Search

Six fine gradings on the real form e 6 , − 14 are described, precisely those ones coming from fine gradings on the complexified algebra. The universal grading groups are Z 2 3 × Z 3 2 , Z 2 6 , Z × Z 2 4 , Z 2 7 , Z × Z 2 5 , and Z 2 × Z 2 3 . [ABSTRACT FROM AUTHOR]

A system of stochastic delayed reaction-diffusion equations with multiplicative noise and deterministic non-autonomous forcing is considered. We first prove the existence and uniqueness of a bi-spatial pullback attractor for these equations when the initial space is C − ρ , 0 , L 2 O and the terminate space is C − ρ , 0 , H 0 1 O . The asymptotic compactness of solutions in C − ρ , 0 , H 0 1 O is established by combining “positive and negative truncations” technique and some new estimates on solutions. Then the periodicity of the random attractors is proved when the stochastic delay equations are forced by periodic functions. Finally, upper semicontinuity of the global random attractors in the delay is obtained as the length of time delay approaches zero. [ABSTRACT FROM AUTHOR]

In this paper, we extend the Remling's theorem on canonical systems that the ω limit points of the Hamiltonian under the shift map are reflectionless on the support of the absolutely continuous part of the spectral measure of a canonical system. [ABSTRACT FROM AUTHOR]

Following Richardson and using results of Kesten on first-passage percolation, we obtain an upper bound on the expected perimeter in an Eden growth process. Using results of the author from a problem in statistical mechanics, we show that the average perimeter of the lattice animals resulting from a very natural family of "growth histories" does not obey a similar bound. [ABSTRACT FROM AUTHOR]

We show that a real eigenfunction of the Schrödinger operator changes sign near some point in ℝn under a suitable assumption on the potential. [ABSTRACT FROM AUTHOR]

In this paper, we first construct a class of weight modules, called ϕ-imaginary Verma modules, for the toroidal Lie algebras. Then a criterion for the irreducibility of the ϕ-imaginary Verma modules is obtained and the irreducible quotients for the reducible ones are studied. Furthermore, we construct a more general class of Zn-graded modules for the toroidal Lie algebras and we discuss their irreducibility. This class of modules includes the ϕ-imaginary Verma modules as special examples [ABSTRACT FROM AUTHOR]

The complete symmetry group classification of the fourth-order Euler–Bernoulli ordinary differential equation, where the elastic modulus and the area moment of inertia are constants and the applied load is a function of the normal displacement, is obtained. We perform the Lie and Noether symmetry analysis of this problem. In the Lie analysis, the principal Lie algebra which is one dimensional extends in four cases, viz. the linear, exponential, general power law, and a negative fractional power law. It is further shown that two cases arise in the Noether classification with respect to the standard Lagrangian. That is, the linear case for which the Noether algebra dimension is one less than the Lie algebra dimension as well as the negative fractional power law. In the latter case the Noether algebra is three dimensional and is isomorphic to the Lie algebra which is sl(2,R). This exceptional case, although admitting the nonsolvable algebra sl(2,R), remarkably allows for a two-parameter family of exact solutions via the Noether integrals. The Lie reduction gives a second-order ordinary differential equation which has nonlocal symmetry. [ABSTRACT FROM AUTHOR]

Galilean conformal algebra (GCA) is an Inönü–Wigner (IW) contraction of a conformal algebra, while Newton–Hooke string algebra is an IW contraction of an Anti-de Sitter (AdS) algebra, which is the isometry of an AdS space. It is shown that the GCA is a boundary realization of the Newton–Hooke string algebra in the bulk AdS. The string lies along the direction transverse to the boundary, and the worldsheet is AdS2. The one-dimensional conformal symmetry so(2,1) and rotational symmetry so(d) contained in the GCA are realized as the symmetry on the AdS2 string worldsheet and rotational symmetry in the space transverse to the AdS2 in AdSd+2, respectively. It follows from this correspondence that 32 supersymmetric GCAs can be derived as IW contractions of superconformal algebras, psu(2,2|4), osp(8|4), and osp(8*|4). We also derive less supersymmetric GCAs from su(2,2|2), osp(4|4), osp(2|4), and osp(8*|2). [ABSTRACT FROM AUTHOR]

This paper deals with the following equation mt=(1/2)(1/mk)xxx-(1/2)(1/mk)x, which is proposed by Z. J. Qiao [J. Math. Phys. 48, 082701 (2007)] and Qiao and Liu [Chaos, Solitons Fractals 41, 587 (2009)]. By adopting the phase analysis method of planar dynamical systems and the theory of the singular traveling wave systems to the traveling wave solutions of the equation, it is shown that for different k, the equation may have infinitely many solitary wave solutions, periodic wave solutions, kink/antikink wave solutions, cusped solitary wave solutions, and breaking loop solutions. We discuss in a detail the cases of k=-2,-12,12,2, and parametric representations of all possible bounded traveling wave solutions are given in the different (c,g)-parameter regions. [ABSTRACT FROM AUTHOR]

The need to consider n-ary algebraic structures, generalizing Lie and Poisson algebras, has become increasingly important in physics, and it should therefore be of interest to study the mathematical concepts related to n-ary algebras. The purpose of this paper is to investigate ternary multiplications (as deformations of n-Lie structures) constructed from the binary multiplication of a Hom–Lie algebra, a linear twisting map, and a trace function satisfying certain compatibility conditions. We show that the relation between the kernels of the twisting maps and the trace function plays an important role in this context and provide examples of Hom–Nambu–Lie algebras obtained using this construction. [ABSTRACT FROM AUTHOR]

We consider integrable open spin chains related to the quantum affine algebras Uq(o(3)) and Uq(A2(2)). We discuss the symmetry algebras of these chains with the local C3 space related to the Birman–Wenzl–Murakami algebra. The symmetry algebra and the Birman–Wenzl–Murakami algebra centralize each other in the representation space H=x1NC3 of the system, and this determines the structure of the spin system spectra. Consequently, the corresponding multiplet structure of the energy spectra is obtained. [ABSTRACT FROM AUTHOR]

In this paper we formulate a geometric theory of thermal stresses. Given a temperature distribution, we associate a Riemannian material manifold to the body, with a metric that explicitly depends on the temperature distribution. A change in temperature corresponds to a change in the material metric. In this sense, a temperature change is a concrete example of the so-called referential evolutions. We also make a concrete connection between our geometric point of view and the multiplicative decomposition of deformation gradient into thermal and elastic parts. We study the stress-free temperature distributions of the finite-deformation theory using curvature tensor of the material manifold. We find the zero-stress temperature distributions in nonlinear elasticity. Given an equilibrium configuration, we show that a change in the material manifold, i.e., a change in the material metric will change the equilibrium configuration. In the case of a temperature change, this means that given an equilibrium configuration for a given temperature distribution, a change in temperature will change the equilibrium configuration. We obtain the explicit form of the governing partial differential equations for this equilibrium change. We also show that geometric linearization of the present nonlinear theory leads to governing equations that are identical to those of the classical linear theory of thermal stresses. [ABSTRACT FROM AUTHOR]

We introduce a partial differential equation whose structure, in many respects, mimics that of the second Painlevé equation. For this equation we give auto-Bäcklund transformations of both ordinary differential equation and partial differential equation types. Our discussion is sufficiently generally phrased to allow us to draw some general conclusions about the connection between auto-Bäcklund transformations and the integrability of ordinary and partial differential equations. [ABSTRACT FROM AUTHOR]

We prove that all the meromorphic solutions of the nonlinear differential equation c0u‴+6u4+c1u″+c2uu′+c3u3+c4u′+c5u2+c6u+c7=0 are elliptic or degenerate elliptic, and we build them explicitly. [ABSTRACT FROM AUTHOR]

We study classes of indecomposable representations of the Diamond Lie algebra: a four-dimensional solvable Lie algebra, which is the central extension of the Poincaré Lie algebra in two dimensions. These representations are obtained first (inspired by Douglas and Premat [Commun. Algebra 35, 1433 (2007)]) by embedding the Diamond Lie algebra in the simple Lie algebras A2 and C2, and then by regarding it as a subalgebra of a truncated current Lie algebra. [ABSTRACT FROM AUTHOR]

We study limiting cases of the two known integrable chiral-type models with three-dimensional configuration space. One of the initial models is the non-Abelian Toda A2(1) model and the other was found by means of the symmetry approach by Meshkov and Demskoi [Theor. Math. Phys. 134, 351 (2003)]. The C-integrability of the reduced models is established by constructing their complete sets of integrals and general solutions. A description of the generalized symmetry algebras of these models is given in terms of operator mapping integrals into symmetries. The integrals of the Liouville-type systems are known to define Miura-type transformations for their generalized symmetries. This fact allowed us to find a few new systems of the Yajima–Oikawa type. We present a recursion operator for one them. [ABSTRACT FROM AUTHOR]

We consider a superintegrable quantum potential in two-dimensional Euclidean space with a second and a third order integral of motion. The potential is written in terms of the fourth Painlevé transcendent. We construct for this system a cubic algebra of integrals of motion. The algebra is realized in terms of parafermionic operators and we present Fock-type representations which yield the corresponding energy spectra. We also discuss this potential from the point of view of higher order supersymmetric quantum mechanics and obtain ground state wave functions. [ABSTRACT FROM AUTHOR]

We classify Lie n-algebras possessing an invariant Lorentzian inner product. [ABSTRACT FROM AUTHOR]

We consider the system x=a(z-a1x3-a2x2-bx), y=-z, ż=-b1x+y+b2z, where a and b are parameters and b1=7/10, b2=6/25, a1=44/3, and a2=41/2. We analyze the existence of local and global analytic first integrals. [ABSTRACT FROM AUTHOR]

The Lie symmetry algebra of a generalized Davey–Stewartson (GDS) system is obtained. The general element of this algebra depends on eight arbitrary functions of time, which has a Kac–Moody–Virasoro loop algebra structure and is isomorphic to that of the standard integrable Davey–Stewartson equations under certain conditions imposed on parameters and arbitrary functions. Then based on the symmetry group direct method proposed by Lou and Ma [J. Phys. A 38, L129 (2005)] the full symmetry groups of the GDS system are obtained. From the full symmetry groups, both the Lie symmetry group and a group of discrete transformations can be obtained. Finally, some exact solutions involving sech-sech2-sech2 and tanh-tanh2-tanh2 type solitary wave solutions are presented by a generalized subequation expansion method. [ABSTRACT FROM AUTHOR]

The presented paper develops a new method for solving Einstein equations explicitly. Not only the solutions are new but also the mathematical framework of their construction which is given by a nonstandard function theory built over nonstandard algebras (in the following denoted as GAPs) and will be called here as “GAP Theory.” It is shown that the GAP theory succeeds in regions, where group theory fails: the group structure is too small to solve Einstein equations explicitly. [ABSTRACT FROM AUTHOR]

We compute necessary conditions on Yetter-Drinfeld modules over the groups SL(2,Fq) and GL(2,Fq) to generate finite dimensional Nichols algebras. This is a first step toward a classification of pointed Hopf algebras with a group of grouplikes isomorphic to one of these groups. [ABSTRACT FROM AUTHOR]

In this paper we prove that the two-dimensional superintegrable systems with quadratic integrals of motion on a manifold can be classified by using the Poisson algebra of the integrals of motion. There are six general fundamental classes of superintegrable systems. Analytic formulas for the involved integrals are calculated in all the cases. All the known superintegrable systems are classified as special cases of these six general classes. [ABSTRACT FROM AUTHOR]

We give sufficient conditions for ergodicity of the Markovian semigroups associated to Dirichlet forms on standard forms of von Neumann algebras constructed by the method proposed by Park. We apply our result to show that the diffusion type Markovian semigroups for quantum spin systems are ergodic in the region of high temperatures where the uniqueness of the KMS state holds. [ABSTRACT FROM AUTHOR]

We construct a new geometric framework based on the concepts of left and right jet-bundles of a classical space-time V in order to analyze the impulsive behavior of a unilateral constraint S. The setup allows deep insights into how one can choose an ideality criterion for the constraint S when the hypothesis of conservation of kinetic energy is assumed. We show that the conservation of kinetic energy alone univocally determines the impulsive reaction when the codimension of S is 1, and that it leaves the impulsive reaction partially undetermined when the codimension of S is greater than 1. If the codimension of S is greater than 1, we prove that an additional minimality requirement determines a physically meaningful constitutive characterization of S. We show that both the Newton-like and the Poisson-like approaches to the description of the reactive impulse are equivalent, in the sense that both give the same results about the ideality criterion. Moreover, we prove that the same results hold using the classical approach based on reflection operators, possible only in case of codimension 1. We present also several physically meaningful examples. [ABSTRACT FROM AUTHOR]

We extend to the K-SAT and p-XOR-SAT optimization problems the results recently achieved, by introducing the concept of random multi-overlap structure, for the Viana-Bray model of diluted mean field spin glass. More precisely we can prove a generalized bound and an extended variational principle for the free energy per site in the thermodynamic limit. Moreover a trial function implementing ultrametric breaking of replica symmetry is exhibited. The ultrametric structure exhibits the same factorization property as the optimal structures for the Viana-Bray model and the Sherrington-Kirkpatrick nondiluted model. [ABSTRACT FROM AUTHOR]

Fermions are expressed by polynomials of canonical generators of the Cuntz algebra O2 and they generate the U(1)-fixed point subalgebra A≡O2U(1) of O2 by the canonical gauge action. We extend the Fock and the infinite wedge representations of A to permutative representations of O2. By these extensions, the boson-fermion correspondence is rewritten by canonical generators of O2. [ABSTRACT FROM AUTHOR]

A basis of N2 projectors, each an N2×N2 matrix with constant elements, is implemented to construct a class of braid matrices R⁁(θ), θ being the spectral parameter. Only odd values of N are considered here. Our ansatz for the projectors Pα appearing in the spectral decomposition of R⁁(θ) leads to exponentials exp(mαθ) as the coefficient of Pα. The sums and differences of such exponentials on the diagonal and the antidiagonal, respectively, provide the (2N2-1) nonzero elements of R⁁(θ). One element at the center is normalized to unity. A class of supplementary constraints imposed by the braid equation leaves 1/2(N+3)(N-1) free parameters mα. The diagonalizer of R⁁(θ) is presented for all N. Transfer matrices t(θ) and L(θ) operators corresponding to our R⁁(θ) are studied. Our diagonalizer signals specific combinations of the components of the operators that lead to a quadratic algebra of N2 constant N×N matrices. The θ dependence factors out for such combinations. R⁁(θ) is developed in a power series in θ. The basic difference arising for even dimensions is made explicit. Some special features of our R⁁(θ) are discussed in a concluding section. [ABSTRACT FROM AUTHOR]

We present an extended version of max-plus algebra which includes the inverse operator of “max.” This algebra enables us to ultradiscretize the system including subtractions and obtain new ultradiscrete equations. The known ultradiscrete equations can also be recovered by this construction. [ABSTRACT FROM AUTHOR]

We establish the exact solution of the nonlinear Schrödinger equation with a delta-function impurity, representing a pointlike defect which reflects and transmits. We solve the problem both at the classical and the second quantized levels. In the quantum case the Zamolodchikov–Faddeev algebra, familiar from the case without impurities, is substituted by the recently discovered reflection-transmission (RT) algebra, which captures both particle–particle and particle–impurity interactions. The off-shell quantum solution is expressed in terms of the generators of the RT algebra and the exact scattering matrix of the theory is derived. [ABSTRACT FROM AUTHOR]

We devise a new simple loop algebra GM and an isospectral problem. By making use of the Tu scheme, the multicomponent generalized Kaup–Newell hierarchy is obtained. Furthermore, an expanding loop algebra FM of the loop algebra GM is presented. Based on FM, the multicomponent integrable couplings system with two arbitrary functions of the multicomponent generalized Kaup–Newell hierarchy are worked out. The method can be applied to other nonlinear evolution equations hierarchy. [ABSTRACT FROM AUTHOR]

The properties of Q-balls in the general case of a sixth order potential have been studied using analytic methods. In particular, for a given potential, the initial field value that leads to the soliton solution has been derived and the corresponding energy and charge have been explicitly evaluated. The proposed scheme is found to work reasonably well for all allowed values of the model parameters. [ABSTRACT FROM AUTHOR]

We present and investigate a new class of quantum channels, what we call “universal collective rotation channels,” that includes the class of collective rotation channels as a special case. The fixed point set and noise commutant coincide for a channel in this class. Computing the precise structure of this C*-algebra is a core problem in a particular noiseless subsystem method of quantum error correction. We prove that there is an abundance of noiseless subsystems for every channel in this class and that the Young tableaux combinatorial machine may be used to explicitly compute these subsystems. [ABSTRACT FROM AUTHOR]

We consider the convex set of positive operator valued measures (POVM) which are covariant under a finite dimensional unitary projective representation of a group. We derive a general characterization for the extremal points, and provide bounds for the ranks of the corresponding POVM densities, also relating extremality to uniqueness and stability of optimized measurements. Examples of applications are given. [ABSTRACT FROM AUTHOR]

A new geometrical framework for tetrad-affine formulation of gravity, pure or coupled with Yang–Mills fields, is proposed. After analyzing the geometrical properties of the new mathematical setting, field equations are deduced from a variational principle in the Poincaré–Cartan formalism. A generalized Noether Theorem is stated and classical relationship between symmetries and conserved quantities are recovered in the newer scheme. Some explicit examples are given. [ABSTRACT FROM AUTHOR]

We identify the hyperbolic Kac Moody algebras for which there exists a Lagrangian of gravity, dilatons, and p-forms which produces a billiard that can be identified with their fundamental Weyl chamber. Because of the invariance of the billiard upon toroidal dimensional reduction, the list of admissible algebras is determined by the existence of a Lagrangian in three space–time dimensions, where a systematic analysis can be carried out since only zero-forms are involved. We provide all highest dimensional parent Lagrangians with their full spectrum of p-forms and dilaton couplings. We confirm, in particular, that for the rank 10 hyperbolic algebra, CE10=A15(2)∧, also known as the dual of B8∧∧, the maximally oxidized Lagrangian is nine-dimensional and involves besides gravity, 2 dilatons, a 2-form, a 1-form, and a 0-form. [ABSTRACT FROM AUTHOR]

It is shown that the algebra of diffeomorphism-invariant charges of the Nambu–Goto string cannot be quantized in the framework of canonical quantization. The argument is shown to be independent of the dimension of the underlying Minkowski space. [ABSTRACT FROM AUTHOR]

Using special quasigraded Lie algebras, that could be viewed as deformations of loop algebras, we obtain new hierarchies of integrable nonlinear equations admitting zero-curvature representations. In particular, we obtain integrable hierarchies that generalize the Heisenberg magnet, Landau–Lifshitz, and anisotropic chiral field hierarchies. We also obtain a new type of so(3) anisotropic chiral field equation along with its higher rank generalization. [ABSTRACT FROM AUTHOR]

This paper aims at presenting the first steps towards a formulation of the Exact Renormalization Group Equation in the Hopf algebra setting of Connes and Kreimer. It mostly deals with some algebraic preliminaries allowing us to formulate perturbative renormalization within the theory of differential equations. The relation between renormalization, formulated as a change of boundary condition for a differential equation, and an algebraic Birkhoff decomposition for rooted trees is explicited. [ABSTRACT FROM AUTHOR]

To analyze linear field equations on a locally homogeneous space–time by means of separation of variables, it is necessary to set up appropriate harmonics according to its symmetry group. In this paper, the harmonics are presented for a spatially compactified Bianchi II cosmological model—the nilgeometric model. Based on the group structure of the Bianchi II group (also known as the Heisenberg group) and the compactified spatial topology, the irreducible differential regular representations and the multiplicity of each irreducible representation, as well as the explicit form of the harmonics are all completely determined. They are also extended to vector harmonics. It is demonstrated that the Klein–Gordon and Maxwell equations actually reduce to systems of ODEs, with an asymptotic solution for a special case. [ABSTRACT FROM AUTHOR]

If a state is passive for uniformly accelerated observers in n-dimensional (n>=2) anti–de Sitter (Ads) space–time (i.e., cannot be used by them to operate a perpetuum mobile), they will (a) register a universal value of the Unruh temperature, (b) discover a PCT symmetry, and (c) find that observables in complementary wedge-shaped regions necessarily commute with each other in this state. The stability properties of such a passive state induce a “geodesic causal structure” on AdS and concommitant locality relations. It is shown that observables in these complementary wedge-shaped regions fulfill strong additional independence conditions. In two-dimensional AdS these even suffice to enable the derivation of a nontrivial, local, covariant net indexed by bounded space–time regions. All these results are model-independent and hold in any theory which is compatible with a weak notion of space–time localization. Examples are provided of models satisfying the hypotheses of these theorems. [ABSTRACT FROM AUTHOR]

The Hughston–Jozsa–Wootters theorem shows that any finite ensemble of quantum states can be prepared “at a distance,” and it has been used to demonstrate the insecurity of all bit commitment protocols based on finite quantum systems without superselection rules. In this paper, we prove a generalized HJW theorem for arbitrary ensembles of states on a C*-algebra. We then use this result to demonstrate the insecurity of bit commitment protocols based on infinite quantum systems, and quantum systems with Abelian superselection rules. [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]

A bijective map r: X2→X2, where X={x1,…,xn} is a finite set, is called a set-theoretic solution of the Yang–Baxter equation (YBE) if the braid relation r12r23r12=r23r12r23 holds in X3. A nondegenerate involutive solution (X,r) satisfying r(xx)=xx, for all x∈X, is called square-free solution. There exist close relations between the square-free set-theoretic solutions of YBE, the semigroups of I-type, the semigroups of skew polynomial type, and the Bieberbach groups, as it was first shown in a joint paper with Michel Van den Bergh. In this paper we continue the study of square-free solutions (X,r) and the associated Yang–Baxter algebraic structures—the semigroup S(X,r), the group G(X,r) and the k-algebra A(k,X,r) over a field k, generated by X and with quadratic defining relations naturally arising and uniquely determined by r. We study the properties of the associated Yang–Baxter structures, and prove a conjecture of the present author that the three notions: a square-free solution of (set-theoretic) YBE, a semigroup of I type, and a semigroup of skew-polynomial-type, are equivalent. This implies that the Yang–Baxter algebra A(k,X,r) is a Poincaré–Birkhoff–Witt-type algebra, with respect to some appropriate ordering of X. We conjecture that every square-free solution of YBE is retractable, in the sense of Etingof–Schedler–Solovyev. [ABSTRACT FROM AUTHOR]

The known link of the division algebras to 10-dimensional spacetime and one leptoquark family is extended to encompass three leptoquark families. [ABSTRACT FROM AUTHOR]

We introduce the free N=1 supersymmetric derivation ring and prove the existence of an exact sequence of supersymmetric rings and linear transformations. We apply necessary and sufficient conditions arising from this exact supersymmetric sequence to obtain the essential relations between conserved quantities, gradients and the N=1 super Korteweg–de Vries (KdV) hierarchy. We combine this algebraic approach with an analytic analysis of the super heat operator. We obtain the explicit expression for the Green’s function of the super heat operator in terms of a series expansion and discuss its properties. The expansion is convergent under the assumption of bounded bosonic and fermionic potentials. We show that the asymptotic expansion when t→0+ of the Green’s function for the superheat operator evaluated over its diagonal generates all the members of the N=1 super KdV hierarchy.© 2004 American Institute of Physics. [ABSTRACT FROM AUTHOR]