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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 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]

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]

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]

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]

This paper examines classical privacy amplification using a universal family of hash functions. In quantum key distribution, the adversary's measurement can wait until the choice of hash functions is announced, and so the adversary's information may depend on the choice. Therefore the existing result on classical privacy amplification, which assumes the independence of the choice from the other random variables, is not applicable to this case. This paper provides a security proof of privacy amplification which is valid even when the adversary's information may depend on the choice of hash functions. The compression rate of the proposed privacy amplification can be taken to be the same as that of the existing one with an exponentially small loss in secrecy of a final key. [ABSTRACT FROM AUTHOR]

In 2008, the conjecture that structural physical approximations (SPAs) to optimal entanglement witnesses are separable states (in general unnormalized) was posed. In an attempt to disprove it, Ha and Kye (2012 arXiv:1210.1088v3) proposed a decomposable entanglement witness, whose SPA is entangled, and argued that it is optimal. In this paper, which is based on a comment to the latter work (Augusiak et al 2013 arXiv:1304.2040v1), we show, both analytically and numerically, that this entanglement witness is not optimal, and as such it is not a counterexample to the conjecture. To this end, we make use of a method for checking the optimality of entanglement witnesses developed already in Lewenstein et al (2000 Phys. Rev. A 62 052310), however, hardly exploited so far in the literature. [ABSTRACT FROM AUTHOR]

n-Lie algebras have close relationships with many important fields in mathematics and mathematical physics. In this paper, we construct (n+1)-Lie algebras by two-dimensional extensions of metric n-Lie algebras, and construct (n + 1)-Lie algebras in terms of n-Lie algebras and certain linear functions. Along this approach, we obtain (n+1)-Lie algebras by n-Lie algebras with the generating index n, and 4-Lie algebras by non-nilpotent 3-Lie algebras with the generating index 4. [ABSTRACT FROM AUTHOR]

In this paper, we consider the transmission of classical information through a class of quantum channels with long-term memory, which are convex combinations of memoryless channels. Hence, the memory of such channels can be considered to be given by a Markov chain which is aperiodic but not irreducible. We prove the coding theorem and weak converse for this class of channels. The main techniques that we employ are a quantum version of Feinstein's fundamental lemma (Feinstein A 1954 IRE Trans. PGIT 4 2-22, Khinchin A I 1957 Mathematical Foundations of Information Theory: II. On the Fundamental Theorems of Information Theory (New York: Dover) chapter IV) and a generalization of Helstrom's theorem (Helstrom CW1976 Quantum detection and estimation theory Mathematics in Science and Engineering vol 123 (London: Academic)). [ABSTRACT FROM AUTHOR]

The focus is on two parties with Hilbert spaces of dimension d, i.e. 'qudits'. In the state space of these two possibly entangled qudits an analogue to the wellknown tetrahedron with the four qubit Bell states at the vertices is presented. The simplex analogue to this magic tetrahedron includes mixed states. Each of these states appears to each of the two parties as the maximally mixed state. Some studies on these states are performed, and special elements of this set are identified. A large number of them are included in the chosen simplex which fits exactly into conditions needed for teleportation and other applications. Its rich symmetry—related to that of a classical phase space—helps to study entanglement, to construct witnesses and perform partial transpositions. This simplex has been explored in detail for d = 3. In this paper, the mathematical background and extensions to arbitrary dimensions are analysed. [ABSTRACT FROM AUTHOR]

We discuss a comment recently made by C F Lo concerning validity of one of the examples used in our paper (2013 J. Phys. A: Math. Theor.46 235301). While we agree with Lo’s statement regarding inapplicability of the example to the proposed schema, we disagree with the claim concerning mathematical correctness of that example. We are not convinced by Lo’s arguments leading to the conclusion that the model is mathematically ill-defined. [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]

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 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]

In a series of papers with Kossakowski, the first author has examined properties of densities for which the positive partial transpose (PPT) property can be readily checked. These densities were also investigated from a different perspective by Baumgartner, Hiesmayr and Narnhofer. In this paper, we show how the support of such densities can be expressed in terms of lines in a finite geometry and how the same structure lends itself to checking the necessary PPT condition and to a novel sufficient condition for separability. [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]

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 association between large shifts of a pointer in a weak measurement and fast oscillations in an associated function involving the pre- and post-selected states has been clarified in a recent paper (Aharonov et al 2011 J. Phys. A: Math. Theor. http://dx.doi.org/10.1088/1751-8113/44/36/365304). Here we explore the association further for the case of an observable with N discrete eigenvalues, by calculating and illustrating how the supershift emerges, even for N = 2, as the uncertainty in the pointer position increases. This happens if the initial pointer wavefunction is Gaussian or Lorentzian but not if it is exponential. [ABSTRACT FROM AUTHOR]

In this paper we demonstrate how one can compute the Veneziano amplitude for bosonic string theory using the Britto-Cachazo-Feng-Witten method. We use an educated ansatz for the cubic amplitude of two tachyons and an arbitrary level string state. [ABSTRACT FROM AUTHOR]

We reply to the comment by van Enter et al on our paper (2011 J. Phys. A: Math. Theor. 44 015002). [ABSTRACT FROM AUTHOR]

Understanding the interaction between electromagnetic waves and matter is vital in applications ranging from classical optics to antenna theory. This paper derives physical limitations on the scattering of electromagnetic vector spherical waves. The assumptions made are that the heterogeneous scatterer is passive, and has constitutive relations which are in convolution form in the time domain and anisotropic in the static limit. The resulting bounds limit the reflection coefficient of the modes over a frequency interval, and can thus be interpreted as limitations on the absorption of power from a single mode. They can be used within a wide range of applications, and are particularly useful for electrically small scatterers. The derivation follows a general approach to derive sum rules and physical limitations on passive systems in convolution form. The time domain versions of the vector spherical waves are used to describe the passivity of the scatterer, and a set of integral identities for Herglotz functions are applied to derive sum rules from which the physical limitations follow. [ABSTRACT FROM AUTHOR]

In their 1995 paper, Jean-Benoit Bost and Alain Connes (BC) constructed a quantum dynamical system whose partition function is the Riemann zeta function z(b), where b is an inverse temperature. We formulate Riemann hypothesis (RH) as a property of the low-temperature Kubo-Martin-Schwinger (KMS) states of this theory. More precisely, the expectation value of the BC phase operator can be written as \phi _{\beta }(q)=N_{q-1}^{\beta -1} \psi _{\beta -1}(N_q), where Nq = [?]qk = 1pk is the primorial number of order q and psb is a generalized Dedekind ps function depending on one real parameter b as \psi _b (q)=q \prod _{p \in \mathcal {P,}p \vert q}\frac{1-1/p^b}{1-1/p}. Fix a large inverse temperature b > 2. The RH is then shown to be equivalent to the inequality N_q |\phi _\beta (N_q)|\zeta (\beta -1) \gt {\rm e}^\gamma \log \log N_q, for q large enough. Under RH, extra formulas for high-temperature KMS states (1.5 < b < 2) are derived.'Number theory is not pure Mathematics. It is the Physics of the world of Numbers.'Alf van der Poorten [ABSTRACT FROM AUTHOR]

The aim of this paper is to give an explicit formula of the invariant distribution of a quasi-birth-and-death process in terms of the block entries of the transition probability matrix using a matrix-valued orthogonal polynomials approach. We will show that the invariant distribution can be computed using the squared norms of the corresponding matrix-valued orthogonal polynomials, no matter if they are or not diagonal matrices. We will give an example where the squared norms are not diagonal matrices, but nevertheless we can compute its invariant distribution. [ABSTRACT FROM AUTHOR]

In this paper, we will prove the existence of the non-isospectral AKNS hierarchy with reality conditions restriction and construct the matrix form Darboux transformation. Using this Darboux transformation, the solutions of the relevant nonlinear equations can be expressed explicitly. [ABSTRACT FROM AUTHOR]

In a recent paper, Miller (2005 J. Phys. A: Math. Gen. 38 3541-5) obtained a new summation formula for the Clausen's series 3F2(1). The aim of this comment is to point out that the summation formula obtained by Miller is not a new one. [ABSTRACT FROM AUTHOR]

Elliptic pseudoanalytic function theory was considered independently by Bers and Vekua decades ago. In this paper, we develop a hyperbolic analogue of pseudoanalytic function theory using the algebra of hyperbolic numbers. We consider the Klein-Gordon equation with a potential. With the aid of one particular solution we factorize the Klein-Gordon operator in terms of two Vekua-type operators. We show that real parts of the solutions of one of these Vekua-type operators are solutions of the considered Klein-Gordon equation. Using hyperbolic pseudoanalytic function theory, we then obtain explicit construction of infinite systems of solutions of the Klein-Gordon equation with potential. Finally, we give some examples of the application of the proposed procedure. [ABSTRACT FROM AUTHOR]