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Motivated by the consideration of integrable systems in three spatial dimensions in Euclidean space with integrals quadratic in the momenta we classify three-dimensional Abelian subalgebras of quadratic elements in the universal enveloping algebra of the Euclidean algebra under the assumption that the Casimir invariant p â†' â‹... l â†' vanishes in the relevant representation. We show by means of an explicit example that in the presence of magnetic field, i.e. terms linear in the momenta in the Hamiltonian, this classification allows for pairs of commuting integrals whose leading order terms cannot be written in the famous classical form of Makarov et al [17]. We consider limits simplifying the structure of the magnetic field in this example and corresponding reductions of integrals, demonstrating that singularities in the integrals may arise, forcing structural changes of the leading order terms. [ABSTRACT FROM AUTHOR]

In Alase et al (2022 J. Phys. A: Math. Theor. 55 244003), Alase et al wrote that 'the constraint of quasi-Hermiticity on observables' is not 'sufficient to extend the standard quantum theory' because 'such a system is equivalent to a standard quantum system.' Three addenda elucidating the current state of the art are found necessary. The first one concerns the project: in the related literature the original 'aim of extending standard quantum theory' has already been abandoned shortly after its formulation. The second comment concerns the method, viz., the study in 'the framework of general probabilistic theories' (GPT). It is noticed that a few other, mathematically consistent GPT-like theories are available. The authors do not mention, in particular, the progress achieved, under the quasi-Hermiticity constraint, in the approach using the effect algebras. We add that this approach already found its advanced realistic applications in the quasi-Hermitian models using the unbounded operators of observables acting in the infinite-dimensional Hilbert spaces. Thirdly, the 'intriguing open question' about 'what possible constraints, if any, could lead to such a meaningful extension' (in the future) is given an immediate tentative answer: the possibility is advocated that the desirable constraint could really be just the quasi-Hermiticity of the observables, provided only that one has in mind its recently developed non-stationary version. [ABSTRACT FROM AUTHOR]

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]

Superintegrable systems in two-dimensional (2D) Darboux spaces were classified and it was found that there exist 12 distinct classes of superintegrable systems with quadratic integrals of motion (and quadratic symmetry algebras generated by the integrals) in the Darboux spaces. In this paper, we obtain exact solutions via purely algebraic means for the energies of all the 12 existing classes of superintegrable systems in four different 2D Darboux spaces. This is achieved by constructing the deformed oscillator realization and finite-dimensional irreducible representation of the underlying quadratic symmetry algebra generated by quadratic integrals respectively for each of the 12 superintegrable systems. We also introduce generic cubic and quintic algebras, generated respectively by linear and quadratic integrals and linear and cubic integrals, and obtain their Casimir operators and deformed oscillator realizations. As examples of applications, we present three classes of new superintegrable systems with cubic symmetry algebras in 2D Darboux spaces. [ABSTRACT FROM AUTHOR]

In a previous paper, we showed how to use the techniques of the group of loops to formulate the loop approach to gravity proposed by Mandelstam in the 1960's. Those techniques allow to overcome some of the difficulties that had been encountered in the earlier treatment. In this approach, gravity is formulated entirely in terms of Dirac observables without constraints, opening attractive new possibilities for quantization. In this paper we discuss the Poisson algebra of the resulting Dirac observables, associated with the intrinsic components of the Riemann tensor. This provides an explicit realization of the non-local algebra of observables for gravity that several authors have conjectured. [ABSTRACT FROM AUTHOR]

A Z 2 × Z 2 -graded Lie superalgebra g is a Z 2 × Z 2 -graded algebra with a bracket [ [ ⋅ , ⋅ ] ] that satisfies certain graded versions of the symmetry and Jacobi identity. In particular, despite the common terminology, g is not a Lie superalgebra. We construct the most general orthosymplectic Z 2 × Z 2 -graded Lie superalgebra o s p (2 m 1 + 1 , 2 m 2 | 2 n 1 , 2 n 2) in terms of defining matrices. A special case of this algebra appeared already in work of Tolstoy in 2014. Our construction is based on the notion of graded supertranspose for a Z 2 × Z 2 -graded matrix. Since the orthosymplectic Lie superalgebra o s p (2 m + 1 | 2 n) is closely related to the definition of parabosons, parafermions and mixed parastatistics, we investigate here the new parastatistics relations following from o s p (2 m 1 + 1 , 2 m 2 | 2 n 1 , 2 n 2) . Some special cases are of particular interest, even when one is dealing with parabosons only. [ABSTRACT FROM AUTHOR]

Yang–Baxter integrable vertex models with a generic Z 2 -staggering can be expressed in terms of composite R -matrices given in terms of the elementary R -matrices. Similarly, integrable open boundary conditions can be constructed through generalized reflection algebras based on these objects and their representations in terms of composite boundary matrices K ± . We show that only two types of staggering yield a local Hamiltonian with integrable open boundary conditions in this approach. The staggering in the underlying model allows for a second hierarchy of commuting integrals of motion (in addition to the one including the Hamiltonian obtained from the usual transfer matrix), starting with the so-called quasi momentum operator. In this paper, we show that this quasi momentum operator can be obtained together with the Hamiltonian for both periodic and open models in a unified way from enlarged Yang–Baxter or reflection algebras in the composite picture. For the special case of the staggered six-vertex model, this allows constructing an integrable spectral flow between the two local cases. [ABSTRACT FROM AUTHOR]

The work is devoted to the study of quantum integrable systems associated with quantum loop algebras. The recently obtained equation for the zero temperature inhomogeneous reduced density operator is analyzed. It is demonstrated that any solution of the corresponding qKZ equation generates a solution to this equation. [ABSTRACT FROM AUTHOR]

In the present paper we study the representations of the Jacobi algebra. More concretely, we define, analogously to the case of semi-simple Lie algebras, the Verma modules over the Jacobi algebra G 2 . We study their reducibility and give explicit construction of the reducible Verma modules exhibiting the corresponding singular vectors. Using this information we give a complete classification of the reducible Verma modules. More than this we exhibit their interrelation of embeddings between these modules. These embeddings are illustrated by diagrams of the embedding patterns so that each reducible Verma module appears in one such diagram. [ABSTRACT FROM AUTHOR]

Construction of superintegrable systems based on Lie algebras have been introduced over the years. However, these approaches depend on explicit realisations, for instance as a differential operators, of the underlying Lie algebra. This is also the case for the construction of their related symmetry algebra which take usually the form of a finitely generated quadratic algebra. These algebras often display structure constants which depend on the central elements and in particular on the Hamiltonian. In this paper, we develop a new approach reexamining the quadratically superintegrable system on the two-sphere for which a symmetry algebra is known to be the Racah algebra R(3). Such a model is related to the 59 two dimensional quadratically superintegrable systems on conformally flat spaces via contractions and limits. We demonstrate that using further polynomials of degree 2, 3 and 4 in the enveloping algebra of su(3) one can generate an algebra based only on abstract commutation relations of su(3) Lie algebra without explicit constraints on the representations or realisations. This construction relies on the maximal Abelian subalgebra, also called MASA, which are the Cartan generators and their commutant. We obtain a new six-dimensional cubic algebra where the structure constant are integer numbers which reduce from a quartic algebra for which the structure constant depend on the Cartan generator and the Casimir invariant. We also present other form of the symmetry algebra using the quadratic and cubic Casimir invariants of su(3). It reduces as the known quadratic Racah algebra R(3) only when using an explicit realization. This algebraic structure describes the symmetry of the quadratically superintegrable systems on the 2 sphere. We also present a contraction to another six-dimensional cubic algebra which would corresponding to the symmetry algebra of a Smorodinsky–Winternitz model. [ABSTRACT FROM AUTHOR]

In this paper, we carry out the algebraic study of integrable differential-difference equations whose field variables take values in an associative (but not commutative) algebra. We adapt the Hamiltonian formalism to nonabelian difference Laurent polynomials and describe how to obtain a recursion operator from the Lax representation of an integrable nonabelian differential-difference system. As an application, we study a family of integrable equations: the nonabelian Narita–Itoh–Bogoyavlensky lattice, for which we construct their recursion operators and Hamiltonian operators and prove the locality of infinitely many commuting symmetries generated from their highly nonlocal recursion operators. Finally, we discuss the nonabelian version of several integrable difference systems, including the relativistic Toda chain and Ablowitz–Ladik lattice. [ABSTRACT FROM AUTHOR]

In this paper, we investigate in detail a superintegrable extension of the singular harmonic oscillator whose wave functions can be expressed in terms of exceptional Jacobi polynomials. We show that this Hamiltonian admits a fourth-order integral of motion and use the classification of such systems to show that the potential gives a rational solution associated with the sixth Painlevé equation. Additionally, we show that the integrals of the motion close to form a cubic algebra and describe briefly deformed oscillator representations of this algebra. [ABSTRACT FROM AUTHOR]

A complete classification of 2D quadratically superintegrable systems with scalar potential on two-dimensional conformally flat spaces has been performed over the years and 58 models, divided into 12 equivalence classes, have been obtained. We will re-examine two pseudo-Hermitian quantum systems E 8 and E 10 from such a classification by a new approach based on extra sets of ladder operators. They correspond in fact to two of those equivalence classes. Those extra ladder operators are exploited to obtain the generating spectrum algebra and the dynamical symmetry one. We will relate the generators of the dynamical symmetry algebra to the Hamiltonian, thus demonstrating that the latter can be written in an algebraic form. We will also link them to the integrals of motion providing the superintegrability property. This demonstrates how the underlying dynamical symmetry algebra allows to write the integrals in terms of its generators and therefore explains the symmetries. Furthermore, we will exploit those algebraic constructions to generate extended sets of states and give the action of the ladder operators on them. We will present polynomials of the Hamiltonian and the integrals of motion that vanish on some of those states, then demonstrating that the sets of states not only contain eigenstates, but also generalized states which are beyond the well-known eigenstates of diagonalizable Hamiltonians and satisfy more complicated polynomial identities. Our approach provides a natural framework for such states. [ABSTRACT FROM AUTHOR]

We present an algorithm to compute the exact critical probability h (n) for an n × ∞ helical square lattice with random and independent site occupancy. The algorithm has time complexity O (n 2 c n) and space complexity O (c n) with c = 2.7459... and allows us to compute h (n) up to n = 24. Since the extrapolation result of h (n) is inconsistent with the current best estimation of pc, we also compute and extend the exact critical probability p c (n) for an n × ∞ cylindrical square lattice to n = 24. Our calculation shows that the current best result of p c = 0.592 746 050 792 10 (2) by Jacobsen (2015 J. Phys. A: Math. Theor. 48 454003) is incorrect and the corrected value should be 0.592 746 050 7896 (1) . [ABSTRACT FROM AUTHOR]

We study solutions of the reflection equation related to the quantum affine algebra U q ( s l n ^) . First, we explain how to construct a family of stochastic integrable vertex models with fixed boundary conditions. Then, we construct upper- and lower-triangular solutions of the reflection equation related to symmetric tensor representations of U q ( s l n ^) with arbitrary spin. We also prove the star–star relation for the Boltzmann weights of the Ising-type model, conjectured by Bazhanov and Sergeev, and use it to verify certain properties of the solutions obtained. [ABSTRACT FROM AUTHOR]

Motivated by the potential connection between metric-affine gravity and linear generalized uncertainty principle (GUP) in the phase space, we develop a covariant form of linear GUP and an associated modified Poincaré algebra, which exhibits distinctive behavior, nearing nullity at the minimal length scale proposed by linear GUP. We use three-torus geometry to visually represent linear GUP within a covariant framework. The three-torus area provides an exact geometric representation of Bekenstein's universal bound. We depart from Bousso's approach, which adapts Bekenstein's bound by substituting the Schwarzschild radius ( r s ) with the radius (R) of the smallest sphere enclosing the physical system, thereby basing the covariant entropy bound on the sphere's area. Instead, our revised covariant entropy bound is described by the area of a three-torus, determined by both the inner radius r s and outer radius R where r s ⩽ R due to gravitational stability. This approach results in a more precise geometric representation of Bekenstein's bound, notably for larger systems where Bousso's bound is typically much larger than Bekensetin's universal bound. Furthermore, we derive an equation that turns the standard uncertainty inequality into an equation when considering the contribution of the three-torus covariant entropy bound, suggesting a new avenue of quantum gravity. [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]

We develop the quantum cluster algebra approach recently introduced by Sun and Yagi to investigate the tetrahedron equation, a three-dimensional generalization of the Yang-Baxter equation. In the case of square quiver, we devise a new realization of quantum Y -variables in terms q -Weyl algebras and obtain a solution that possesses three spectral parameters. It is expressed in various forms, comprising four products of quantum dilogarithms depending on the signs in decomposing the quantum mutations into the automorphism part and the monomial part. For a specific choice of them, our formula precisely reproduces Sergeev's R matrix, which corresponds to a vertex formulation of the Zamolodchikov-Bazhanov-Baxter model when q is specialized to a root of unity. [ABSTRACT FROM AUTHOR]

This study investigates the scaling behavior of the ground-state entanglement entropy in a model of free fermions on folded cubes. An analytical expression is derived in the large-diameter limit, revealing a strict adherence to the area law. The absence of the logarithmic enhancement expected for free fermions is explained using a decomposition of folded cubes in chains based on its Terwilliger algebra and s o (3) − 1 . The entanglement Hamiltonian and its relation to Heun operators are also investigated. [ABSTRACT FROM AUTHOR]

We consider intertwining relations of the triangular q-Onsager algebra, and obtain generic triangular boundary K-operators in terms of the Borel subalgebras of Uq(sl2). These K-operators solve the reflection equation. [ABSTRACT FROM AUTHOR]

There are different methods of discretizing integrable systems. We consider semi-discrete analog of two-dimensional Toda lattices associated to the Cartan matrices of simple Lie algebras that was proposed by Habibullin in 2011. This discretization is based on the notion of Darboux integrability. Generalized Toda lattices are known to be Darboux integrable in the continuous case (that is, they admit complete families of characteristic integrals in both directions). We prove that semi-discrete analogs of Toda lattices associated to the Cartan matrices of all simple Lie algebras are Darboux integrable. By examining the properties of Habibullin's discretization we show that if a function is a characteristic integral for a generalized Toda lattice in the continuous case, then the same function is a characteristic integral in the semi-discrete case as well. We consider characteristic algebras of such integral-preserving discretizations of Toda lattices to prove the existence of complete families of characteristic integrals in the second direction. [ABSTRACT FROM AUTHOR]

The aim of this work is to study the relation between S -expansions and other transformations of Lie algebras. In particular, we prove that contractions, deformations and central extensions of Lie algebras are preserved by S -expansions. We also provide several examples and give conditions so transformations of reduced subalgebras of S -expanded algebras are preserved by the S -expansion procedure. [ABSTRACT FROM AUTHOR]

We show, using the methods of geometric algebra, that Runge–Lenz vector in the Kepler problem is a 3-dimensional projection of SO(4) moment map that acts on the phase space of 4-dimensional particle motion. Thus, Runge–Lenz vector is a consequence of geometric symmetry of R 4 × R 4 phase space. [ABSTRACT FROM AUTHOR]

We consider the Plancherel measure on irreducible components of tensor powers of the spinor representation of s o 2 n + 1 . The irreducible representations correspond to the generalized Young diagrams. With respect to this measure the probability of an irreducible representation is the product of its multiplicity and dimension, divided by the total dimension of the tensor product. We study the limit shape of the generalized Young diagram when the tensor power N and the rank n of the algebra tend to infinity with N / n fixed. We derive an explicit formula for the limit shape and prove convergence to it in probability. We prove central limit theorem for global fluctuations around the limit shape. [ABSTRACT FROM AUTHOR]

In this paper we consider the slightly L2-supercritical gKdV equations , with the nonlinearity and . In the previous work of the author, we know that there exists a stable self-similar blow-up dynamics for slightly L2-supercritical gKdV equations. Such solutions can be viewed as solutions with a single blow-up point. In this paper we will prove the existence of solutions with multiple blow-up points, and give a description of the formation of the singularity near the blow-up time. [ABSTRACT FROM AUTHOR]

This paper presents a derivation of the possible residual symmetries of rational K-matrices which are invertible in the ‘classical limit’ (the spectral parameter goes to infinity). This derivation uses only the boundary Yang–Baxter equation and the asymptotic expansions of the R-matrices. The result proves the previous assumption of the literature: if the original and the residual symmetry algebras are and then there exists a Lie-algebra involution of for which the invariant sub-algebra is . In addition, we study a K-matrix which is not invertible in the ‘classical limit’. It is shown that its symmetry algebra is not reductive but a semi-direct sum of reductive and solvable Lie-algebras. [ABSTRACT FROM AUTHOR]

Numerical dosimetry studies require the development of accurate numerical 3D models of the human body. This paper proposes a novel method for building 3D heterogeneous young children models combining results obtained from a semi-automatic multi-organ segmentation algorithm and an anatomy deformation method. The data consist of 3D magnetic resonance images, which are first segmented to obtain a set of initial tissues. A deformation procedure guided by the segmentation results is then developed in order to obtain five young children models ranging from the age of 5 to 37 months. By constraining the deformation of an older child model toward a younger one using segmentation results, we assure the anatomical realism of the models. Using the proposed framework, five models, containing thirteen tissues, are built. Three of these models are used in a prospective dosimetry study to analyze young child exposure to radiofrequency electromagnetic fields. The results lean to show the existence of a relationship between age and whole body exposure. The results also highlight the necessity to specifically study and develop measurements of child tissues dielectric properties. [ABSTRACT FROM AUTHOR]

In the general case of deformed Heisenberg algebra leading to the minimal length, we present a definition of the δ ′(x) potential as a linear kernel of potential energy operator in momentum representation. We find exactly the energy level and corresponding eigenfunction for δ ′(x) and δ (x) âˆ' δ ′(x) potentials in deformed space with arbitrary function of deformation. The energy spectrum for different partial cases of deformation function is analysed. [ABSTRACT FROM AUTHOR]

We start a systematic investigation of possible isometries of the asymptotically de Sitter solutions to Einstein equations. We reformulate the Killing equation as conformal equations for the initial data at I + . This allows for partial classification of possible symmetry algebras. In particular, if they are not maximal, they may be at most four-dimensional. We provide several examples. As a simple corollary it is shown that the only spacetime in which the Killing horizon intersects I + (after a conformal completion) is locally the de Sitter Universe. [ABSTRACT FROM AUTHOR]

It is known that any Poisson manifold can be embedded into a bigger space which admits a description in terms of a global symplectic structure. Such a procedure is known as a symplectic realization and has a number of important applications like the quantization of the original Poisson manifold. In the present paper we extend the above idea to the case of quasi-Poisson structures which should not necessarily satisfy the Jacobi identity. For any given quasi-Poisson structure we provide a closed recursive formula for local embedding functions and Darboux coordinates. Our construction is illustrated for the examples of the constant R-flux algebra, quasi-Poisson structure isomorphic to the commutator algebra of imaginary octonions and the non-geometric M-theory R-flux background. In all cases we derive explicit formulas for the symplectic realization and the corresponding expression for Darboux coordinates. [ABSTRACT FROM AUTHOR]

It was recently conjectured that the AGT correspondence between the —instanton counting on and the two-dimensional field theories with the conformal symmetry algebra can be considered as a root of unity limit of its K-theoretic analogue. From this point of view, the algebra and a special basis in its representation are limits of the Ding–Iohara algebra and the Macdonald polynomials respectively. In this paper we confirm this conjecture for the special case r = 1. We uncover the implicit symmetry in this limit. We also found that the vertex operators in the special basis have factorized AFLT form. [ABSTRACT FROM AUTHOR]

Image Rotation and Subtraction (IRS) is a high-contrast imaging technique that can be used to suppress the speckle noise and facilitate the direct detection of exoplanets. IRS is different from Angular Differential Imaging (ADI), in that it will subtract a copy of the image with 180° rotated around its point-spread function (PSF) center, rather than the subtraction of the median of all of the PSF images. Since the planet itself will be rotated to the other side of the PSF, IRS does not suffer from planet self-subtraction. In this paper, we have introduced an optimization algorithm to IRS (OIRS), that can provide an extra contrast gain at small angular separations. The performance of OIRS has been demonstrated with ADI data. We then made a comparison of the signal-to-noise ratio (S/N) achieved by algorithms of locally optimized combinations of images and OIRS. Finally, we found that the OIRS algorithm can deliver a better S/N for small angular separations. [ABSTRACT FROM AUTHOR]

In this paper, we study the evolution of the vortex filament equation, with X(s, 0) being a regular planar polygon. Using algebraic techniques, supported by full numerical simulations, we give strong evidence that X(s, t) is also a polygon at any rational time; moreover, it can be fully characterized, up to a rigid movement, by a generalized quadratic Gauß sum.We also study the fractal behaviour of X(0, t), relating it with the so-called Riemann's non-differentiable function, that was proved by Jaffard to be a multifractal. [ABSTRACT FROM AUTHOR]

This paper discusses the basic properties of various versions of the two-variable elliptic genus with special attention to the equivariant elliptic genus. The main applications are to the elliptic genera attached to non-compact GITs, including the theories regarding the elliptic genera of phases on N  =  2 introduced in Witten (1993 Nucl. Phys. B 403 159–222). [ABSTRACT FROM AUTHOR]

The spacetime in the interior of a black hole can be described by an homogeneous line element, for which the Einsteinâ€"Hilbert action reduces to a one-dimensional mechanical model. We have shown in Geiller et al (2021 SciPost Phys. 10 022) that this model exhibits a symmetry under the (2 + 1)-dimensional PoincarĂ© group. Here we extend the PoincarĂ© transformations to the infinite-dimensional BMS3 group. Although the black hole model is not invariant under those extended transformations, we can write it as a geometric action for BMS3, where the configuration space variables are elements of the algebra b m s 3 and the equations of motion transform as coadjoint vectors. The BMS3 symmetry breaks down to its PoincarĂ© subgroup, which arises as the stabilizer of the vacuum orbit. This symmetry breaking is analogous to what happens with the Schwarzian action in AdS2 JT gravity, although in the present case there is no direct interpretation in terms of boundary symmetries. This observation, together with the fact that other lower-dimensional gravitational models (such as the BTZ black hole) possess the same broken BMS3 symmetries, provides yet another illustration of the ubiquitous role played by this group. [ABSTRACT FROM AUTHOR]

This paper studies the entropy of tree-shifts of finite type with and without boundary conditions. We demonstrate that computing the entropy of a tree-shift of finite type is equivalent to solving a system of nonlinear recurrence equations. Furthermore, the entropy of the binary Markov tree-shifts over two symbols is either 0 or . Meanwhile, the realization of a class of reals including multinacci numbers is elaborated, which indicates that tree-shifts are capable of rich phenomena. By considering the influence of three different types of boundary conditions, say, the periodic, Dirichlet, and Neumann boundary conditions, the necessary and sufficient conditions for the coincidence of entropy with and without boundary conditions are addressed. [ABSTRACT FROM AUTHOR]

The fully packed loop (FPL) model is a statistical model related to the integrable vertex model. In this paper we study the continuum limit of the FPL. With the appropriate weight of non-contractible loops, we give evidence of an extended W3 symmetry in the continuum. The partition function on the torus is calculated exactly, yielding new modular invariants of W3 characters. The full conformal field theory spectrum is obtained, and is found to be in excellent agreement with exact diagonalisation. [ABSTRACT FROM AUTHOR]

The authors discuss various topics to show the impact of the articles by A. J. Macfarlane and L. C. Biedenharn in which q-boson operators play a significant role. Topics include the extensions and generalizations of the q-boson algebra, position and momentum operators and q-deformed quantum mechanics, and representations of quantum algebras and q-functions. Also mentioned are the applications of q-boson operators to various problems in physics.

In this paper we produce seven new algebras as confluences of the Cherednik algebra of type and we characterise their spherical-sub-algebras. The limit of the spherical sub-algebra of the Cherednik algebra of type is the monodromy manifold of the Painlevé VI equation (Oblomkov 2004 Int. Math. Res. Not. 2004 877–912). Here we prove that by considering the limits of the spherical sub-algebras of our new confluent algebras, one obtains the monodromy manifolds of all other Painlevé differential equations. Moreover, we introduce confluent versions of the Zhedanov algebra and prove that each of them (quotiented by their Casimir) is isomorphic to the corresponding spherical sub-algebra of our new confluent Cherednik algebras. We show that in the basic representation our confluent Zhedanov algebras act as symmetries of certain elements of the q-Askey scheme, thus setting a stepping stone towards the solution of the open problem of finding the corresponding quantum algebra for each element of the q-Askey scheme. These results establish a new link between the theory of the Painlevé equations and the theory of the q-Askey scheme making a step towards the construction of a representation theoretic approach for the Painlevé theory. [ABSTRACT FROM AUTHOR]

We consider the classical Hamiltonian of the interior of the Schwarzschild black hole in Ashtekar–Barbero connection formalism. Then, inspired by generalized uncertainty principle models, we deform the classical canonical algebra and derive the effective dynamics of the model under this modification. We show that such a deformation leads to the resolution of the singularity of the black hole and a minimum nonzero radius for the infalling two-spheres, provided that the deformation parameters are chosen to be negative. [ABSTRACT FROM AUTHOR]

We show the existence of families of orthonormal, future directed bases which allow to cast every skew-symmetric endomorphism of () in a single canonical form depending on a minimal number of parameters. This canonical form is shared by every pair of elements in differing by an orthochronous Lorentz transformation, i.e. it defines the orbits of the orthochronous Lorentz group O+(1, n) under the adjoint action on its algebra. Using this form, we obtain the quotient topology of. From known relations between and the conformal Killing vector fields (CKVFs) of the sphere , a canonical form for CKVFs follows immediately. This form is used to find adapted coordinates to an arbitrary CKVF that covers all cases at the same time. We do the calculation for even n and obtain the case of odd n as a consequence. Finally, we employ the adapted coordinates to obtain a wide class of TT-tensors for n = 3, which provide Cauchy data at conformally flat null infinity. Specifically, this class of data is characterized for generating Λ > 0-vacuum spacetimes with two-symmetries, one of which axial, admitting a conformally flat. The class of data is infinite dimensional, depending on two arbitrary functions of one variable as well as a number of constants. Moreover, it contains the data for the Kerr–de Sitter spacetime, which we explicitly identify within. [ABSTRACT FROM AUTHOR]

This article is the third in a series, the aim of which is to use Lie group theory to obtain exact analytic solutions of delay ordinary differential systems (DODSs). Such a system consists of two equations involving one independent variable x and one dependent variable y. As opposed to ordinary differential equations (ODEs) the variable x figures in more than one point (we consider the case of two points, x and x−). The dependent variable y and its derivatives figure in both x and x−. Two previous articles were devoted to first-order DODSs, here we concentrate on a large class of second-order ones. We show that within this class the symmetry algebra can be of dimension n with 0 ⩽ n ⩽ 6 for nonlinear DODSs and must be infinite-dimensional for linear or linearizable ones. The symmetry algebras can be used to obtain exact particular group invariant solutions. As a specific application we present some exact solutions of a DODS model of traffic flow. [ABSTRACT FROM AUTHOR]

Quantum entanglement breaking channels are a fundamental class of quantum operations; originally investigated for quantum information theoretic reasons, their study has since grown to touch on many aspects of quantum information science. Here we investigate the nullspace structures of entanglement breaking channels and we derive a pair of related applications. We show that every operator space of trace zero matrices is the nullspace of an entanglement breaking channel. We derive a test for mixed unitarity of quantum channels based on complementary channel behaviour and entanglement breaking channel nullspaces. We identify conditions that guarantee the existence of private algebras for certain classes of entanglement breaking channels. [ABSTRACT FROM AUTHOR]

In a previous paper (2010 Phys. Rev. A 82 042702; 2014 Phys. Rev. A 89 069901(E)), one of us (EAGA) calculated the resonant contribution to , the effective number of electrons available for annihilation by a positron with wave number k0, in the scattering of a heavy positron by H2. The mass of the positron was increased just sufficiently for a bound state to occur. This calculation was carried out using the Kohn variational method. An alternative method is to use the close-coupled equations for the system under consideration. We compare our results with those obtained by Gribakin and Lee (2006 Phys. Rev. Lett.97 193201). There is a resonant contribution to from the vibrationally excited quasibound state which may be described by a Breit–Wigner resonance formula arising naturally from the close-coupling analysis if a certain additional assumption is made. There is also a separate resonant contribution to from the open channel function influenced by the quasibound state, and a cross term. It is shown that the contribution from the quasibound state is very similar to the expression for the resonant contribution obtained by Gribakin and Lee. Comparison is made with other treatments, for example, the close-coupling calculation of Nishimura and Gianturco (2003 Phys. Rev. Lett.90 183201). [ABSTRACT FROM AUTHOR]

We conjecture a closed-form expression for the Schur limit of the superconformal index of two infinite series of Argyres–Douglas (AD) superconformal field theories (SCFTs): the and the theories. While these SCFTs can be realized at special points on the Coulomb branch of certain gauge theories, their superconformal R symmetries are emergent, and hence their indices cannot be evaluated by localization. Instead, we construct the and indices by using a relation to two-dimensional q-deformed Yang–Mills theory and data from the class construction. Our results generalize the indices derived from the torus partition functions of the two-dimensional chiral algebras associated with the and SCFTs. As checks of our conjectures, we study the consistency of our results with an S-duality recently discussed by us in collaboration with Giacomelli and Papageorgakis, we reproduce known Higgs branch relations, we check consistency with a series of renormalization group flows, and we verify that the small S1 limits of our indices reproduce expected Cardy-like behavior. We will discuss the S1 reduction of our indices in a separate paper. [ABSTRACT FROM AUTHOR]

In this paper, we defined a new multi-component B type Kadomtsev-Petviashvili (BKP) hierarchy that takes values in a commutative subalgebra of After this, we give the gauge transformation of this commutative multicomponent BKP (CMBKP) hierarchy. Meanwhile, we construct a new constrained CMBKP hierarchy that contains some new integrable systems, including coupled KdV equations under a certain reduction. After this, the quantum torus symmetry and quantum torus constraint on the tau function of the commutative multi-component BKP hierarchy will be constructed. [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]

The construction of generalized Bäcklund transformation for the An affine Toda hierarchy is proposed in terms of gauge transformation acting on the zero curvature representation. Such construction is based upon the graded structure of the underlying affine algebra which induces a classification of generalized Bäcklund transformations. Moreover, explicit examples for sl(3) and sl(4) lead to uncover interesting composition properties of various types of Bäcklund transformations. The universality character of the gauge-Bäcklund transformation method is extended to all equations of the hierarchy. Such interesting property provides a systematic framework to construct Bäcklund transformations to higher flow equations. Explicit example for the simplest higher flow of the sl(3) hierarchy is presented. [ABSTRACT FROM AUTHOR]

The constraint algebra is derived in the second order tetrad Hamiltonian formalism of the bigravity. This is done by a straightforward calculation without involving any insights, implicit functions, and Dirac brackets. The tetrad approach is the only way to present the bigravity action as a linear functional of lapses and shifts and the Hassan–Rosen transform (characterized as 'a complicated redefinition of the shift variable' according to the authors) appears here not as an ansatz but as fixing of a Lagrange multiplier. A comparison of this approach with the other ones is provided. [ABSTRACT FROM AUTHOR]