On non-linear partial differential equations with an infinite-dimensional conditional symmetry

The invariance of non-linear partial differential equations under a certain infinite-dimensional Lie algebra AN(z) in N spatial dimensions is studied. The special case A₁(2) was introduced in [J. Stat. Phys. 75 (1994) 1023] and contains the Schrödinger Lie algebra sch₁ as a Lie subalgebra. It is shown that there is no second-order equation which is invariant under the massless realizations of AN(z). However, a large class of strongly non-linear partial differential equations is found which are conditionally invariant with respect to the massless realization of AN(z) such that the well-known Monge-Ampère equation is the required additional condition. New exact solutions are found for some representatives of this class.     

On Smital properties

Let A be an algebra and J⊂A an ideal of subsets of a group 〈X,+〉 with an invariant topology τ. We say that a triple 〈A,J,τ〉 has the Smital property if, for any set A∈A?J and any set D dense in τ, the set c(A+D) belongs to J. In the paper we compare this property and similar ones with the well-known Steinhaus type properties. We consider several weak and strong versions of Smital properties.     

Some exact solutions of the ideal MHD equations through symmetry reduction method

We use the symmetry reduction method based on Lie group theory to obtain some exact solutions, the so-called invariant solutions, of the ideal magnetohydrodynamic equations in (3+1) dimensions. In particular, these equations are invariant under a Galilean-similitude Lie algebra for which the classification by conjugacy classes of r-dimensional subalgebras (1?r?4) was already known. We restrict our study to the three-dimensional Galilean-similitude subalgebras that give us systems composed of ordinary differential equations. Here, some examples of these solutions are presented with a brief physical interpretation.     

Invariant bilinear forms on a vertex algebra

In this paper we construct a linear space that parameterizes all invariant bilinear forms on a given vertex algebra with values in a arbitrary vector space. Also we prove that every invariant bilinear form on a vertex algebra is symmetric. This is a generalization of the result of Li (J. Pure Appl. Algebra 96(3) (1994) 279), who proved this for the case when the vertex algebra is non-negatively graded and has finite dimensional homogeneous components.As an application, we introduce a notion of a radical of a vertex algebra. We prove that a radical-free vertex algebra A is non-negatively graded, and its component A₀ of degree 0 is a commutative associative algebra, so that all structural maps and operations on A are A₀-linear. We also show that in this case A is simple if and only if A₀ is a field.     

The Injective Leavitt Complex

For a finite quiver Q without sinks, we consider the corresponding finite dimensional algebra A with radical square zero. We construct an explicit compact generator for the homotopy category of acyclic complexes of injective A-modules. We call such a generator the injective Leavitt complex of Q. This terminology is justified by the following result: the differential graded endomorphism algebra of the injective Leavitt complex of Q is quasi-isomorphic to the Leavitt path algebra of Q. Here, the Leavitt path algebra is naturally \(\mathbb {Z}\)-graded and viewed as a differential graded algebra with trivial differential.     

Associate symmetries: A novel procedure for finding contact symmetries

A new method for finding contact symmetries is proposed for both ordinary and partial differential equations. Symmetries more general than Lie point are often difficult to find owing to an increased dependency of the infinitesimal functions on differential quantities. As a consequence, the invariant surface condition is often unable to be “split” into a reasonably sized set of determining equations, if at all. The problem of solving such a system of determining equations is here reduced to the problem of finding its own point symmetries and thus subsequent similarity solutions to these equations. These solutions will (in general) correspond to some subset of symmetries of the original differential equations. For this reason, we have termed such symmetries associate symmetries. We use this novel method of associate symmetries to determine new contact symmetries for a non-linear PDE and a second order ODE which could not previously be found using computer algebra packages; such symmetries for the latter are particularly difficult to find. We also consider a differential equation with known contact symmetries in order to illustrate that the associate symmetry procedure may, in some cases, be able to retrieve all such symmetries.     

One family of conformally Hamiltonian systems

We propose a method for constructing conformally Hamiltonian systems of dynamical equations whose invariant measure arises from the Hamiltonian equations of motion after a change of variables including a change of time. As an example, we consider the Chaplygin problem of the rolling ball and the Veselova system on the Lie algebra e*(3) and prove their complete equivalence.     

Euler-Bernoulli beams from a symmetry standpoint-characterization of equivalent equations

We completely solve the equivalence problem for Euler-Bernoulli equation using Lie symmetry analysis. We show that the quotient of the symmetry Lie algebra of the Bernoulli equation by the infinite-dimensional Lie algebra spanned by solution symmetries is a representation of one of the following Lie algebras: 2A₁, A₁⊕A₂, 3A₁, or A3,3⊕A₁. Each quotient symmetry Lie algebra determines an equivalence class of Euler-Bernoulli equations. Save for the generic case corresponding to arbitrary lineal mass density and flexural rigidity, we characterize the elements of each class by giving a determined set of differential equations satisfied by physical parameters (lineal mass density and flexural rigidity). For each class, we provide a simple representative and we explicitly construct transformations that maps a class member to its representative. The maximally symmetric class described by the four-dimensional quotient symmetry Lie algebra A3,3⊕A₁ corresponds to Euler-Bernoulli equations homeomorphic to the uniform one (constant lineal mass density and flexural rigidity). We rigorously derive some non-trivial and non-uniform Euler-Bernoulli equations reducible to the uniform unit beam. Our models extend and emphasize the symmetry flavor of Gottlieb's iso-spectral beams [H.P.W. Gottlieb, Isospectral Euler-Bernoulli beam with continuous density and rigidity functions, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 413 (1987) 235-250].     

Reflexive linear transformations

A linear operator A is called reflexive if the only operators that leave invariant the invariant subspaces of A are the operators in the weak closure of the algebra of polynomials in A. In this note we completely characterize reflexive operators on finite-dimensional spaces.     

Weakly almost periodic functionals on certain Banach algebras

For a Lau algebra A, we study the Banach space WAP(A) of all weakly almost periodic functionals on A to obtain some equivalent conditions for the existence of topological left invariant means on a topological left introverted subspace X of A contained in WAP(A). Finally, we consider relations between the existence of a topological left invariant mean on X and a common fixed point property.     

Morita context of weak Hopf algebras

Let H be a finite-dimensional weak Hopf algebra and A a left H-module algebra with its invariant subalgebra A~H.We prove that the smash product A#H is an A-ring with a grouplike character, and give a criterion for A#H to be Frobenius over A. Using the theory of A-rings, we mainly construct a Morita context connecting the smash product A#H and the invariant subalgebra A~H , which generalizes the corresponding results obtained by Cohen, Fischman and Montgomery.     

Dynamical Systems Associated with Crossed Products

In this paper, we consider both algebraic crossed products of commutative complex algebras A with the integers under an automorphism of A, and Banach algebra crossed products of commutative C ^*-algebras A with the integers under an automorphism of A. We investigate, in particular, connections between algebraic properties of these crossed products and topological properties of naturally associated dynamical systems. For example, we draw conclusions about the ideal structure of the crossed product by investigating the dynamics of such a system. To begin with, we recall results in this direction in the context of an algebraic crossed product and give simplified proofs of generalizations of some of these results. We also investigate new questions, for example about ideal intersection properties of algebras properly between the coefficient algebra A and its commutant A′. Furthermore, we introduce a Banach algebra crossed product and study the relation between the structure of this algebra and the topological dynamics of a naturally associated system.     

Using impulses to control the convergence toward invariant surfaces of continuous dynamical systems

Let us consider a smooth invariant surface S of a given ordinary differential equations system. In this work we develop an impulsive control method in order to assure that the trajectories of the controlled system converge toward the surface S. The method approach is based on a property of a certain class of invariant surfaces whose the dynamics associated to their transverse directions can be described by a non-autonomous linear system. This fact allows to define an impulsive system which drives the trajectories toward the surface S. Also, we set up a definition of local stability exponents which can be associated to such kind of invariant surface.     

Fully invariant and verbal congruence relations

A congruence relation θ on an algebra A is fully invariant if every endomorphism of A preserves θ. A congruence θ is verbal if there exists a variety ${\mathcal{V}}$ such that θ is the least congruence of A such that ${{\bf A}/\theta \in \mathcal{V}}$ . Every verbal congruence relation is known to be fully invariant. This paper investigates fully invariant congruence relations that are verbal, algebras whose fully invariant congruences are verbal, and varieties for which every fully invariant congruence in every algebra in the variety is verbal.     

Ideal Turaev-Viro invariants

Turaev-Viro invariants are defined via state sum polynomials associated to a special spine or a triangulation of a compact 3-manifold. By evaluation of the state sum at any solution of the so-called Biedenharn-Elliott equations, one obtains a homeomorphism invariant of the manifold (“numerical Turaev-Viro invariant”). The Biedenharn-Elliott equations define a polynomial ideal. The key observation of this paper is that the coset of the state sum polynomial with respect to that ideal is a homeomorphism invariant of the manifold (“ideal Turaev-Viro invariant”), stronger than the numerical Turaev-Viro invariants. Using computer algebra, we obtain computational results on several examples of ideal Turaev-Viro invariants, for all closed orientable irreducible manifolds of complexity at most 9.     

Some Algebraic Operators and the Invariant Subspace Problem

We consider the resolvent algebra ${R_A=\{T\in\mathcal{L} (X):\sup_{m \geq 0}\|(1+\,mA)T(1+\,mA)^{-1}\| < \infty \}}$ , and Deddens?? algebra ${B_A= \{T\in \mathcal{B}(H) : \sup_{n\geq 0}\|A^nTA^{-n}\|<\infty\}}$ . It is shown that both R _A and B _A?CI possess non-trivial invariant subspaces when A is an algebraic operator of degree 2. This assertion becomes stronger than the existence of a hyper-invariant subspace for R _A whenever R _A ?? {A}??. Investigation of the relationship between these two algebras is addressed for different classes of operators A. Also, a complete characterization of the algebra R _A when A is an algebraic operator is given. For the finite dimensional case, we present an elementary example showing that R _A contains properly {A}?? whenever A has an eigenvalue other than zero.     

Generalized Exponents and Forms

We consider generalized exponents of a finite reflection group acting on a real or complex vector space V. These integers are the degrees in which an irreducible representation of the group occurs in the coinvariant algebra. A basis for each isotypic component arises in a natural way from a basis of invariant generalized forms. We investigate twisted reflection representations (V tensor a linear character) using the theory of semi-invariant differential forms. Springer’s theory of regular numbers gives a formula when the group is generated by dim V reflections. Although our arguments are case-free, we also include explicit data and give a method (using differential operators) for computing semi-invariants and basic derivations. The data give bases for certain isotypic components of the coinvariant algebra.     

Torsion Abelian afi-Groups

This paper is devoted to the study of Abelian afi-groups. A subgroup A of an Abelian group G is called its absolute ideal if A is an ideal of any ring on G. We will call an Abelian group an afi-group if all of its absolute ideals are fully invariant subgroups. In this paper, we will describe afi-groups in the class of fully transitive torsion groups (in particular, separable torsion groups) and divisible torsion groups.     

Transitive algebras and reductive algebras on reproducing analytic Hilbert spaces

In this paper, we consider the well-known transitive algebra problem and reductive algebra problem on vector valued reproducing analytic Hilbert spaces. For an analytic Hilbert space H(k) with complete Nevanlinna-Pick kernel k, it is shown that both transitive algebra problem and reductive algebra problem on multiplier invariant subspaces of H(k)⊗Cm have positive answer if the algebras contain all analytic multiplication operators. This extends several known results on the problems.