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By using the so-called double-complex function method, a doubleness symmetry for each member of the class of stationary axisymmetric general symplectic gravity models is found and exploited so that some double-complex (n+1)×(n+1) matrix Ernst-like potential for any non-negative integer n can be constructed and the associated motion equations can be extended into a double-complex matrix Ernst-like form. Then double symmetry symplectic groups Sp(2(n+1), R(J)) of the theories are given and verified that their actions can be realized concisely by double-complex matrix form generalizations of the fractional linear transformation on the Ernst potential. These results demonstrate that the theories under consideration possess more and richer symmetry structures. The special cases n=0 and n=1 correspond, respectively, to the pure Einstein gravity and the Einstein–Maxwell-dilaton–axion theories. Moreover, as an application, for each n=0,1,2,..., an infinite chain of double-solutions of the general symplectic gravity model is obtained, which shows that the double-complex method is more effective. Some of the results in this paper cannot be obtained by the usual (nondouble) scheme.© 2003 American Institute of Physics. [ABSTRACT FROM AUTHOR]

Proposes methods for determining the Green's matrix for problems admitting Hamiltonians that have infinite symmetric tridiagonal matrix form on some basis representation. Recurrence relation from the Jacobi-matrix; Matrix elements of the Green's operator; Solution to the recurrence relation by continued fractions.

Using adjoint representation of Lie superalgebras, we obtain the matrix form of super-Jacobi and mixed super-Jacobi identities of Lie superbialgebras. By direct calculations of these identities, and use of automorphism supergroups of two and three dimensional Lie superalgebras, we obtain and classify all two and three dimensional Lie superbialgebras. [ABSTRACT FROM AUTHOR]

Complete Hamiltonian operators have been obtained in terms of internal coordinates on the basis of the fiber bundle theory in geometry [J. Math Phys. 44, 4411 (2003)]. In this article, the full Hamiltonian is specialized for a rigid and for a semirigid molecule. For the rigid molecule, all internal coordinates are fixed at constants, so that the Hamiltonian operator comes to take an ordinary matrix form, and accordingly, the Schrödinger equation becomes an algebraic eigenvalue equation. The eigenvalues then provide rotational energy spectra of the rigid molecule. For the semirigid molecule, the full Hamiltonian is expanded in the vicinity of an equilibrium position into a power series in an infinitesimal parameter, to which the perturbation method is applied to obtain energy spectra in the form of a power series in the infinitesimal parameter. Indeed, the energy spectra are calculated to the second order term in the infinitesimal parameter in both the cases where the unperturbed energy spectra are nondegenerate and where those are degenerate. It then turns out that the amount of the energy shift caused by the rotation, which is the sum of the pure rotational and the rotation-vibration coupling terms, is proportional to the total angular momentum eigenvalue. It is also observed that a vibrational energy shift occurs simultaneously, which is caused by the metric defined on the internal space. No symmetry is assumed on the shape of the molecule in equilibrium. [ABSTRACT FROM AUTHOR]

This paper presents a geometric discretization of elasticity when the ambient space is Euclidean. This theory is built on ideas from algebraic topology, exterior calculus, and the recent developments of discrete exterior calculus. We first review some geometric ideas in continuum mechanics and show how constitutive equations of linearized elasticity, similar to those of electromagnetism, can be written in terms of a material Hodge star operator. In the discrete theory presented in this paper, instead of referring to continuum quantities, we postulate the existence of some discrete scalar-valued and vector-valued primal and dual differential forms on a discretized solid, which is assumed to be a triangulated domain. We find the discrete governing equations by requiring energy balance invariance under time-dependent rigid translations and rotations of the ambient space. There are several subtle differences between the discrete and continuous theories. For example, power of tractions in the discrete theory is written on a layer of cells with a nonzero volume. We obtain the compatibility equations of this discrete theory using tools from algebraic topology. We study a discrete Cosserat medium and obtain its governing equations. Finally, we study the geometric structure of linearized elasticity and write its governing equations in a matrix form. We show that, in addition to constitutive equations, balance of angular momentum is also metric dependent; all the other governing equations are topological. [ABSTRACT FROM AUTHOR]

We introduce the concept of Miura maps between parameter-dependent algebraic curves of hyper-elliptic type. These Miura maps induce Miura maps between Stäckel systems defined (on the extended phase space) by the considered algebraic curves. This construction yields a new way of generating multi-Hamiltonian representations for Stäckel systems. [ABSTRACT FROM AUTHOR]

Key properties of a physical system depend on whether it is gapped, i.e., whether its spectral gap has a positive lower bound that is independent of system size. Here, we provide a prescription for renormalizing a spin chain Hamiltonian in such a way that the renormalized Hamiltonian is gapped if and only if the original Hamiltonian is gapped. Then, we articulate a set of conditions that guarantees the renormalized Hamiltonian is gapped. These conditions are built on a certain strong notion of decaying correlations involving an operator norm of non-commuting terms in the renormalized Hamiltonian. We apply the method to show that two interesting models, with forms motivated by quantum circuits, are gapped. We also confirm the generality of the method by successfully applying it to a somewhat different case, the well-known Affleck-Kennedy-Lieb-Tasaki (AKLT) model. [ABSTRACT FROM AUTHOR]

Cubic blocks are studied assembled from linear operators R acting in the tensor product of d linear "spin" spaces. Such operator is associated with a linear transformation A in a vector space over a field F of a finite characteristic p, like "permutation-type" operators studied by Hietarinta [J. Phys. A: Math. Gen. 30, 4757–4771 (1997)]. One small difference is that we do not require A and, consequently, R to be invertible; more importantly, no relations on R are required of the type of Yang–Baxter or its higher analogues. It is shown that, in d = 3 dimensions, a pn × pn × pn block decomposes into the tensor product of operators similar to the initial R. One generalization of this involves commutative algebras over F and allows to obtain, in particular, results about spin configurations determined by a four-dimensional R. Another generalization deals with introducing Boltzmann weights for spin configurations; it turns out that there exists a non-trivial self-similarity involving Boltzmann weights as well. [ABSTRACT FROM AUTHOR]

Focuses on the derivation of Fredholm integral equations for the electrostatics anisotropic inclusions in an anisotropic matrix. Extension of the use of the renormalization method for the evaluation of lattice sums; Formulation of integral operators on matrix form; Algorithm for anisotropic disks in an isotropic matrix.

The eigenvalue problem of the angular part of generalized spheroidal wave equations (SWE) for describing the scattering of a charged particle on two Coulomb centers with different charges is cast in a matrix form which is then solved by a standard matrix procedure. This simple mathematical method is shown to be the most efficient and accurate among the others given for obtaining the solution to the ordinary SWE and the generalized SWE. An application to a complete solution for describing the scattering of a charged particle by a dipole is presented. The mathematical method for the analytical solution to the radial part of SWE obtained by series expansion in Coulomb wave functions for the two-center Coulomb scattering is found to be more efficient and accurate than the numerical methods. In the conclusion this work provides an analytical solution to the generalized spheroidal wave equation and the Green’s function of one-electron diatomic molecules. [ABSTRACT FROM AUTHOR]

By using the explicit expression for the coefficients bKk(LM|l)[Eq. (16) in J. Math. Phys. 26, 3193 (1985), which is referred to hereafter as Part II], a closed expression for the coefficients cij(NLM|l) appearing in Löwdin’s alpha function (1/r)αl(NLM|a,r) for a Slater-type atomic orbital is obtained. By manipulating its expression mathematically, it was manifested that cij(NLM|l) vanishes if i≥N+L-M+l. For the specific cases that each of i and j takes 0 or 1, cij(NLM|l) take the same value. From the symmetry relation on bKk(LM|l) [Eq. (6) in Part II], it was proved that cij(NLM|l)=cji(NLM|l), provided M=L. Through some manipulation, the recursion formula for cij(NLM|l) in N was derived, while that about M was obtained by introducing the recursion formula for bKk(LM|l) in M [Eq. (24) in Part II] into the expression for cij(NLM|l). For the special cases where N and M both are equal to L, the expression for cij(NLM|l) is given in a compact and symmetric form. The use of this expression leads to a recursion formula for cij(LLL|l) in L. Especially, cij(LLL|l) for L=0 is expressed by a single term, and a recursion formula for cij(000|l) in l was obtained in a very simple form. The systematic use of these recursion formulas leads to a procedure for evaluating cij(NLM|l) successively. Finally, cij(NLM|l) evaluated by this procedure, with the restriction that 2≥N≥L≥0 and 2≥min{L,l}≥M≥0, are given in a matrix form. [ABSTRACT FROM AUTHOR]

The Chandler–Gibson theory of N-body scattering is used to define a new K matrix for N-body quantum scattering systems. The half-on-shell K matrix has the proper channel thresholds and may be computed using a K-matrix form of the CG equations. The on-shell K matrix is an Hermitian matrix for all energies and is related to a unitary scattering matrix via a Cayley transform. A B-spline solution method is developed and applied to a two-body and a three-body test problem. The three-body numerical calculations are within 0.5% of the exact solution both below and above the breakup threshold. [ABSTRACT FROM AUTHOR]

A model of a planar tri-atomic molecule is presented, which is integrable in the Born–Oppenheimer adiabatic approximation. The molecular Hamiltonian is the sum of a nuclear vibrational energy operator and an electronic Hamiltonian, where vibrations of nuclei are defined to be motions with vanishing total angular momentum in the center-of-mass system, and where the electronic Hamiltonian is assumed to be a traceless 2 × 2 Hermitian matrix defined on R ̇ 3 , the shape space of the planar three-body system. Once an eigenvalue of the electronic Hamiltonian is chosen, vibrational-electronic interaction is introduced through covariant differential operators acting on sections of the eigen-line bundle associated with the chosen eigenvalue. The Hamiltonian for nuclear motion coupled with electronic state is then described in terms of these covariant differential operators together with the chosen eigenvalue as a potential for nuclear motion. The eigenvalues of the nuclear Hamiltonian are evaluated for bound states. In the case that vibrational-electronic interaction is restricted to small vibrational-electronic one around a symmetric configuration of the nuclei, a remark is made on a relation to a well-known Hamiltonian describing the dynamic Jahn–Teller effect for a planar tri-atomic molecule X3. [ABSTRACT FROM AUTHOR]

In this paper, we consider the connection between generalized conditional symmetries (GCSs) and pre-Hamiltonian operators. The set of GCSs of an evolutionary partial differential equations system is divided into a union of many linear subspaces by different characteristic operators, and we consider the mappings between two of them, which generalize the recursion operators of symmetries and the pre-Hamiltonian operators. Finally, we give a systematic method to construct infinitely many GCSs for integrable systems, including the Gelfand–Dickey hierarchy and the AKNS-D hierarchy. All time flows in one integrable hierarchy, admitting infinitely many common GCSs. [ABSTRACT FROM AUTHOR]

The Hamiltonian approach to isomonodromic deformation systems is extended to include generic rational covariant derivative operators on the Riemann sphere with irregular singularities of arbitrary Poincaré rank. The space of rational connections with given pole degrees carries a natural Poisson structure corresponding to the standard classical rational R-matrix structure on the dual space L * g l (r) of the loop algebra L g l (r). Nonautonomous isomonodromic counterparts of isospectral systems generated by spectral invariants are obtained by identifying deformation parameters as Casimir elements on the phase space. These are shown to coincide with higher Birkhoff invariants determining local asymptotics near to irregular singular points, together with the pole loci. Pairs consisting of Birkhoff invariants, together with the corresponding dual spectral invariant Hamiltonians, appear as "mirror images" matching, at each pole, the negative power coefficients in the principal part of the Laurent expansion of the fundamental meromorphic differential on the associated spectral curve with the corresponding positive power terms in the analytic part. Infinitesimal isomonodromic deformations are shown to be generated by the sum of the Hamiltonian vector field and an explicit derivative vector field that is transversal to the symplectic foliation. The Casimir elements serve as coordinates complementing those along the symplectic leaves, defining a local symplectomorphism between them. The explicit derivative vector fields preserve the Poisson structure and define a flat transversal connection, spanning an integrable distribution whose leaves may be identified as the orbits of a free Abelian local group action. The projection of infinitesimal isomonodromic deformation vector fields to the quotient manifold under this action gives commuting Hamiltonian vector fields corresponding to the spectral invariants dual to the Birkhoff invariants and the pole loci. [ABSTRACT FROM AUTHOR]

This paper concerns the existence of transonic shocks for steady exothermically reacting Euler flows in an almost flat nozzle with the small rate of the exothermic reaction. One of the key points is to quantitatively determine the position of the shock front in the nozzle. We focus on contributions of the perturbation of the flat nozzle and the exothermic reaction in determining the position of the shock front by comparing the orders of σ and κ, where σ represents the scale of the perturbation of the flat nozzle and κ represents the rate of the exothermic reaction. To this end, a free boundary problem for the linearized reacting Euler system is proposed to catch an approximating position of the shock front and the associated approximating shock solution, with which the existence of a shock solution close to it can be established via a nonlinear iteration scheme. One of the key steps is to solve the nonlinear equation derived from the solvability condition for the elliptic sub-problem in the domain of the subsonic flow behind the shock front, which determines the free boundary of the proposed problem. Four typical cases are analyzed, which describe possible interactions between the geometry of the nozzle boundary and the exothermic reaction. The results also manifest that the exothermic reaction has a stabilization effect on transonic shocks in the nozzles. [ABSTRACT FROM AUTHOR]

In this article, we introduce b-semitoric systems as a generalization of semitoric systems, specifically tailored for b-symplectic manifolds. The objective of this article is to furnish a collection of examples and investigate the distinctive characteristics of these systems. A b-semitoric system is a four-dimensional b-integrable system that satisfies certain conditions: one of its momentum map components is proper and generates an effective global S1-action and all singular points are non-degenerate and devoid of hyperbolic components. To illustrate this concept, we provide five examples of b-semitoric systems by modifying the coupled spin oscillator and the coupled angular momenta, and we also classify their singular points. Additionally, we describe the dynamics of these systems through the image of their respective momentum maps. [ABSTRACT FROM AUTHOR]

In this paper, we constructively solve the Riemann problem for the relativistic Euler equations with the logarithmic equation of state. The concentration and cavitation phenomena are observed and analyzed during the process of vanishing pressure in the Riemann solutions. It is rigorously proved that, as the pressure vanishes, they tend to the two kinds of Riemann solutions to the zero-pressure relativistic Euler equations, which include a delta shock formed by a weighted δ-measure and a vacuum state. [ABSTRACT FROM AUTHOR]

Admissible structure constants related to the dual Lie superalgebras of particular Lie superalgebra (풞3 + 풜) are found by straightforward calculations from the matrix form of super Jacobi and mixed super Jacobi identities which are obtained from adjoint representation. Then, by making use of the automorphism supergroup of the Lie superalgebra (풞3 + 풜), the Lie superbialgebra structures on the Lie superalgebra (풞3 + 풜) are obtained and classified into inequivalent 31 families.We also determine all corresponding coboundary and bi-r-matrix Lie superbialgebras. The quantum deformations associated with some Lie superbialgebras (풞3 + 풜) are obtained, together with the corresponding deformed Casimir elements. As an application of these quantum deformations, we construct a deformed integrable Hamiltonian system from the representation of the Hopf superalgebra Uλ(풞p=12,ϵ⊕ 풜1,1) ( (풞3 + 풜) ). [ABSTRACT FROM AUTHOR]

We construct classes of Z 2 × Z 2 -graded Lie algebras corresponding to the classical Lie algebras in terms of their defining matrices. For the Z 2 × Z 2 -graded Lie algebra of type A, the construction coincides with the previously known class. For the Z 2 × Z 2 -graded Lie algebra of types B, C, and D, our construction is new and gives rise to interesting defining matrices closely related to the classical ones but undoubtedly different. We also give some examples and possible applications of parastatistics. [ABSTRACT FROM AUTHOR]

We study complex product structures on quadratic vector spaces and on quadratic Lie algebras analyzing the Lagrangian and orthogonal splittings associated with them. We show that a Manin triple equipped with generalized metric G + B such that B is an O -operator with extension G of mass −1 can be turned into another Manin triple that admits also an orthogonal splitting in Lie ideals. Conversely, a quadratic Lie algebra orthogonal direct sum of a pair of anti-isomorphic Lie algebras, following similar steps as in the previous case, can be turned into a Manin triple admitting an orthogonal splitting into Lie ideals. [ABSTRACT FROM AUTHOR]

In this paper, we study the boundedness of atmospheric Ekman flows with classical boundary conditions. We consider the system with a two-layer eddy viscosity, consisting of a constant eddy viscosity in the upper layer and a continuous eddy viscosity in the lower layer. We analyze the boundedness of the solution by using the logarithmic matrix norm. [ABSTRACT FROM AUTHOR]

A charged harmonic oscillator in a magnetic field, Landau problems, and an oscillator in a noncommutative space share the same mathematical structure in their Hamiltonians. We have considered a two-dimensional anisotropic harmonic oscillator with arbitrarily time-dependent parameters (effective mass and frequencies), placed in an arbitrarily time-dependent magnetic field. A class of quadratic invariant operators (in the sense of Lewis and Riesenfeld) have been constructed. The invariant operators ( I ̂) have been reduced to a simplified representative form by a linear canonical transformation [the group S p (4 , R) ]. An orthonormal basis of the Hilbert space consisting of the eigenvectors of I ̂ is obtained. In order to obtain the solutions of the time-dependent Schrödinger equation corresponding to the system, both the geometric and dynamical phase-factors are constructed. A generalized Peres–Horodecki separability criterion (Simon's criterion) for the ground state corresponding to our system has been demonstrated. [ABSTRACT FROM AUTHOR]

Coupled-cavity traveling wave tube (CCTWT) is a high power microwave vacuum electronic device used to amplify radio frequency signals. CCTWTs have numerous applications, including radar, radio navigation, space communication, television, radio repeaters, and charged particle accelerators. Microwave-generating interactions in CCTWTs take place mostly in coupled resonant cavities positioned periodically along the electron beam axis. Operational features of a CCTWT, particularly the amplification mechanism, are similar to those of a multicavity klystron. We advance here a Lagrangian field theory of CCTWTs with the space being represented by one-dimensional continuum. The theory integrates into it the space-charge effects, including the so-called debunching (electron-to-electron repulsion). The corresponding Euler–Lagrange field equations are ordinary differential equations with coefficients varying periodically in the space. Utilizing the system periodicity, we develop instrumental features of the Floquet theory, including the monodromy matrix and its Floquet multipliers. We use them to derive closed form expressions for a number of physically significant quantities. Those include, in particular, dispersion relations and the frequency dependent gain foundational to the RF signal amplification. Serpentine (folded, corrugated) traveling wave tubes are very similar to CCTWTs, and our theory applies to them also. [ABSTRACT FROM AUTHOR]

We use the Tomita–Takesaki modular theory and the Kubo–Ando operator mean to write down a large class of multi-state quantum f-divergences and prove that they satisfy the data processing inequality. For two states, this class includes the (α, z)-Rényi divergences, the f-divergences of Petz, and the Rényi Belavkin-Staszewski relative entropy as special cases. The method used is the interpolation theory of non-commutative L ω p spaces, and the result applies to general von Neumann algebras, including the local algebra of quantum field theory. We conjecture that these multi-state Rényi divergences have operational interpretations in terms of the optimal error probabilities in asymmetric multi-state quantum state discrimination. [ABSTRACT FROM AUTHOR]

The Lie superalgebra p s l (2 | 2) is recognized as a quite special algebra. In mathematics, it has the vanishing Killing form and allows for the three-dimensional central extension. In physics, it shows up as the symmetry of the one-dimensional Hubbard model and the asymptotic S-matrix of the anti-de Sitter/conformal field theory (AdS/CFT) correspondence. In this paper, we present the Drinfeld realization of the Yangian algebra associated with the centrally extended Lie superalgebra p s l (2 | 2). Furthermore, we show that it possesses Hopf algebra structures, particularly the coproducts. The idea to prove the existence of manifest coproducts is as follows. First, we shall introduce them to Levendorskii's realization, a system of a finite truncation of Drinfeld generators. Second, we show that Levendorskii's realization is isomorphic to the Drinfeld realization by induction. [ABSTRACT FROM AUTHOR]

We make a number of simplifications in Gour and Friedland's proof of local additivity of the minimum output entropy of a quantum channel. We follow them in reframing the question as one about the entanglement entropy of bipartite states associated with a dB × dE matrix. We use a different approach to reduce the general case to that of a square positive definite matrix. We use the integral representation of the log to obtain expressions for the first and second derivatives of the entropy, and then exploit the modular operator and functional calculus to streamline the proof following their underlying strategy. We also extend this result to the maximum relative entropy with respect to a fixed reference state, which has important implications for studying the superadditivity of the capacity of a quantum channel to transmit classical information. [ABSTRACT FROM AUTHOR]

We study a class of S U (N) Wigner 6j symbols involving two fundamental representations and derive explicit formulas for all 6j symbols in this class. Our formulas express the 6j symbols in terms of the dimensions of the involved representations, and they are thereby functions of N. We view these explicit formulas as a first step toward efficiently decomposing S U (N) color structures in terms of group invariants. [ABSTRACT FROM AUTHOR]

Two modified Boussinesq equations along with their Lax pairs are proposed by introducing the Miura transformations. The modified good Boussinesq equation with initial condition is investigated by the Riemann–Hilbert method. Starting with the three-order Lax pair of this equation, the inverse scattering transform is formulated and the Riemann–Hilbert problem is established, and the properties of the reflection coefficients are presented. Then, the formulas of long-time asymptotics to the good Boussinesq equation and its modified version are given based on the Deift–Zhou approach of nonlinear steepest descent analysis. It is demonstrated that the results from the long-time asymptotic analysis are in excellent agreement with the numerical solutions. This is the first result on the long-time asymptotic behaviors of the good Boussinesq equation with qxx-term and its modified version. [ABSTRACT FROM AUTHOR]

We study the solutions to the Klein–Gordon equation for the massive scalar field in the universal covering space of a two-dimensional anti-de Sitter space. For certain values of the mass parameter, we impose a suitable set of boundary conditions, which make the spatial component of the Klein–Gordon operator self-adjoint. This makes the time-evolution of the classical field well defined. Then, we use the transformation properties of the scalar field under the isometry group of the theory, namely, the universal covering group of S L (2 , R) , in order to determine which self-adjoint boundary conditions are invariant under this group and which lead to the positive-frequency solutions forming a unitary representation of this group and, hence, to a vacuum state invariant under this group. Then, we examine the cases where the boundary condition leads to an invariant theory with a non-invariant vacuum state and determine the unitary representation to which the vacuum state belongs. [ABSTRACT FROM AUTHOR]

The aim of this study is to solve the Riemann problem of isentropic magnetogasdynamics equations for a more realistic version of the extended Chaplygin gas model. The analysis demonstrates that under some special circumstances, delta shock and vacuum appear in the solution, describing the phenomena of concentration and cavitation, respectively. By examining the limiting behavior, it is obtained that solutions coincide with corresponding Riemann solutions of the transport equations when both the magnetic field and pressure drop to zero. [ABSTRACT FROM AUTHOR]

The classical equations of motion of a charged particle in a spherically symmetric distribution of magnetic monopoles can be transformed into a system of linear equations, thereby providing a type of integrability. In the case of a single monopole, the solution was given long ago by Poincaré. In the case of a uniform distribution of monopoles, the solution can be expressed in terms of parabolic cylinder functions (essentially the eigenfunctions of an inverted harmonic oscillator). This solution is relevant to recent studies of nonassociative star products, symplectic lifts of twisted Poisson structures, and fluids and plasmas of electric and magnetic charges. [ABSTRACT FROM AUTHOR]

Classical and quantum walks on some finite paths are introduced. It is shown that these walks have explicit solutions given in terms of exceptional Krawtchouk polynomials, and their properties are explored. In particular, fractional revival is shown to take place in the corresponding quantum walks. [ABSTRACT FROM AUTHOR]

In integrable models, stationary equations for higher symmetries serve as one of the main sources of reductions consistent with dynamics. We apply this method to the non-Abelian two-dimensional Toda lattice. It is shown that already the stationary equation of the simplest higher flow gives a non-trivial non-autonomous constraint that reduces the Toda lattice to a non-Abelian analog of pumped Maxwell–Bloch equations. The Toda lattice itself is interpreted as an auto-Bäcklund transformation acting on the solutions of this system. Further self-similar reduction leads to non-Abelian analogs of the Painlevé III equation. [ABSTRACT FROM AUTHOR]

We consider a problem of separation of variables for Lax-integrable Hamiltonian systems governed by gl(n) ⨂ gl(n)-valued classical r-matrices r(u, v). We find a new class of classical non-skew-symmetric non-dynamical gl(n) ⨂ gl(n)-valued r-matrices rJ(u, v) for which the "magic recipe" of Sklyanin [Prog. Theor. Phys. Suppl. 118, 35 (1995)] in the theory of variable separation is applicable, i.e., for which standard separating functions A(u) and B(u) of Gekhtman [Commun. Math. Phys. 167, 593 (1995)] and Scott ["Classical functional Bethe ansatz for SL(N): Separation of variables for the magnetic chain," arXiv:hep-th 940303] produce a complete set of canonical coordinates satisfying the equations of separation. We illustrate the corresponding separation of variable theory by the example of the anisotropically deformed An Toda models proposed in the work of Skrypnyk [J. Phys. A: Math. Theor. 38, 9665–9680 (2005)] and governed by the r-matrices rJ(u, v) and by the generalized Gaudin models [T. Skrypnyk, Phys. Lett. A 334(5–6), 390 (2005)] governed by the same classical r-matrices. The n = 2 and n = 3 cases are considered in detail. [ABSTRACT FROM AUTHOR]

We interpret the elements of the exceptional Lie algebra e 8 (− 24) as objects in the Standard Model, including lepton and quark spinors with the usual properties, the Standard Model Lie algebra s u (3) ⊕ s u (2) ⊕ u (1) , and the Lorentz Lie algebra s o (3 , 1). Our construction relies on identifying a complex structure on spinors and then working in the enveloping algebra. The resulting model naturally contains Grand Unified Theories based on SO(10) (Georgi), SU(5) (Georgi–Glashow), and SU(4) ×SU(2) ×SU(2) (Pati–Salam). We then briefly speculate on the role of the remaining elements of e 8 and propose a mechanism leading to exactly three generations of particles. [ABSTRACT FROM AUTHOR]

We introduce the notion of a p-adic quantum bit (p-qubit) in the context of the p-adic quantum mechanics initiated and developed after the seminal paper of Volovich [Theor. Math. Phys. 71, 574 (1987)]. In this approach, physics takes place in a three-dimensional p-adic space rather than Euclidean space. Based on our prior work describing the p-adic special orthogonal group [Di Martino et al., arXiv:2104.06228 [math.NT] (2021)], we outline a program to classify its continuous unitary projective representations, which can be interpreted as a theory of p-adic angular momentum. The p-adic quantum bit arises from the irreducible representations of minimal nontrivial dimension two, of which we construct examples for all primes p. [ABSTRACT FROM AUTHOR]

Multicavity Klystron (MCK) is a high power microwave vacuum electronic device used to amplify radio frequency (RF) signals. MCKs have numerous applications, including radar, radio navigation, space communication, television, radio repeaters, and charged particle accelerators. The microwave-generating interactions in klystrons take place mostly in coupled resonant cavities positioned periodically along the electron beam axis. Importantly, there is no electromagnetic coupling between cavities. The cavities are coupled only by the flow of bunched electrons drifting from one cavity to the next. We advance here a Lagrangian field theory of MCKs with the space being represented by a one-dimensional continuum. The theory integrates into it the space-charge effects, including the so-called debunching (electron-to-electron repulsion). The corresponding Euler–Lagrange equations are ordinary differential equations with coefficients varying periodically in the space. Utilizing the system periodicity, we develop the instrumental features of the Floquet theory, including the monodromy matrix and its Floquet multipliers. We use them to derive closed form expressions for a number of physically significant quantities. Those include, in particular, the dispersion relations and the frequency dependent gain foundational to the RF signal amplification. We assume that MCKs operate in the voltage amplification mode associated with the maximal gain. [ABSTRACT FROM AUTHOR]

We complete Dyson's dream by cementing the links between symmetric spaces and classical random matrix ensembles. Previous work has focused on a one-to-one correspondence between symmetric spaces and many but not all of the classical random matrix ensembles. This work shows that we can completely capture all of the classical random matrix ensembles from Cartan's symmetric spaces through the use of alternative coordinate systems. In the end, we have to let go of the notion of a one-to-one correspondence. We emphasize that the KAK decomposition traditionally favored by mathematicians is merely one coordinate system on the symmetric space, albeit a beautiful one. However, other matrix factorizations, especially the generalized singular value decomposition from numerical linear algebra, reveal themselves to be perfectly valid coordinate systems that one symmetric space can lead to many classical random matrix theories. We establish the connection between this numerical linear algebra viewpoint and the theory of generalized Cartan decompositions. This, in turn, allows us to produce yet more random matrix theories from a single symmetric space. Yet, again, these random matrix theories arise from matrix factorizations, though ones that we are not aware have appeared in the literature. [ABSTRACT FROM AUTHOR]

The nonrelativistic motion of a charged particle around a dyon in (9 + 1) spacetime is known as the nine-dimensional McIntosh–Cisneros–Zwanziger–Kepler problem. This problem has been solved exactly by the variable-separation method in three different coordinate systems: spherical, parabolic, and prolate spheroidal. In the present study, we establish a relationship between the variable separation and the algebraic structure of SO(10) symmetry. Each of the spherical, parabolic, or prolate spheroidal bases is proved to be a set of eigenfunctions of a corresponding nonuplet of algebraically independent integrals of motion. This finding also helps us establish connections between the bases by the algebraic method. This connection, in turn, allows calculating complicated integrals of confluent Heun, generalized Laguerre, and generalized Jacobi polynomials, which are important in physics and analytics. [ABSTRACT FROM AUTHOR]

In tri-partite systems, there are three basic biseparability, A-BC, B-CA, and C-AB, according to bipartitions of local systems. We begin with three convex sets consisting of these basic biseparable states in the three-qubit system, and consider arbitrary iterations of intersections and/or convex hulls of them to get convex cones. One natural way to classify tri-partite states is to consider those convex sets to which they belong or do not belong. This is especially useful to classify partial entanglement of mixed states. We show that the lattice generated by those three basic convex sets with respect to convex hull and intersection has infinitely many mutually distinct members to see that there are infinitely many kinds of three-qubit partial entanglement. To do this, we consider an increasing chain of convex sets in the lattice and exhibit three-qubit Greenberger–Horne–Zeilinger diagonal states distinguishing those convex sets in the chain. [ABSTRACT FROM AUTHOR]

We review Kitaev's celebrated "periodic table" for topological phases of condensed matter, which identifies ground states (Fermi projections) of gapped periodic quantum systems up to continuous deformations. We study families of projections that depend on a periodic crystal momentum and respect the symmetries that characterize the various classes of topological insulators. Our aim is to classify such families in a systematic, explicit, and constructive way: we identify numerical indices for all symmetry classes and provide algorithms to deform families of projections whose indices agree. Aiming at simplicity, we illustrate the method for zero- and one-dimensional systems and recover the (weak and strong) topological invariants proposed by Kitaev and others. [ABSTRACT FROM AUTHOR]

The set of bounded observables for a quantum system is represented by the set of bounded self-adjoint operators S(H) on a complex Hilbert space H, and the quantum effects for a physical system can be described by the set E(H) of positive contractive operators on a complex Hilbert space H. In this note, by the techniques of operator block and spectral, we give the simpler representation of A∧P and obtained the new necessary and sufficient conditions for A∨P, for A∈S(H) and P∈P(H), where P(H) is the set of all orthogonal projection operators on H. In particular, we get that if A∨P exists, then A∨P∈E(H) for A∈E(H) and P∈P(H). In addition, we consider the relations between the existence of A∨B, A-∨B-, and A+∨B+, where A+, B+, A-, and B- are the positive and negative parts of A,B∈S(H). [ABSTRACT FROM AUTHOR]

In this work, the double and triple pole soliton solutions for the Gerdjikov–Ivanov type of the derivative nonlinear Schrödinger equation with zero boundary conditions (ZBCs) and nonzero boundary conditions (NZBCs) are studied via the Riemann–Hilbert (RH) method. With spectral problem analysis, we first obtain the Jost function and scattering matrix under ZBCs and NZBCs. Then, according to the analyticity, symmetry, and asymptotic behavior of the Jost function and scattering matrix, the RH problem (RHP) with ZBCs and NZBCs is constructed. Furthermore, the obtained RHP with ZBCs and NZBCs can be solved in the case that reflection coefficients have double or triple poles. Finally, we derive the general precise formulas of N-double and N-triple pole solutions corresponding to ZBCs and NZBCs, respectively. In addition, the asymptotic states of the one-double pole soliton solution and the one-triple pole soliton solution are analyzed when t tends to infinity. The dynamical behaviors for these solutions are further discussed by image simulation. [ABSTRACT FROM AUTHOR]

In this paper, we have illustrated the construction of a real structure on a fuzzy sphere S * 2 in its spin-1/2 representation. Considering the SU(2) covariant Dirac and chirality operator on S * 2 given by U. C. Watamura and Watamura [Commun. Math. Phys. 183, 365–382 (1997) and Commun. Math. Phys. 212, 395–413 (2000)], we have shown that the real structure is consistent with other spectral data for KO dimension-4 fulfilling the zero order condition, where we find it necessary to enlarge the symmetry group from SO(3) to the full orthogonal group O(3). However, the first order condition is violated, thus paving the way to construct a toy model for an SU(2) gauge theory to capture some features of physics beyond the standard model following Chamseddine et al. (J. High Energy Phys. 2013, 132). [ABSTRACT FROM AUTHOR]

We show that if the Wigner function of a (possibly mixed) quantum state decays toward infinity faster than any polynomial in the phase space variables x and p, then so do all its derivatives, i.e., it is a Schwartz function on phase space. This is equivalent to the condition that the Husimi function is a Schwartz function, that the quantum state is a Schwartz operator in the sense of Keyl et al. [Rev. Math. Phys. 28(03), 1630001 (2016)], and, in the case of a pure state, that the wavefunction is a Schwartz function on configuration space. We discuss the interpretation of this constraint on Wigner functions and provide explicit bounds on Schwartz seminorms. [ABSTRACT FROM AUTHOR]