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In this paper, we study the reducibility of time quasiperiodic perturbations of the quantum harmonic or anharmonic oscillator in one space dimension. We modify known algorithms obtaining a reducibility result which allows us to deal with perturbations of order strictly larger than the ones considered in all the previous papers. [ABSTRACT FROM AUTHOR]

In theories such as teleparallel gravity and its extensions, the frame basis replaces the metric tensor as the primary object of study. A choice of coordinate system, frame basis, and spin-connection must be made to obtain a solution from the field equations of a given teleparallel gravity theory. It is worthwhile to express solutions in an invariant manner in terms of torsion invariants to distinguish between different solutions. In this paper, we discuss the symmetries of teleparallel gravity theories, describe the classification of the torsion tensor and its covariant derivative, and define scalar invariants in terms of the torsion. In particular, we propose a modification of the Cartan–Karlhede algorithm for geometries with torsion (and no curvature or nonmetricity). The algorithm determines the dimension of the symmetry group for a solution and suggests an alternative frame-based approach to calculating symmetries. We prove that the only maximally symmetric solution to any theory of gravitation admitting a non-zero torsion tensor is Minkowski space. As an illustration, we apply the algorithm to six particular exact teleparallel geometries. From these examples, we notice that the symmetry group of the solutions of a teleparallel gravity theory is potentially smaller than their metric-based analogs in general relativity. [ABSTRACT FROM AUTHOR]

Perturbations to Mei symmetry and the numerical algorithm of disturbed nonholonomic systems are studied under total variational discretization. The discrete equations on regular lattices of nonholonomic systems in the undisturbed and the disturbed cases are presented. The determining equations of Mei symmetry are established for undisturbed and disturbed systems. The exact invariants of Noether type led by Mei symmetry for undisturbed nonholonomic systems are given under infinitesimal transformations of Lie groups. For discrete disturbed nonholonomic systems, the condition of existence of adiabatic invariants led by perturbation to Mei symmetry and their forms are presented. The numerical simulations demonstrate that the geometric numerical algorithm has a higher precision and longer time stability than the standard numerical method. [ABSTRACT FROM AUTHOR]

The Helon model identifies standard model quarks and leptons with framed braids composed of three ribbons joined together at both ends by a connecting node (disk). These surfaces with boundary are called braided 3-belts. The twisting and braiding of ribbons composing braided 3-belts are interchangeable, and any braided 3-belt can be written in a pure twist form with trivial braiding, specified by a vector of three multiples of half integers [a, b, c], a topological invariant. This paper identifies the set of braided 3-belts that can be written in a braid only form in which all twisting is eliminated instead. For these braids, an algorithm to calculate the braid word is determined which allows the braid word of every braided 3-belt to be written in a canonical form. It is furthermore demonstrated that the set of braided 3-belts does not form a group due to a lack of isogeny. [ABSTRACT FROM AUTHOR]

Recently a simple differential algorithm to compute all the zeros of a generic polynomial was introduced. In this paper an analogous, but finite-difference, algorithm is introduced and discussed. At the end of the paper a minor generalization of the differential algorithm is also mentioned. [ABSTRACT FROM AUTHOR]

This paper is endeavored to present a new version of the LSMR algorithm for solving the linear least squares problem in quaternion field, by means of direct quaternion arithmetics rather than the usually used real or complex representation methods. The present new algorithm is based on the classical Golub-Kahan bidiagonalization process, but is instead of using two QR factorizations. It has several advantages as follows: (i) does not make the scale of the problem dilate exponentially, compared to the conventional complex representation or real representation methods, (ii) has monotonic and smooth convergence behavior, compared to the Q-LSQR algorithm, and (iii) the new algorithm is more straightforward, and there is no expensive matrix inversion or decomposition. It may reduce the number of iterations in some cases. The performances of the algorithm are illustrated by some numerical experiments. [ABSTRACT FROM AUTHOR]

We discuss an embedding of su(n) rank two antisymmetric supercharges in the su(2, 2|dn) superalgebra, where dn = n(n − 1)/2. We describe an algorithm to construct the explicit form of the generators of the superalgebra. [ABSTRACT FROM AUTHOR]

Intersection numbers of twisted cocycles arise in mathematics in the field of algebraic geometry. Quite recently, they appeared in physics: Intersection numbers of twisted cocycles define a scalar product on the vector space of Feynman integrals. With this application, the practical and efficient computation of intersection numbers of twisted cocycles becomes a topic of interest. An existing algorithm for the computation of intersection numbers of twisted cocycles requires in intermediate steps the introduction of algebraic extensions (for example, square roots) although the final result may be expressed without algebraic extensions. In this article, I present an improvement of this algorithm, which avoids algebraic extensions. [ABSTRACT FROM AUTHOR]

We establish a polynomial-time approximation algorithm for partition functions of quantum spin models at high temperature. Our algorithm is based on the quantum cluster expansion of Netočný and Redig and the cluster expansion approach to designing algorithms due to Helmuth, Perkins, and Regts. Similar results have previously been obtained by related methods, and our main contribution is a simple and slightly sharper analysis for the case of pairwise interactions on bounded-degree graphs. [ABSTRACT FROM AUTHOR]

We generalize an efficient exact synthesis algorithm for single-qubit unitaries over the Clifford+T gate set which was presented by Kliuchnikov, Maslov, and Mosca [Quantum Inf. Comput. 13(7,8), 607-630 (2013)]. Their algorithm takes as input an exactly synthesizable single-qubit unitary--one which can be expressed without error as a product of Clifford and T gates--and outputs a sequence of gates which implements it. The algorithm is optimal in the sense that the length of the sequence, measured by the number of T gates, is smallest possible. In this paper, for each positive even integer n, we consider the "Clifford-cyclotomic" gate set consisting of the Clifford group plus a z-rotation by π/n. We present an efficient exact synthesis algorithm which outputs a decomposition using the minimum number of π/n z-rotations. For the Clifford+T case n = 4, the group of exactly synthesizable unitaries was shown to be equal to the group of unitaries with entries over the ring ℤ[eiπ/n, 1/2]. We prove that this characterization holds for a handful of other small values of n but the fraction of positive even integers for which it fails to hold is 100%. [ABSTRACT FROM AUTHOR]

Separated variables for a classical GL(3) magnetic chain are coordinates of a generic positive divisor D of degree n on a genus g nonhyperelliptic algebraic curve. Because n > g, this divisor D has unique representative ρ(D) in the Jacobian, which can be constructed by using dim|D| = n − g steps of Abel’s algorithm. We study the properties of the corresponding chain of divisors and prove that the classical GL(3) magnetic chain is a superintegrable system with dim|D| = 2 superintegrable Hamiltonians. [ABSTRACT FROM AUTHOR]

We consider the entropy dimension of G-shifts of finite type for the case where G is a non-Abelian monoid. Entropy dimension tells us whether a shift space has zero topological entropy. Suppose the Cayley graph CG of G has a finite representation (that is, {CgG : g ∈ G} is a finite set up to graph isomorphism), and relations among generators of G are determined by a matrix A. We reveal an association between the characteristic polynomial of A and the finite representation of the Cayley graph. After introducing an algorithm for the computation of the entropy dimension, the set of entropy dimensions is related to a collection of matrices in which the sum of each row of every matrix is bounded by the number of leaves of the graph. Furthermore, the algorithm extends to G having finitely many free-followers. [ABSTRACT FROM AUTHOR]

An algorithm to compute Connes spectral distance, adaptable to the Hilbert-Schmidt operatorial formulation of non-commutative quantum mechanics, was developed earlier by introducing the appropriate spectral triple and used to compute infinitesimal distances in the Moyal plane, revealing a deep connection between geometry and statistics. In this paper, using the same algorithm, the Connes spectral distance has been calculated in the Hilbert-Schmidt operatorial formulation for the fuzzy sphere whose spatial coordinates satisfy the su(2) algebra. This has been computed for both the discrete and the Perelemov’s SU(2) coherent state. Here also, we get a connection between geometry and statistics which is shown by computing the infinitesimal distance between mixed states on the quantum Hilbert space of a particular fuzzy sphere, indexed by n ∈ Z/2. [ABSTRACT FROM AUTHOR]

The paper is concerned with relations between local symmetries of systems of partial differential equations (PDE) within trees of nonlocally related PDE systems. It is shown that potential systems arising from a given system through linearly independent conservation laws are nonlocally related to each other. Further, a theorem is proven stating that for a PDE system which has precisely n linearly independent local conservation laws, any local symmetry of the PDE system is a projection of some local symmetry of the n-plet potential system. Moreover, a criterion is presented to determine whether or not a specific local symmetry of a given PDE system is a projection of some local symmetry of a specific potential system. Examples are considered. Finally, a formula for a symmetry of a given PDE system in terms of a local symmetry of a nonlocally related subsystem is given. The formula can be used to determine whether a symmetry of the subsystem yields a local or a nonlocal symmetry of the given system, without the need to undertake a full symmetry classification and comparison between the given system and the subsystem. [ABSTRACT FROM AUTHOR]

In this paper, we propose an algorithm to construct coherent states for an exactly solvable position dependent mass Schrödinger equation. We use point canonical transformation method and obtain ground state eigenfunction of the position dependent mass Schrödinger equation. We fix the ladder operators in the deformed form and obtain explicit expression of the deformed superpotential in terms of mass distribution and its derivative. We also prove that these deformed operators lead to minimum uncertainty relations. Further, we illustrate our algorithm with two examples, in which the coherent states given for the second example are new. [ABSTRACT FROM AUTHOR]

The objective of this paper is to create a new kind of dynamical systems—a quantum-classical hybrid—that would preserve superposition and entanglement of random solutions while allowing one to measure their state variables by using classical methods. Such an optimal combination of characteristics is a perfect match for quantum-inspired computing. The model is represented by a modified Madelung equation in which the quantum potential is replaced by a different, specially chosen “computational” potential. As a result, the dynamics attains both quantum and classical properties. Similarities and differences of the proposed model with quantum systems are outlined. As an application, an algorithm for the global maximum of an arbitrary integrable function is proposed. The idea of the proposed algorithms is very simple: based on the quantum-inspired maximizer, introduce a positive function to be maximized as the probability density to which the solution is attracted. Then, the larger value of this function will have the higher probability to appear. Special attention is paid to the simulation of integer programming, NP-complete problems and information retrieval. [ABSTRACT FROM AUTHOR]

Reduction of a state of a quantum system to a subsystem gives partial quantum information about the true state of the total system. In connection with optimal state determination for two qubits, the question was raised about the maximum number of pairwise complementary reductions. The main result of the paper tells that the maximum number is 4, that is, if A1,A2,...,Ak are pairwise complementary (or quasiorthogonal) subalgebras of the algebra M4(C) of all 4×4 matrices and they are isomorphic to M2(C), then k≼4. The proof is based on a Cartan decomposition of SU(4). In the way to the main result, contributions are made to the understanding of the structure of complementary reductions. [ABSTRACT FROM AUTHOR]

In this paper we construct quantum mechanical observables of a single free particle that lives on the surface of the two-sphere S2 by implementing the Fedosov *-formalism. The Fedosov * is a generalization of the Moyal star product on an arbitrary symplectic manifold. After their construction we show that they obey the standard angular momentum commutation relations in ordinary nonrelativistic quantum mechanics. The purpose of this paper is threefold. One is to find an exact, nonperturbative solution of these observables. The other is to verify that the commutation relations of these observables correspond to angular momentum commutation relations. The last is to show a more general computation of the observables in Fedosov *-formalism; essentially an undeformation of Fedosov’s algorithm. [ABSTRACT FROM AUTHOR]

By means of complex representation and companion vector, in this paper we introduce a definition of rank of a quaternion matrix, study the problems of quaternionic linear equations, and obtain an algorithm for quaternionic linear equations in quaternionic quantum theory. [ABSTRACT FROM AUTHOR]

Recent works have shown that quantum computers can polynomially speed up certain SAT-solving algorithms even when the number of available qubits is significantly smaller than the number of variables. Here, we generalize this approach. We present a framework for hybrid quantum-classical algorithms which utilize quantum computers significantly smaller than the problem size. Given an arbitrarily small ratio of the quantum computer to the instance size, we achieve polynomial speedups for classical divide-and-conquer algorithms, provided that certain criteria on the time- and space-efficiency are met. We demonstrate how this approach can be used to enhance Eppstein's algorithm for the cubic Hamiltonian cycle problem and achieve a polynomial speedup for any ratio of the number of qubits to the size of the graph. [ABSTRACT FROM AUTHOR]

General solutions of nonlinear ordinary differential equations (ODEs) are in general difficult to find; although, powerful integrability techniques exist in the literature for this purpose. It has been shown that in some scalar cases particular solutions may be found with little effort if it is possible to factorize the equation in terms of first-order differential operators. In our present study, we use this factorization technique to address the problem of finding solutions of a system of general two-coupled Liénard-type nonlinear differential equations. We describe a generic algorithm to identify specific classes of Liénard-type systems for which solutions may be found. We demonstrate this method by identifying a class of two-coupled equations for which the particular solution can be found by solving a Bernoulli equation. This class of equations include coupled generalization of the modified Emden equation. We further deduce the general solution of a class of coupled ODEs using the factorization procedure discussed in this paper. [ABSTRACT FROM AUTHOR]

Quantum entanglement plays a central role in quantum information processing. A main objective of the theory is to classify different types of entanglement according to their interconvertibility through manipulations that do not require additional entanglement to perform. While bipartite entanglement is well understood in this framework, the classification of entanglements among three or more subsystems is inherently much more difficult. In this paper, we study pure state entanglement in systems of dimension 2xmxn. Two states are considered equivalent if they can be reversibly converted from one to the other with a nonzero probability using only local quantum resources and classical communication (SLOCC). We introduce a connection between entanglement manipulations in these systems and the well-studied theory of matrix pencils. All previous attempts to study general SLOCC equivalence in such systems have relied on somewhat contrived techniques which fail to reveal the elegant structure of the problem that can be seen from the matrix pencil approach. Based on this method, we report the first polynomial-time algorithm for deciding when two 2xmxn states are SLOCC equivalent. We then proceed to present a canonical form for all 2xmxn states based on the matrix pencil construction such that two states are equivalent if and only if they have the same canonical form. Besides recovering the previously known 26 distinct SLOCC equivalence classes in 2x3xn systems, we also determine the hierarchy between these classes. [ABSTRACT FROM AUTHOR]

In this paper, we present a method for solving a time-harmonic acoustic inverse scattering problem for a sound-soft crack in R2. Based on the integral equation method, our method splits the nonlinear, severely ill-posed inverse problem into a linear well-posed direct problem and a nonlinear ill-posed problem. At this setting, not only the regularized Newton’s method can still be used to solve the inverse problem numerically but also with the advantage of keeping the structure of the algorithm simple and efficient. [ABSTRACT FROM AUTHOR]

In this paper, we present a study of discrete time quantum walks whose underlying graph is a d-dimensional lattice. The dynamical behavior of these systems is of current interest because of their applications in quantum information theory as tools to design quantum algorithms. We assume that, at each step of the walk evolution, the coin transformation is allowed to change so that we can use it as a control variable to drive the evolution in a desired manner. We give an exact description of the possible evolutions and of the set of possible states that can be achieved with such a system. In particular, we show that it is possible to go from a state where there is probability 1 for the walker to be found in a vertex to a state where all the vertices have equal probability. We also prove a number of properties of the set of admissible states in terms of the number of steps needed to obtain them. We provide explicit algorithms for state transfer in low dimensional cases as well as results that allow to reduce algorithms on two-dimensional lattices to algorithms on the one-dimensional lattice, the cycle. [ABSTRACT FROM AUTHOR]

In this paper, we derive a new ten-point multisymplectic scheme for the regularized long-wave equation from its Bridges’ multisymplectic form. The new scheme is an explicit scheme in the sense that it does not need iteration. We discuss some properties of the new scheme. The performance and the efficiency of the new scheme are illustrated by solving several test examples. The obtained results are presented and compared with previous methods. Numerical results indicate that the multisymplectic scheme cannot only obtain satisfied solutions for the regularized long-wave equation but also keep three invariants of motion which are evaluated to determine the conservation properties of the algorithm very well. [ABSTRACT FROM AUTHOR]

We present graphics processing unit accelerated implementations of Markov chain algorithms to sample random tilings, dimers, and the six vertex model. [ABSTRACT FROM AUTHOR]

It is well known that the equations governing the evolution of scalar, electromagnetic, and gravitational perturbations of the background geometry of a Schwarzschild black hole can be reduced to a single master equation. We use Kovacic’s algorithm to obtain all Liouvillian solutions, i.e., essentially all solutions in terms of quadratures, of this master equation. We prove that the algebraically special Liouvillian solutions χ and χ ∫ d r * χ 2 , initially found by Chandrasekhar in the gravitational case, are the only Liouvillian solutions to the master equation. We show that the Liouvillian solution χ ∫ d r * χ 2 is a product of elementary functions, one of them being a polynomial solution P to an associated confluent Heun equation. P admits a finite expansion both in terms of truncated confluent hypergeometric functions of the first kind, and also in terms of associated Laguerre polynomials. Remarkably both expansions entail not constant coefficients but appropriate function coefficients instead. We highlight the relation of these results with inspiring new developments. Our results set the stage for deriving similar results in other black hole geometries 4-dim and higher. [ABSTRACT FROM AUTHOR]

In this work, a complete tree of nonlocally related partial differential equations of Chaplygin gas equations is constructed. This tree includes the systems, which are obtained through local conservation laws and the local symmetry based method. We classify the nonlocal symmetries from potential systems as well as inverse potential systems (IPSs). Furthermore, we propose a systematic algorithm for identification of nonlocal symmetries through IPSs by combining the ideas in the studies of Bluman and Yang [J. Math. Phys. 54, 093504 (2013)] and Yang and Cheviakov [J. Math. Phys. 55, 083514 (2014)]. Finally, we obtain a new exact solution through nonlocal symmetry analysis and physical behavior of solution is presented. [ABSTRACT FROM AUTHOR]

In this paper, we rigorously prove that the complete set of Riemann tensor invariants given by Sneddon [J. Math. Phys. 40, 5905 (1999)] is both minimal and complete. Furthermore, we provide a two-stage algorithm for the explicit construction of polynomial syzygies relating any dependent Riemann tensor invariant to members of the complete set. [ABSTRACT FROM AUTHOR]

Landscape paradigm is ubiquitous in physics and other natural sciences, but it has to be supplemented with both quantitative and qualitatively meaningful tools for analyzing the topography of a given landscape. We here consider dynamic explorations of the relief and introduce as basic topographic features “wells of duration T and altitude y.” We determine an intrinsic exploration mechanism governing the evolutions from an initial state in the well up to its rim in a prescribed time, whose finite-difference approximations on finite grids yield a constructive algorithm for determining the wells. Our main results are thus (i) a quantitative characterization of landscape topography rooted in a dynamic exploration of the landscape, (ii) an alternative to stochastic gradient dynamics for performing such an exploration, (iii) a constructive access to the wells, and (iv) the determination of some bare dynamic features inherent to the landscape. The mathematical tools used here are not familiar in physics: They come from set-valued analysis (differential calculus of set-valued maps and differential inclusions) and viability theory (capture basins of targets under evolutionary systems) that have been developed during the last two decades; we therefore propose a minimal Appendix exposing them at the end of this paper to bridge the possible gap. [ABSTRACT FROM AUTHOR]

In this paper, we give a realization of crystal bases of the fundamental representations over Uq(Cn(1)) in terms of Young diagrams k introduced by Premat. Further, we give a generalized Lascoux–Leclerc–Thibon algorithm for computing the global bases. [ABSTRACT FROM AUTHOR]

By means of complex representation and companion vector, this paper studies the problems of eigenvalues and eigenvectors of quaternion matrices, and gives a technique of computing the eigenvalues and eigenvectors of the quaternion matrices in quaternionic quantum mechanics. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The paper deals with inhomogeneous medium Green’s functions in the phase-space domain by which the phase-space (local) spectral distributions of the field, scattered by a high contrast object due a genetic time-harmonic incidence, are evaluated. Two forms of phase-space Green’s functions are considered: one that links induced sources in the configuration-space to phase-space distributions of the scattered field, while the other one directly links the phase-space distribution of the incident field to phase-space distributions of the scattered field. The scattering mechanism is described in terms of local samplings of the object function which are localized in the object domain according to the scattered- and incidence-processing parameters. Applications in the field of inverse scattering may be expected to yield fast and efficient algorithms, due to the capability of analytically evaluating (forward) scattering Green’s functions. © 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]

The aim of this paper is to develop a constraint algorithm for time-dependent Lagrangian systems which permit us to solve the motion equations. This algorithm extends the Gotay and Nester algorithm for autonomous Lagrangians which is, in fact, a particular case. To do this the almost stable tangent geometry of the evolution space and the notion of cosymplectic structure are used. [ABSTRACT FROM AUTHOR]

A numerical algorithm that computes the decomposition of any finite-dimensional unitary reducible representation of a compact Lie group is presented. The algorithm, which does not rely on an algebraic insight into the group structure, is inspired by quantum mechanical notions. After generating two adapted states (these objects will be conveniently defined in Definition II.1) and after appropriate algebraic manipulations, the algorithm returns the block matrix structure of the representation in terms of its irreducible components. It also provides an adapted orthonormal basis. The algorithm can be used to compute the Clebsch--Gordan coefficients of the tensor product of irreducible representations of a given compact Lie group. The performance of the algorithm is tested on various examples: the decomposition of the regular representation of two finite groups and the computation of Clebsch--Gordan coefficients of two examples of tensor products of representations of SU(2). [ABSTRACT FROM AUTHOR]

A q-difference lattice Boussinesq equation and its confluent form are proposed. With the help of determinant identity, their molecule solutions are constructed. It is shown that the q-difference lattice Boussinesq equation can be used as a numerical convergence acceleration algorithm to compute the approximation limt → ∞f(t). Numerical examples with the application of this algorithm are presented. [ABSTRACT FROM AUTHOR]

Recently, Jafarizadeh et al. [ J. Phys. A: Math. Theor. 40, 4949 (2007)] have given a method for calculation of effective resistance (two-point resistance) on distance-regular networks, where the calculation was based on stratification introduced by Jafarizadeh and Salimi [J. Phys. A 39, 1 (2006)] and Stieltjes transform of the spectral distribution (Stieltjes function) associated with the network. Also,Jafarizadeh et al. [ J. Phys. A: Math. Theor. 40, 4949 (2007)] have shown that effective resistances between a node α and all nodes β belonging to the same stratum with respect to α (Rαβ(i), β belonging to the ith stratum with respect to α) are the same. In this work, an algorithm for recursive calculation of the effective resistances in an arbitrary distance-regular resistor network is provided, where the derivation of the algorithm is based on the Bose–Mesner algebra, stratification of the network, spectral techniques, and Christoffel–Darboux identity. It is shown that the effective resistance on a distance-regular network is a strictly increasing function of the shortest path distance defined on the network. In other words, the effective resistance Rαβ(m+1) is strictly larger than Rαβ(m). The link between effective resistance and random walks on distance-regular networks is discussed, where average commute time and its square root (called Euclidean commute time) as distance are related to effective resistance. Finally, for some important examples of finite distance-regular networks, effective resistances are calculated. [ABSTRACT FROM AUTHOR]

The aim of the paper is to provide a constructive method for recovering a unitary matrix from experimental data. Since there is a natural immersion of unitary matrices within the set of double stochastic ones, the problem to solve is to find necessary and sufficient criteria that separate the two sets. A complete solution is provided for the three-dimensional case, accompanied by a χ2 test necessary for the reconstruction of a unitary matrix from error affected data. [ABSTRACT FROM AUTHOR]

An algorithmic method to exploit a general class of infinitesimal symmetries for reducing stochastic differential equations is presented, and a natural definition of reconstruction, inspired by the classical reconstruction by quadratures, is proposed. As a side result, the well-known solution formula for linear one-dimensional stochastic differential equations is obtained within this symmetry approach. The complete procedure is applied to several examples with both theoretical and applied relevance. [ABSTRACT FROM AUTHOR]

We consider here special Poisson brackets given by the "averaging" of local multidimensional Poisson brackets in the Whitham method. For the brackets of this kind it is natural to ask about their canonical forms, which can be obtained after transformations preserving the "physical meaning" of the field variables. We show here that the averaged bracket can always be written in the canonical form after a transformation of "Hydrodynamic Type" in the case of absence of annihilators of initial bracket. However, in general case the situation is more complicated. As we show here, in more general case the averaged bracket can be transformed to a "pseudo-canonical" form under some special ("physical") requirements on the initial bracket. [ABSTRACT FROM AUTHOR]

Entanglement is a resource to overcome the natural restriction of operations used for state manipulation to Local Operations assisted by Classical Communication (LOCC). Hence, a bipartite maximally entangled state is a state which can be transformed deterministically into any other state via LOCC. In the multipartite setting no such state exists. There, rather a whole set, the Maximally Entangled Set of states (MES), which we recently introduced, is required. This set has on the one hand the property that any state outside of this set can be obtained via LOCC from one of the states within the set and on the other hand, no state in the set can be obtained from any other state via LOCC. Recently, we studied LOCC transformations among pure multipartite states and derived the MES for three and generic four qubit states. Here, we consider the non-generic four qubit states and analyze their properties regarding local transformations. As already the most coarse grained classification, due to Stochastic LOCC (SLOCC), of four qubit states is much richer than in case of three qubits, the investigation of possible LOCC transformations is correspondingly more difficult. We prove that most SLOCC classes show a similar behavior as the generic states, however we also identify here three classes with very distinct properties. The first consists of the GHZ and W class, where any state can be transformed into some other state non-trivially. In particular, there exists no isolation. On the other hand, there also exist classes where all states are isolated. Last but not least we identify an additional class of states, whose transformation properties differ drastically from all the other classes. Although the possibility of transforming states into local-unitary inequivalent states by LOCC turns out to be very rare, we identify those states (with exception of the latter class) which are in the MES and those, which can be obtained (transformed) non-trivially from (into) other states respectively. These investigations do not only identify the most relevant classes of states for LOCC entanglement manipulation, but also reveal new insight into the similarities and differences between separable and LOCC transformations and enable the investigation of LOCC transformations among arbitrary four qubit states. [ABSTRACT FROM AUTHOR]

Boson realizations map operators and states of groups to transformations and states of bosonic systems. We devise a graph-theoretic algorithm to construct the boson realizations of the canonical SU(n) basis states, which reduce the canonical subgroup chain, for arbitrary n. The boson realizations are employed to construct Dfunctions, which are the matrix elements of arbitrary irreducible representations, of SU(n) in the canonical basis. We demonstrate that our D-function algorithm offers significant advantage over the two competing procedures, namely, factorization and exponentiation. [ABSTRACT FROM AUTHOR]

It is shown that, if h(x) is any continuous function defined on some interval [-a,b][contained_within](-1,1) of the real axis, then, in general, its best L2 approximant, in the class of functions holomorphic and bounded by unity in the unit disk of the complex plane, is a finite Blaschke product. An upper bound is placed on the number of factors of the latter and a method for its construction is given. The paper contains a discussion of the use of these results in performing a stable analytic continuation of a set of data points under a condition of uniform boundedness, as well as some numerical examples. [ABSTRACT FROM AUTHOR]

Vector coherent state theory is used to give an algorithm for constructing the so(5) irreps of type [N,0] and [N,N] in the physical so(3) basis. The construction is remarkably simple and permits the matrices of the so(5) operators, for irreps of low-dimension and/or small multiplicities, to be computed by hand. The construction is applied to the [2,0] and [2,2] irreps to illustrate the process. [ABSTRACT FROM AUTHOR]

A direct and systematic algorithm is proposed to find one-dimensional optimal system for the group invariant solutions, which is attributed to the classification of its corresponding one-dimensional Lie algebra. Since the method is based on different values of all the invariants, the process itself can both guarantee the comprehensiveness and demonstrate the inequivalence of the optimal system, with no further proof. To leave the algorithm clear, we illustrate each stage with a couple of well-known examples: the Korteweg-de Vries equation and the heat equation. Finally, we apply our method to the Novikov equation and use the found optimal system to investigate the corresponding invariant solutions. [ABSTRACT FROM AUTHOR]

The no-signaling approach to nonlocality deals with separable and inseparable multiparty correlations in the same set of probability states without conflicting causality. The set of half-spaces describing the polytope of no-signaling probability states that are admitted by the most general class of Bell scenarios is formulated in full detail. An algorithm for determining the skeleton that solves the no-signaling description is developed upon a new strategy that is partially pivoting and partially incremental. The algorithm is formulated rigorously and its implementation is shown to be effective to deal with the highly degenerate no-signaling descriptions. Several applications of the algorithm as a tool for the study of quantum nonlocality are mentioned. Applied to a large set of bipartite Bell scenarios, we found that the corresponding no-signaling polytopes have a striking high degeneracy that grows up exponentially with the size of the Bell scenario. [ABSTRACT FROM AUTHOR]

We give a new proof of a formula for the fusion rules for type A2 due to Bégin, Mathieu, and Walton. Our approach is to symbolically evaluate the Kac-Walton algorithm. [ABSTRACT FROM AUTHOR]

A covariant algorithm for deriving the conserved quantities for natural Hamiltonian systems is combined with the non-relativistic framework of Eisenhart, and of Duval, in which the classical trajectories arise as geodesics in a higher dimensional space-time, realized by Brinkmann manifolds. Conserved quantities which are polynomial in the momenta can be built using time-dependent conformal Killing tensors with flux. The latter are associated with terms proportional to the Hamiltonian in the lower dimensional theory and with spectrum generating algebras for higher dimensional quantities of order 1 and 2 in the momenta. Illustrations of the general theory include the Runge-Lenz vector for planetary motion with a time-dependent gravitational constant G(t), motion in a time-dependent electromagnetic field of a certain form, quantum dots, the Hénon-Heiles and Holt systems, respectively, providing us with Killing tensors of rank that ranges from one to six. [ABSTRACT FROM AUTHOR]

We give an algorithm which produces a unique element of the Clifford group on n qubits (Cn) from an integer 0 ≤ i < |Cn| (the number of elements in thegroup). The algorithm involves O(n³) operations and provides, in addition to a canonical mapping from the integers to group elements g, a factorization of g into a sequence of at most 4n symplectic transvections. The algorithm can be used to efficiently select random elements of Cn which are often useful in quantum information theory and quantum computation. We also give an algorithm for the inverse map, indexing a group element in time O(n³). [ABSTRACT FROM AUTHOR]

We introduce the confluent version of the quantum-mechanical supersymmetry formalism for the Dirac equation with a pseudoscalar potential. Application of the formalism to spectral problems is discussed, regularity conditions for the transformed potentials are derived, and normalizability of the transformed solutions is established. Our findings extend and complement former results [L. M. Nieto, A. A. Pecheritsin, and B. F. Samsonov, "Intertwining technique for the one-dimensional stationary Dirac equation," Ann. Phys.305, 151-189 (2003)]. [ABSTRACT FROM AUTHOR]