the flexibility of the data representation and compare the applicability of two machine

We show that a coalescence equation exhibits a variety of critical behaviors, depending on the initial condition. This equation was introduced a few years ago to understand a toy model studied by Derrida and Retaux to mimic the depinning transition in presence of disorder. It was shown recently that this toy model exhibits the same critical behaviors as the equation studied in the present work. Here

We exploit the gauge equivalence between the Hirota equation and the extended continuous Heisenberg equation to investigate how nonlocality properties of one system are inherited by the other. We provide closed generic expressions for nonlocal multi-soliton solutions for both systems. By demonstrating that a specific auto-gauge transformation for the extended continuous Heisenberg equation becomes

The anomalous quantum Hall conductivity in the 2 + 1 D topological insulators in the absence of interactions may be expressed as the topological invariant composed of the two-point Green function. For the noninteracting system this expression is the alternative way to represent the TKNN invariant. It is widely believed that in the presence of interactions the Hall conductivity is given by the same

We address the problem of compressing density operators defined on a finite dimensional Hilbert space which assumes a tensor product decomposition. In particular, we look for an efficient procedure for learning the most likely density operator, according to ‘Jaynes’ principle, given a chosen set of partial information obtained from the unknown quantum system we wish to describe. For complexity reasons

We show that D = 4 Minkowski space is an emergent concept related to a class of operators in extended Hilbert space with no positive-definite scalar product. We start with the idea of position-like and momentum-like operators (Plewa 2019 J. Phys. A: Math. Theor. 52 375401), introduced discussing a connection between quantum entanglement and geometry predicted by ER = EPR conjecture. We examine eigenequations

This article presents a finite dimensional unbreakable ##IMG## [http://ej.iop.org/images/1751-8121/53/19/195701/aab7f68ieqn3.gif] {$\mathcal{PT}$} -symmetric waveguide system with linear and nonlinear coupling. In traditional ##IMG## [http://ej.iop.org/images/1751-8121/53/19/195701/aab7f68ieqn4.gif] {$\mathcal{PT}$} -symmetric systems, a balance (or a symmetric state) among the waveguides with loss

We propose a variation of the Guiol–Machado–Schinazi (GMS) model of evolution of species. In our version, as in the GMS model, at each birth, the new species in the system is labeled with a random fitness, but in our variation, to each extinction event is associated a random threshold and all species with fitness below the threshold are removed from the system. We present necessary and suficient criteria

In this topical review we consider stochastic processes under resetting, which have attracted a lot of attention in recent years. We begin with the simple example of a diffusive particle whose position is reset randomly in time with a constant rate r , which corresponds to Poissonian resetting, to some fixed point (e.g. its initial position). This simple system already exhibits the main features of

We study a general ansatz for an odd supersymmetric version of the Kronecker elliptic function, which satisfies the genus one Fay identity. The obtained result is used for construction of the odd supersymmetric analogue for the classical and quantum elliptic R -matrices. They are shown to satisfy the classical Yang–Baxter equation and the associative Yang–Baxter equation. The quantum Yang–Baxter equation

The Dirac oscillator in a homogeneous magnetic field exhibits a chirality phase transition at a particular (critical) value of the magnetic field. Recently, this system has also been shown to be exactly solvable in the context of noncommutative quantum mechanics featuring the interesting phenomenon of re-entrant phase transitions. In this work we provide a detailed study of the thermodynamics of such

Using stochastic methods, general formulas for average kinetic and potential energies for anharmonic, undamped (frictionless), classical oscillators are derived. It is demonstrated that for potentials of | x | ν , ( ν > 0) type energies are equipartitioned for the harmonic potential only. For potential wells weaker than parabolic potential energy dominates, while for potentials stronger than parabolic

As a contribution requested by the editors of a memorial volume for Peter G O Freund (1936–2018) we recall the lively particle theory group at the Enrico Fermi Institute of the University of Chicago in the late sixties, of which Peter was a memorable member. We also discuss a period some twenty years later when our and Peter’s research overlapped on the topic of p -Adic strings.

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We discuss a class of models that generalize the two-state Landau–Zener Hamiltonian to both the multistate and multitime evolution. It is already known that the corresponding quantum mechanical evolution can be understood in great detail. Here, we present an approach to classify such solvable models, namely, to identify all their independent families for a given number N of interacting states and prove

We consider a special type of restricted Boltzmann machine (RBM), namely a Gaussian-spherical RBM where the visible units have Gaussian priors while the vector of hidden variables is constrained to stay on an ##IMG## [http://ej.iop.org/images/1751-8121/53/18/184002/aab79f3ieqn1.gif] {${\mathbb{L}}_{2}$} sphere. The spherical constraint having the advantage to admit exact asymptotic treatments, various

Many interesting problems in fields ranging from telecommunications to computational biology can be formalized in terms of large underdetermined systems of linear equations with additional constraints or regularizers. One of the most studied, the compressed sensing problem, consists in finding the solution with the smallest number of non-zero components of a given system of linear equations ##IMG##

For one-dimensional random Schrödinger operators, the integrated density of states is known to be given in terms of the (averaged) rotation number of the Prüfer phase dynamics. This paper develops a controlled perturbation theory for the rotation number around an energy at which all the transfer matrices commute and are hyperbolic. Such a hyperbolic critical energy appears in random hopping models

In this paper, we take up an old thread of development concerning the characterization of supersymmetric theories without any use of anticommuting variables that goes back to one of the authors’ very early work (Nicolai H 1980 Nucl. Phys. B 176 419). Our special focus here will be on the formulation of supersymmetric Yang–Mills theories, extending previous results beyond D = 4 dimensions. This perspective

This paper is concerned with the study of a nonholonomic system with parametric excitation, the Suslov problem with variable gyrostatic momentum. A detailed analysis is made of the problem of the existence of regimes with unbounded growth of energy (an analog of Fermi’s acceleration). The existence of trajectories is shown for which the angular velocity of the rigid body increases indefinitely and

We analyse the evolution of superoscillations in a relativistic wavepacket. A simple superoscillating wavepacket is set up and is allowed to evolve freely according to both the Klein–Gordon equation and the Dirac equation. The superoscillations evolve anisotropically and decay after a time. Both the lifetime and anisotropy can be understood in terms of the interaction of contributions to the wavepacket

We analyse the formation and the dynamics of quantum turbulence in a two-dimensional Bose–Einstein condensate with a Josephson junction barrier modeled using the Gross–Pitaevskii equation. We show that a sufficiently high initial superfluid density imbalance leads to randomisation of the dynamics and generation of turbulence, namely, the formation of a quasi-1D dispersive shock consisting of a train