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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]

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 1989 Mason and Newman (Commun. Math. Phys. 121 659–68) proved that there is a 1-1-correspondence between self-dual metrics satisfying Einstein vacuum equation (in complex case or in neutral signature) and pairs of commuting parameter depending vector fields X 1 (λ) , X 2 (λ) which are divergence free with respect to some volume form. Earlier Plebański (1975 J. Math. Phys. 16 2395–402) showed instances of such vector fields depending of one function of four variables satisfying the so-called I or II Plebański heavenly PDEs. Other PDEs leading to Mason–Newman vector fields are also known in the literature: Park (1992 Int. J. Mod. Phys. A 7 1415), Husain (1994 Phys. Rev. Lett. 72 800), Grant (1993 Phys. Rev. D 48 2606–12), Schief (Phys. Lett. A 223 55–62). In this paper we discuss these matters in the context of the web theory, i.e. theory of collections of foliations on a manifold, understood from the point of view of Nijenhuis operators. In particular we show how to apply this theory for constructing new 'heavenly' PDEs based on different normal forms of Nijenhuis operators in 4D, which are integrable similarly to their predecessors. Relation with the Hirota dispersionless systems of PDEs and the corresponding Veronese webs, which was recently observed by Konopelchenko–Schief–Szereszewski, is established in all the cases. We also discuss some higher dimensional generalizations of the 'heavenly' PDEs and the existence of related vacuum Einstein metrics in 4D-case. [ABSTRACT FROM AUTHOR]

In the paper, we are concerned with the nonlinear Cauchy problem on the Vlasov–Poisson–Landau/Boltzmann system around global Maxwellians in a torus or a finite channel. The main goal is to establish the global existence and large time behaviour of small amplitude solutions for a class of low regularity initial data. The molecular interaction type is restricted to the case of hard potentials for two classical collision operators because of the effect of the self-consistent forces. The result extends the one by Duan–Liu–Sakamoto-Strain (Duan et al 2021 Commun. Pure Appl. Math. 74 932–1020) for the pure Landau/Boltzmann equation to the case of the VPL and VPB systems. [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]

In this paper, we consider a Levenbergâ€"Marquardt method with general regularization terms that are uniformly convex on bounded sets to solve the ill-posed inverse problems in Banach spaces, where the forward mapping might not Gâteaux differentiable and the image space is unnecessarily reflexive. The method therefore extends the one proposed by Jin and Yang in (2016 Numer. Math. 133 655â€"684) for smooth inverse problem setting with globally uniformly convex regularization terms. We prove a novel convergence analysis of the proposed method under some standing assumptions, in particular, the generalized tangential cone condition and a compactness assumption. All these assumptions are fulfilled when investigating the identification of the heat source for semilinear elliptic boundary-value problems with a Robin boundary condition, a heat source acting on the boundary, and a possibly non-smooth nonlinearity. Therein, the Clarke subdifferential of the non-smooth nonlinearity is employed to construct the family of bounded operators that is a replacement for the non-existing Gâteaux derivative of the forward mapping. The efficiency of the proposed method is illustrated with a numerical example. [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 investigate existence and nonexistence of global in time nonnegative solutions to the semilinear heat equation, with a reaction term of the type e μ t u p (μ ∈ R , p > 1) , posed on cones of the hyperbolic space. Under a certain assumption on µ and p, related to the bottom of the spectrum of − Δ in H n , we prove that any solution blows up in finite time, for any nontrivial nonnegative initial datum. Instead, if the parameters µ and p satisfy the opposite condition we have: (a) blow-up when the initial datum is large enough, (b) existence of global solutions when the initial datum is small enough. Hence our conditions on the parameters µ and p are optimal. We see that blow-up and global existence do not depend on the amplitude of the cone. This is very different from what happens in the Euclidean setting (Bandle and Levine 1989 Trans. Am. Math. Soc. 316 595–622), and it is essentially due to a specific geometric feature of H n . [ABSTRACT FROM AUTHOR]

This qualitative, questionnaire and interview-based study explores how pre-service physics teachers think about the role that mathematics plays in the teaching and learning of physics at university and school level and whether these views change during their pre-service teacher education. Many of the pre-service teachers were aware of the complex relationship between these two subjects at university level, noting that success in mathematics can often mask a lack of conceptual understanding in physics and that there can be a disconnect between the physics and mathematical aspects of undergraduate courses. At school level, many stressed the importance of a focus on conceptual understanding and that technical competence in mathematics lessons does not always transfer to physics lessons. Almost all the pre-service physics teachers changed their views during the year, often in response to their classroom experiences. As they became more attuned to the difficulties students faced with respect to the mathematical challenges involved in learning physics, many took a more pragmatic position that balanced the role of mathematics in physics with acceptance that they must respond to student needs. We suggest that these changing views can be framed in terms of two re-orientations. A disciplinary re-orientation where the role that mathematics plays in order to be successful in physics is reassessed, and a pedagogical re-orientation that attends to pragmatic, teaching considerations. We recommend that direct attention to the role of mathematics in school physics should be an integral part of pre-service physics teacher education in order to encourage these re-orientations. [ABSTRACT FROM AUTHOR]

We prove some qualitative properties for singular solutions to a class of strongly coupled system involving a Grossâ€"Pitaevskii-type nonlinearity. Our main theorems are vectorial fourth order counterparts of the classical results due to Serrin (1964 Acta Math. 111 247â€"252), Lions (1980 J. Differ. Equ. 38 441â€"450), Aviles (1987 Commun. Math. Phys. 108 177â€"192), and Gidas and Spruck (1981 Commun. Pure Appl. Math. 34 525â€"598). On the technical level, we use the moving sphere method to classify the limit blow-up solutions to our system. Besides, applying asymptotic analysis, we show that these solutions are indeed the local models near the isolated singularity. We also introduce a new fourth order nonautonomous Pohozaev functional, whose monotonicity properties yield improvement for the asymptotics results due to Soranzo (1997 Potential Anal. 6 57â€"85). [ABSTRACT FROM AUTHOR]

In Jayanti and Trivisa (2022 J. Math. Fluid Mech. 24 46), the authors proved the existence of local-in-time weak solutions to a model of superfluidity. The system of governing equations was derived in Pitaevskii (1959 Sov. Phys. JETP 8 282â€"287) and couples the nonlinear Schrödinger equation and the Navierâ€"Stokes equations. In this article, we prove a weakâ€"strong type uniqueness theorem for these weak solutions. Only some of their regularity properties are used, allowing room for improved existence theorems in the future, with compatible uniqueness results. [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]