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A class of left bialgebroids whose underlying algebra A ♯ H is a smash product of a bialgebra H with a braided commutative Yetter–Drinfeld H -algebra A has recently been studied in relation to models of field theories on noncommutative spaces. In Borowiec and Pachoł (2017 J. Phys. A: Math. Theor. 50 055205) a proof has been presented that the bialgebroid A F ♯ H F where HF and AF are the twists of H and A by a Drinfeld 2-cocycle F = ∑ F 1 ⊗ F 2 is isomorphic to the twist of bialgebroid A ♯ H by the bialgebroid 2-cocycle ∑ 1 ♯ F 1 ⊗ 1 ♯ F 2 induced by F. They assume H is quasitriangular, which is reasonable for many physical applications. However the proof and the entire paper take for granted that the coaction and the prebraiding are both given by special formulas involving the R-matrix. There are counterexamples of Yetter–Drinfeld modules over quasitriangular Hopf algebras which are not of this special form. Nevertheless, the main result essentially survives. We present a proof with a general coaction and the correct prebraiding, and even without the assumption of quasitriangularity. [ABSTRACT FROM AUTHOR]

Godrčche, in Comment on 'Fluctuation dominated phase ordering at a mixed order transition' [2021 J. Phys. A: Math. Theor. 54 038001], has commented on our recent paper Fluctuation dominated phase ordering at a mixed order transition (2019 J. Phys. A: Math. Theor. 52 254001). This comment concerns the prefactor of the cusp-like small-argument singularity of the scaled spin-spin correlation function at criticality. We remark that the approach used in our paper is adequate for computing the cusp exponent, which is what is really needed to establish fluctuation-dominated phase ordering. Computing the precise value of the prefactor of the cusp singularity is irrelevant for this purpose, or for the physics behind the relation between the mixed order phase transition and the fluctuation-dominated phase ordering—the understanding of which was the main purpose of our paper. [ABSTRACT FROM AUTHOR]

The authors replies to the comment made by Yang and Zhou (2024 J. Phys. A: Math. Theor.) on his 2015 paper entitled 'Critical points of Potts and O(N) models from eigenvalue identities in periodic Temperley–Lieb algebras' (Jacobsen 2015 J. Phys. A: Math. Theor. 48 454003). [ABSTRACT FROM AUTHOR]

In this paper, we present the construction of all nonstandard integrable systems in magnetic fields whose integrals have leading order structure corresponding to the case (i) of theorem 1 in Marchesiello and Šnobl (2022 J. Phys. A: Math. Theor. 55 145203). We find that the resulting systems can be written as one family with several parameters. For certain limits of these parameters the system belongs to intersections with already known standard systems separating in Cartesian and / or cylindrical coordinates and the number of independent integrals of motion increases, thus the system becomes minimally superintegrable. These results generalize the particular example presented in section 3 of Marchesiello and Šnobl (2022 J. Phys. A: Math. Theor. 55 145203). [ABSTRACT FROM AUTHOR]

It is well-known that any Lennard–Jones type potential energy must have a periodic ground state given by a triangular lattice in dimension 2. In this paper, we describe a computer-assisted method that rigorously shows such global minimality result among 2-dimensional lattices once the exponents of the potential have been fixed. The method is applied to the widely used classical (12 , 6) Lennard–Jones potential, which is the main result of this work. Furthermore, a new bound on the inverse density (i.e. the co-volume) for which the triangular lattice is minimal is derived, improving those found in (Bétermin and Zhang 2015 Commun. Contemp. Math. 17 1450049) and (Bétermin 2016 SIAM J. Math. Anal. 48 3236–3269). The same results are also shown to hold for other exponents as additional examples and a new conjecture implying the global optimality of a triangular lattice for any parameters is stated. [ABSTRACT FROM AUTHOR]

This paper studies the conductance on the universal homology covering space Z of 2D orbifolds in a strong magnetic field, thereby removing the rationality constraint on the magnetic field in earlier works (Avron et al 1994 Phys. Rev. Lett. 73 3255–3257; Mathai and Wilkin 2019 Lett. Math. Phys. 109 2473–2484; Prieto 2006 Commun. Math. Phys. 265 373–396) in the literature. We consider a natural Landau Hamiltonian on Z and study its spectrum which we prove consists of a finite number of low-lying isolated points and calculate the von Neumann degree of the associated holomorphic spectral orbibundles when the magnetic field B is large, and obtain fractional quantum numbers as the conductance. [ABSTRACT FROM AUTHOR]

Let us say that a three-qubit state u 000|000âź© + u 001|001âź© + â‹Ż + u 111|111âź© is real if all its amplitudes u rst are real numbers. We will prove that for every real three-qubit state | Ď• âź©, there exist three angles θ 0, θ 1 and θ 2 such that R y (θ 2) ⊗ R y (θ 1) ⊗ R y (θ 0)| Ď• âź© is a three-qubit of the form λ 1|000âź© + λ 2|011âź© + λ 3|101âź© + λ 4|110âź© + λ 5|111âź© with the λ i real numbers. In contrast with the general case, the case of three-qubits with complex amplitudes, we proved that for three qubit states, the dimension of the real entanglement space (the space obtained by identifying real qubit states with local orthogonal gates, instead of local unitary gates) is 4 and in this paper we find four linearly independent polynomial invariants of degree 4 which are not possible to find for the different Schmidt representations of three qubit states. See (AcĂ-n et al 2000 Phys. Rev. Lett. 85 1560; AcĂ-n et al 2001 J. Phys. A: Math. Gen. 34 6725; Carteret et al 2000 J. Math. Phys. 41 7932; Sudbery 2001 J. Phys. A: Math. Gen. 34 643). [ABSTRACT FROM AUTHOR]

We consider the elliptic Gaudin-type model in an external magnetic field (Skrypnyk T 2005 Phys. Lett. A 334 390–9; Skrypnyk T 2005 Phys. Lett. A 347 266–7; Skrypnyk T 2006 J. Geom. Phys. 57 53–67; Skrypnyk T 2006 J. Math. Phys. 47 ; Skrypnyk T 2007 J. Phys. A 40 1611–23; Skrypnyk T 2019 Nucl. Phys. B 941 225–48) associated with non-skew-symmetric elliptic r -matrix (Skrypnyk T 2005 Phys. Lett. A 334 390–9; Skrypnyk T 2005 Phys. Lett. A 347 266–7; Skrypnyk T 2006 J. Geom. Phys. 57 53–67; Skrypnyk T 2006 J. Math. Phys. 47). Using them we construct a new integrable fermion Hamiltonian of the Richardson type. We use the modified algebraic Bethe ansatz obtained for integrable models with the considered elliptic r -matrix in (Skrypnyk T 2023 Nucl. Phys. B 988 116102) and find the spectrum of the obtained Richardson-type Hamiltonian in terms of solutions of the modified Bethe equations. The obtained results generalize our previous results on Richardson-type models (Skrypnyk T 2022 Nucl. Phys. B 975 115679). [ABSTRACT FROM AUTHOR]

Riemann surfaces are among the simplest and most basic geometric objects. They appear as key players in many branches of physics, mathematics, and other sciences. Despite their widespread significance, how to compute distances between pairs of points on compact Riemann surfaces is surprisingly unknown, unless the surface is a sphere or a torus. This is because on higher-genus surfaces, the distance formula involves an infimum over infinitely many terms, so it cannot be evaluated in practice. Here we derive a computable distance formula for a broad class of Riemann surfaces. The formula reduces the infimum to a minimum over an explicit set consisting of finitely many terms. We also develop a distance computation algorithm, which cannot be expressed as a formula, but which is more computationally efficient on surfaces with high genuses. We illustrate both the formula and the algorithm in application to generalized Bolza surfaces, which are a particular class of highly symmetric compact Riemann surfaces of any genus greater than 1. [ABSTRACT FROM AUTHOR]

We extend the method for constructing symmetry operators of higher order for two-dimensional quantum Hamiltonians by Kalnins et al (2010 J. Phys. A: Math. Theor. 43 265205). This expansion method expresses the integral in a finite power series in terms of lower degree integrals so as to exhibit it as a first-order differential operators. One advantage of this approach is that it does not require the a priori knowledge of the explicit eigenfunctions of the Hamiltonian nor the action of their raising and lowering operators as in their recurrence approach (Kalnins et al 2011 SIGMA 7 031). We obtain insight into the two-dimensional Hamiltonians of radial oscillator type with general second-order differential operators for the angular variable. We then re-examine the Hamiltonian of Tremblay et al (2009 J. Phys. A: Math. Theor. 42 242001) as well as a deformation discovered by Post et al (2011 J. Phys. A: Math. Theor. 44 505201) which possesses reflection operators. We will extend the analysis to spaces of constant curvature. We present explicit formulas for the integrals and the symmetry algebra, the Casimir invariant and oscillator realizations with finite-dimensional irreps which fill a gap in the literature. [ABSTRACT FROM AUTHOR]

In the initial version of the article (Klotz and Maldonado 2021 J. Phys. A: Math. Theor. 54 445201), an upper bound on the ropelength of (p, 2) torus links was found by considering a helix wrapped around a circle. The exact expression for the bound was determined using a small-angle approximation which was unjustified. Here we provide the correct expression, which has the same asymptotic behavior as the original. [ABSTRACT FROM AUTHOR]

We analyse a recent application of homotopy perturbation method to some models for population balances. We show that the only result of the application of this implementation of the perturbation approach is the Taylor series expansion of the solution to the integro-differential equations. [ABSTRACT FROM AUTHOR]

We study the Laurent property, the irreducibility and co-primeness of discrete integrable and non-integrable equations. First we study a discrete integrable equation related to the Somos-4 sequence, and also a non-integrable equation as a comparison. We prove that the conditions of irreducibility and co-primeness hold only in the integrable case. Next, we generalize our previous results on the singularities of the discrete Korteweg–de Vries (dKdV) equation. In our previous paper (Kanki et al 2014 J. Phys. A: Math. Theor.47 065201) we described the singularity confinement test (one of the integrability criteria) using the Laurent property, and the irreducibility, and co-primeness of the terms in the bilinear dKdV equation, in which we only considered simplified boundary conditions. This restriction was needed to obtain simple (monomial) relations between the bilinear form and the nonlinear form of the dKdV equation. In this paper, we prove the co-primeness of the terms in the nonlinear dKdV equation for general initial conditions and boundary conditions, by using the localization of Laurent rings and the interchange of the axes. We assert that co-primeness of the terms can be used as a new integrability criterion, which is a mathematical re-interpretation of the confinement of singularities in the case of discrete equations. [ABSTRACT FROM AUTHOR]

The damped and parametrically driven nonlinear Dirac equation with arbitrary nonlinearity parameter is analyzed, when the external force is periodic in space and given by , both numerically and in a variational approximation using five collective coordinates (time dependent shape parameters of the wave function). Our variational approximation satisfies exactly the low-order moment equations. Because of competition between the spatial period of the external force , and the soliton width ls, which is a function of the nonlinearity as well as the initial frequency of the solitary wave, there is a transition (at fixed ) from trapped to unbound behavior of the soliton, which depends on the parameters r and K of the external force and the nonlinearity parameter . We previously studied this phenomena when (Quintero et al 2019 J. Phys. A: Math. Theor. 52 285201) where we showed that for the soliton oscillates in an effective potential, while for it moves uniformly as a free particle. In this paper we focus on the dependence of the transition from oscillatory to particle behavior and explicitly compare the curves of the transition regime found in the collective coordinate approximation as a function of r and K when at fixed value of the frequency . Since the solitary wave gets narrower for fixed as a function of , we expect and indeed find that the regime where the solitary wave is trapped is extended as we increase . [ABSTRACT FROM AUTHOR]

We complete the derivation of the sausage model NLIE by giving a proof of the crucial relation (3.24) of the original paper based on the analytic properties of Q and . [ABSTRACT FROM AUTHOR]

Numerical studies are presented to assess error estimates for a separable (Hartree) approximation for dynamically evolving composite quantum systems which exhibit distinct scales defined by their mass and frequency ratios. The relevant error estimates were formally described in our previous work Burghardt et al (2021 J. Phys. A: Math. Theor. 54 414002). Specifically, we consider a representative two-dimensional tunneling system where a double well and a harmonic coordinate are cubically coupled. The time-dependent Hartree approximation is compared with a fully correlated solution, for different parameter regimes. The impact of the coupling and the resulting correlations are quantitatively assessed in terms of a time-dependent reaction probability along the tunneling coordinate. We show that the numerical error is correctly predicted on moderate time scales by a theoretically derived error estimate. [ABSTRACT FROM AUTHOR]

In this paper we prove that given a weakly almost periodic measure μ supported inside some model set with closed window W, then the strongly almost periodic component and the null weakly almost periodic component μ0 are both supported inside . As a consequence we prove that given any translation bounded measure ω, supported inside some model set, then each of the pure point diffraction spectrum and the continuous diffraction spectrum is either trivial or has a relatively dense support. [ABSTRACT FROM AUTHOR]

The Witten–Dijkgraaf–Verlinde–Verlinde (or WDVV) equations, as one would expect from an integrable system, has many symmetries, both continuous and discrete. One class—the so-called Legendre transformations—were introduced by Dubrovin. They are a discrete set of symmetries between the stronger concept of a Frobenius manifold, and are generated by certain flat vector fields. In this paper this construction is generalized to the case where the vector field (called here the Legendre field) is non-flat but satisfies a certain set of defining equations. One application of this more general theory is to generate the induced symmetry between almost-dual Frobenius manifolds whose underlying Frobenius manifolds are related by a Legendre transformation. This also provides a map between rational and trigonometric solutions of the WDVV equations. [ABSTRACT FROM AUTHOR]

It has been shown that many networks associated with complex systems are small-world (they have both a large local clustering coefficient and a small diameter) and also scale-free (the degrees are distributed according to a power law). Moreover, these networks are very often hierarchical, as they describe the modularity of the systems that are modeled. Most of the studies for complex networks are based on stochastic methods. However, a deterministic method, with an exact determination of the main relevant parameters of the networks, has proven useful. Indeed, this approach complements and enhances the probabilistic and simulation techniques and, therefore, it provides a better understanding of the modeled systems. In this paper we find the radius, diameter, clustering coefficient and degree distribution of a generic family of deterministic hierarchical small-world scale-free networks that has been considered for modeling real-life complex systems. [ABSTRACT FROM AUTHOR]

We show how the derivation of the Derrida–Lebowitz–Speer–Spohn equation can be prolonged to obtain a new equation, generalizing the models obtained in the paper by these authors. We then investigate its properties from both an analytical and numerical perspective. Specifically, a numerical method is presented to approximate solutions of the prolonged equation. Using this method, we investigate the relationship between the solutions of the prolonged equation and the Tracy–Widom GOE distribution. [ABSTRACT FROM AUTHOR]

State sum constructions, such as Kuperberg’s algorithm, give partition functions of physical systems, like lattice gauge theories, in various dimensions by associating local tensors or weights with different parts of a closed triangulated manifold. Here we extend this construction by including matter fields to build partition functions in both two and three space–time dimensions. The matter fields introduce new weights to the vertices and they correspond to Potts spin configurations described by an -module with an inner product. Performing this construction on a triangulated manifold with a boundary we obtain transfer matrices which are decomposed into a product of local operators acting on vertices, links and plaquettes. The vertex and plaquette operators are similar to the ones appearing in the quantum double models (QDMs) of Kitaev. The link operator couples the gauge and the matter fields, and it reduces to the usual interaction terms in known models such as gauge theory with matter fields. The transfer matrices lead to Hamiltonians that are frustration-free and are exactly solvable. According to the choice of the initial input, that of the gauge group and a matter module, we obtain interesting models which have a new kind of ground state degeneracy that depends on the number of equivalence classes in the matter module under gauge action. Some of the models have confined flux excitations in the bulk which become deconfined at the surface. These edge modes are protected by an energy gap provided by the link operator. These properties also appear in ‘confined Walker–Wang’ models which are 3D models having interesting surface states. Apart from the gauge excitations there are also excitations in the matter sector which are immobile and can be thought of as defects like in the Ising model. We only consider bosonic matter fields in this paper. [ABSTRACT FROM AUTHOR]

We apply machine-learning to the study of dessins d'enfants. Specifically, we investigate a class of dessins which reside at the intersection of the investigations of modular subgroups, Seiberg–Witten (SW) curves and extremal elliptic K3 surfaces. A deep feed-forward neural network with simple structure and standard activation functions without prior knowledge of the underlying mathematics is established and imposed onto the classification of extension degree over the rationals, known to be a difficult problem. The classifications reached 0.92 accuracy with 0.03 standard error relatively quickly. The SW curves for those with rational coefficients are also tabulated. [ABSTRACT FROM AUTHOR]

Fernández (2020) has found a number of serious typos in Kaur et al (2019) that must be corrected. A list of corrections is enumerated below. All of these are typographical mistakes in the transcription of the equations and have no bearing on the results or conclusions of the work. Once the typos are corrected the only point remaining to address in the comment by Fernández (2020) is the connection between the homotopy perturbation method (HPM) and the Taylor series approach. Indeed we have acknowledged this relationship in Kaur et al (2019) (see discussion, section 5) and have cited several references to that effect, including Melzak (1957a,1957b), Lushnikov (1973) and Song and Poland (1992). In Kaur et al (2019) we offer HPM as an alternative to the series expansion method, apply it to initial conditions other than monodisperse, and present a number of new solutions that have not been reported in the previously literature. [ABSTRACT FROM AUTHOR]

Fring et al (2010 J. Phys. A: Math. Theor. 43 345401) have introduced a new set of noncommutative space-time commutation relations in two space dimensions. It had been shown that any fundamental objects introduced in this space-space noncommutativity are string-like. Taking this result into account, we generalize the seminal work of Fring et al to the case that there is also a maximal length from position-dependent noncommutativity and a minimal momentum arising from generalized versions of Heisenberg’s uncertainty relations. The existence of maximal length is related to the presence of an extra, first order term in particle’s length that provides the basic difference of our analysis with theirs. This maximal length breaks up the well known singularity problem of space time. We establish different representations of this noncommutative space and finally we study some basic and interesting quantum mechanical systems in these new variables. [ABSTRACT FROM AUTHOR]

The externally driven nonlinear Dirac (NLD) equation with scalar-scalar self-interaction studied in (2016 J. Phys. A: Math. Theor. 49 065402) is revisited. By using a variational method and an ansatz with five collective coordinates, the dynamics of the NLD solitons is well described. It is shown that this new ansatz possesses certain advantages, namely the canonical momentum agrees with the field momentum, the energy associated to the collective coordinate equations agrees with the energy of the NLD soliton, whereas the ansatz with either three or four collective coordinates does not. Thus the study of the whole phase space of the system is enhanced. It is also shown that this approach is equivalent to the method of moments: the time variation of the charge, the momentum, the energy, and the first moment of the charge. The advantages of the new ansatz are illustrated by means of numerical simulations. [ABSTRACT FROM AUTHOR]

Recent developments have brought the possibility of achieving scalable quantum networks and quantum devices closer. From the computational point of view these emerging technologies become relevant when they are no longer classically simulatable. Hence a pressing challenge is the construction of practical methods to verify the correctness of the outcome produced by universal or non-universal quantum devices. A promising approach that has been extensively explored is the scheme of verification via encryption through blind quantum computation. We present here a new construction that simplifies the required resources for any such verifiable protocol. We obtain an overhead that is linear in the size of the input (computation), while the security parameter remains independent of the size of the computation and can be made exponentially small (with a small extra cost). Furthermore our construction is generic and could be applied to any universal or non-universal scheme with a given underlying graph. [ABSTRACT FROM AUTHOR]

We investigate and compare three distinguished geometric measures of bipartite quantum correlations that have been recently introduced in the literature: the geometric discord, the measurement-induced geometric discord, and the discord of response, each one defined according to three contractive distances on the set of quantum states, namely the trace, Bures, and Hellinger distances. We establish a set of exact algebraic relations and inequalities between the different measures. In particular, we show that the geometric discord and the discord of response based on the Hellinger distance are easy to compute analytically for all quantum states whenever the reference subsystem is a qubit. These two measures thus provide the first instance of discords that are simultaneously fully computable, reliable (since they satisfy all the basic Axioms that must be obeyed by a proper measure of quantum correlations), and operationally viable (in terms of state distinguishability). We apply the general mathematical structure to determine the closest classical-quantum state of a given state and the maximally quantum-correlated states at fixed global state purity according to the different distances, as well as a necessary condition for a channel to be quantumness breaking. [ABSTRACT FROM AUTHOR]

Gauge-invariant twistor variables are found for the massive spinning particle with -extended local worldline supersymmetry, in spacetime dimensions The twistor action is manifestly Lorentz invariant but the anticommuting spin variables appear exactly as in the non-relativistic limit. This allows a simple confirmation that the quantum spinning particle has either spin one or spin zero, and that is quantum inconsistent for [ABSTRACT FROM AUTHOR]

For symbolic dynamical systems we use the Carathéodory construction as described in (Pesin 1997 Dimension Theory in Dynamical Systems, ConTemporary Views and Applications (Chicago: University of Chicago Press)) to introduce the notions of q-topological and q-metric entropies. We describe some basic properties of these entropies and in particular, discuss relations between q-metric entropy and local metric entropy. Both q-topological and q-metric entropies are new invariants respectively under homeomorphisms and metric isomorphisms of dynamical systems. [ABSTRACT FROM AUTHOR]

Recent advancements in the synthesis of anisotropic macromolecules and nanoparticles have spurred an immense interest in theoretical and computational studies of self-assembly. The cornerstone of such studies is the role of shape in self-assembly and in inducing complex order. The problem of identifying different types of order that can emerge in such systems can, however, be challenging. Here, we revisit the problem of quantifying orientational order in systems of building blocks with non-trivial rotational symmetries. We first propose a systematic way of constructing orientational coordinates for such symmetric building blocks. We call the arising tensorial coordinates strong orientational coordinates (SOCs) as they fully and exclusively specify the orientation of a symmetric object. We then use SOCs to describe and quantify local and global orientational order, and spatiotemporal orientational correlations in systems of symmetric building blocks. The SOCs and the orientational order parameters developed in this work are not only useful in performing and analyzing computer simulations of symmetric molecules or particles, but can also be utilized for the efficient storage of rotational information in long trajectories of evolving many-body systems. [ABSTRACT FROM AUTHOR]

We investigate type IIB strings on with mixed Ramond–Ramond and Neveu–Schwarz–Neveu–Schwarz flux. By suitably gauge-fixing the closed string Green–Schwarz action of this theory, we derive the off-shell symmetry algebra and its representations. We use these to determine the non-perturbative worldsheet S matrix of fundamental excitations in the theory. The analysis involves both massive and massless modes in complete generality. The S matrix we find involves a number of phase factors, which in turn satisfy crossing equations that we also determine. We comment on the nature of the heaviest modes of the theory, but leave their identification either as composites or bound-states to a future investigation. [ABSTRACT FROM AUTHOR]

A recently introduced one-dimensional two-particle Bose–Hubbard model with a single impurity (Zhang et al 2012 Phys. Rev. Lett.109 116405) is studied on finite lattices. The model possesses a discrete reflection symmetry and we demonstrate that all eigenstates odd under this symmetry can be obtained with a generalized Bethe ansatz if periodic boundary conditions~(BC) are imposed. Furthermore, we provide numerical evidence that this holds true for open BC as well. The model exhibits backscattering at the impurity site—which usually destroys integrability—yet there exists an integrable subspace. We investigate the non-integrable even sector numerically and find a class of states which have almost the Bethe ansatz form. These weakly diffractive states correspond to a weak violation of the non-local Yang–Baxter relation which is satisfied in the odd sector. We introduce a method based on the Prony algorithm to check whether a numerically obtained wave function has the Bethe form or not, and if so, to extract its parameters. This technique is applicable to a wide variety of other lattice models. [ABSTRACT FROM AUTHOR]

We perform an exact renormalization group analysis of a fermionic system on a three-dimensional lattice, showing the persistence of the Weyl semimetallic phase up the quantum critical point discriminating between the semimetallic and insulating phases. The interaction produces finite modifications of the anisotropic Fermi velocity and of the overlap of the true (dressed) quasiparticle with the free electron wave function. The optical conductivity is the one of Weyl fermions with anisotropic Fermi velocities replacing the light velocity, provided that the frequencies are below a certain energy scale, which is vanishing at the quantum critical point; above such scale, a different behavior is found, dominated by the quadratic corrections to the dispersion relation. [ABSTRACT FROM AUTHOR]

The short- and long-scale behaviour of tangled wave vortices (nodal lines) in random three-dimensional (3D) wave fields is studied via computer experiment. The zero lines are tracked in numerical simulations of periodic superpositions of 3D complex plane waves. The probability distribution of local geometric quantities such as curvature and torsion are compared to previous analytical and new Monte Carlo results from the isotropic Gaussian random wave model. We further examine the scaling and self-similarity of tangled wave vortex lines individually and in the bulk, drawing comparisons with other physical systems of tangled filaments. [ABSTRACT FROM AUTHOR]

We establish an intriguing correspondence between a special set of classical solutions of the modified sinh-Gordon equation (i.e., Hitchinʼs ‘self-duality’ equations) on a punctured Riemann sphere and a set of stationary states in the finite-volume Hilbert space of the integrable 2D quantum field theory introduced by VA Fateev. An application of this correspondence to the problem of non-perturbative quantization of classically integrable nonlinear sigma models is briefly discussed. A detailed account of the results announced in this communication is contained in separate publications (Bazhanov and Lukyanov 2014 arXiv:1310.4390 [hep-th] and Bazhanov et al 2014 J. High Energy Phys. JHEP09(2014)147). [ABSTRACT FROM AUTHOR]

Starting from exact analytical results on singular values and complex eigenvalues of products of independent Gaussian complex random N × N matrices, also called the Ginibre ensemble, we rederive the Lyapunov exponents for an infinite product. We show that for a large number t of product matrices, the distribution of each Lyapunov exponent is normal, and we compute its t-dependent variance as well as corrections in a large-t expansion. Originally Lyapunov exponents are defined for the singular values of the product matrix that represents a linear time evolution. Surprisingly a similar construction for the moduli of the complex eigenvalues yields the very same exponents and normal distributions to leading order. We discuss a general mechanism for 2 × 2 matrices showing why the singular values and the radii of complex eigenvalues collapse onto the same value in the large-t limit. We thereby rederive Newmanʼs triangular law which has a simple interpretation as the radial density of complex eigenvalues in the circular law, and study the commutativity of the two limits and on the global scale and on the local scale. We show as a mathematical byproduct that a particular asymptotic expansion of a Meijer G-function with large index leads to a Gaussian. [ABSTRACT FROM AUTHOR]

We show that the dispersionless DKP hierarchy (the dispersionless limit of the Pfaff lattice) admits a suggestive reformulation through elliptic functions. We also consider one-variable reductions of the dispersionless DKP hierarchy and show that they are described by an elliptic version of the Löwner equation. With a particular choice of the driving function, the latter appears to be closely related to the Painlevé VI equation with a special choice of parameters. [ABSTRACT FROM AUTHOR]

We introduce a new family of quasi-exactly solvable (QES) generalized isotonic oscillators which are based on the Hermite exceptional orthogonal polynomials. We obtain exact closed-form expressions for the energies and wavefunctions as well as the allowed potential parameters for the first two members of the family using the Bethe ansatz method. Numerical calculations of the energies reveal that member potentials have multiple QES eigenstates and the number of states for higher members are parameter dependent. [ABSTRACT FROM AUTHOR]

The asymptotic behaviour of solutions to was investigated by Bender et al (2014 J. Phys. A: Math. Theor. 47 235204). They found, for example, a relation between the initial value and the number of maxima that the solution exhibited. We present an alternative derivation of the asymptotic results that looks at the solutions in the regions and , and confirms the behaviour found previously for larger values of a. This method uses the small amplitude and high frequency of the oscillatory behaviour in the region . [ABSTRACT FROM AUTHOR]