In this article, we study the oscillation of solutions to a nonlinear forced fractional differential equation. The fractional derivative is defined in the sense of the modified Riemann-Liouville derivative. Based on a transformation of variables and properties of the modified Riemann-liouville derivative, the fractional differential equation is transformed into a second-order ordinary different...

We explore the role of a special class of optimization problems involving controlled Liouville equations. We present some new results on the controllability of the Liouville equation and discuss the optimal control of its moments for some important special cases. The problems treated suggest a new domain of applicability for optimal control, one which can include constraints more appropriate fo...

We consider a singular Sturm-Liouville expression with the indefinite weight sgnx. To this expression there is naturally a self-adjoint operator in some Krein space associated. We characterize the local definitizability of this operator in a neighbourhood of ∞. Moreover, in this situation, the point ∞ is a regular critical point. We construct an operator A = (sgnx)(−d2/dx2 + q) with non-real sp...

The Liouville equation, first Bogoliubov hierarchy and Vlasov equations with derivatives of non-integer order are derived. Liouville equation with fractional derivatives is obtained from the conservation of probability in a fractional volume element. This equation is used to obtain Bogoliubov hierarchy and fractional kinetic equations with fractional derivatives. Statistical mechanics of fracti...

In this Letter, we combine the recently introduced transfer tensor method with the mixed quantum-classical Liouville method. The resulting protocol provides an accurate, general, flexible and robust new route for simulating the reduced dynamics of the quantum subsystem for arbitrarily long times, starting with computationally feasible short-time mixed quantum-classical Liouville dynamical maps....

Let (M,F) be a compact codimension one foliated manifold whose leaves are endowed with Riemannian metrics and consider continuous functions on M that are harmonic along the leaves of F . If every such function is constant on leaves we say that (M,F) has the Liouville property. Our main result is that codimension-one foliated bundles over compact negatively curved manifolds satisfy the Liouville...

We compute N -point correlation functions of non-unitary (2k−1, 2) minimal matter coupled to 2D quantum gravity on a sphere using the continuum Liouville field approach. A gravitational dressing of the matter primary field with the minimum conformal weight is used as the cosmological operator. Our results are in agreement with the correlation functions of the one-matrix model at the k-th critic...

Large N matrix models modified by terms of the form g(TrΦn)2 generate random surfaces which touch at isolated points. Matrix model results indicate that, as g is increased to a special value gt, the string susceptibility exponent suddenly jumps from its conventional value γ to γ γ−1 . We study this effect in Liouville gravity and attribute it to a change of the interaction term from Oeα+φ for g...

It is known that Liouville theory can be represented as an SL(2,R) gauged WZW model. We study a two dimensional field theory which can be obtained by analytically continuing some of the variables in the SL(2,R) gauged WZW model. We can derive Liouville theory from the analytically continued model, ( which is a gauged SL(2, C)/SU(2) model, ) in a similar but more rigorous way than from the origi...

A random vector (Xl' •.. , X n ), with positive components, is said to have a Liouville distribution if its joint probability density aC I an-l function is of the form f (xl + ... + x n ) xl •.. x n with the a i all positive. Examples of these are the Dirichlet and inverted Dirichlet distribution. In this paper, a comprehensive treatment of the Liouville distributions is provided. The results d...