Hermite–Hadamard-type inequalities for Riemann–Liouville fractional integrals via two kinds of convexity

In this article, two fundamental integral identities including the second-order derivatives of a given function via Riemann–Liouville fractional integrals are established. With the help of these two fractional-type integral identities, all kinds of Hermite–Hadamard-type inequalities involving left-sided and right-sided Riemann–Liouville fractional integrals for m-convex and (s,?m)-convex functions, respectively. Our methods considered here may be a stimulant for further investigations concerning Hermite–Hadamard-type inequalities involving Hadamard fractional integrals.     

HYPOELLIPTICITY OF NONSINGULAR CLOSED 1-FORMS ON COMPACT MANIFOLDS

ABSTRACT

Hypoellipticity of nonsingular closed 1-forms on compact manifolds is characterized by a diophantine condition on the generators of the group of periods of the form.     

A simple approach for determining the eigenvalues of the fourth-order Sturm–Liouville problem with variable coefficients

In this paper, a simple and efficient approach is presented to compute the eigenvalues of the fourth-order Sturm–Liouville equations with variable coefficients. By transforming the governing differential equation to a system of algebraic equation, we can get the corresponding polynomial characteristic equations for kinds of boundary conditions based on the polynomial expansion and integral technique. Moreover, the lower and higher-order eigenvalues can be determined simultaneously from the multi-roots. Several examples for estimating eigenvalues are given. The convergence and effectiveness of the method are confirmed by comparing numerical results with the exact and other existing numerical results.     

Extremality Results for Discontinuous Sturm‐Liouville Boundary Value Problems

In this paper we derive extremalit and comparison results for explicit and implicit boundary value problems of Sturm‐Liouville differential equations. Differential equations and boundary conditions ma involve discontinuous and functional dependencies.     

Generalizations of the Virasoro algebra and matrix Sturm–Liouville operators

We study the series of Lie algebras generalizing the Virasoro algebra introduced in [V. Yu, Ovsienko, C. Roger, Functional Anal. Appl. 30 (4) (1996)]. We show that the coadjoint representation of each of these Lie algebras has a natural geometrical interpretation by matrix differential operators generalizing the Sturm–Liouville operators.     

Multifractality in Uniform Hyperbolic Lattices and in Quasi-Classical Liouville Field Theory

We introduce a deterministic model defined on a two dimensional hyperbolic lattice. This model provides an example of a non random system whose multifractal behaviour has a number theoretic origin. We determine the multifractal exponents, discuss the termination of multifractality and conjecture the geometric origin of the multifractal behavior in Liouville quasi-classical field theory.     

Approach to equilibrium of coupled harmonic oscillator systems. II

The approach to equilibrium of a finite segment of an infinite chain of harmonically coupled masses is studied in several variations. The chain is taken as completely free, or it is bound atx ₀ =0; ordinary coordinates and momenta or Schrödinger variables are used to treat the dynamics; and the inital distribution of heat-bath variables is chosen to be canonical or noncanonical. Equipartition of energy is found in all cases. Brownian drifts are obtained for the free chain with ordinary variables, but when this is excluded, the equilibrium entropy is found to be canonical and extensive when the initial heat bath is canonical, but less than canonical and slightly nonextensive when the initial heat bath is noncanonical. The modifications of the entropy do not contribute to the heat capacity of the system.Supported in part by the United States Atomic Energy Commission.     

A Generalized Coordinate-Momentum Representation in Quantum Mechanics

We obtain a one-parameter family of (q, p)-representations of quantum mechanics; the Wigner distribution function and the distribution function we previously derived are particular cases in this family. We find the solutions o the evolution equations or the microscopic classical and quantum distribution functions in the form of integrals over paths in a phase space. We show that when varying canonical variables in the Green’s function of the quantum Liouville equation, we must use the total increment o the action functional in its path-integral representation, whereas in the Green’s function of the classical Liouville equation, the linear part o the increment is sufficient. A correspondence between the classical and quantum schemes holds only under a certain choice of the value of the distribution family parameter. This value corresponds to the distribution unction previously found.__________Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 143, No. 3, pp. 401–416, June, 2005.     

Vertical Cohomologies and Their Application to Completely Integrable Hamiltonian Systems

Some functorial and topological properties of vertical cohomologies and their application to completely integrable Hamiltonian systems are studied.     

Formal structure of kinetic theory

We study the Liouville equation in the domain of small deviations from absolute equilibrium. The solution is expressed in terms of amplitudes ofn-body additive functions which are orthogonal with respect to the Gibbs weight factor. In the memory operator approach the memory operators are formally exact continued fractions. We show that with the isolation in the Liouville operator of a one-body additive operatorL _o, any memory operator can be written alternatively as an exact infinite series, each term of which can be calculated exactly. This yields improvements of the dressed particle approximation. We discuss the choice ofL _o, which is in general time dependent. The theory is developed both for smooth potentials and for hard spheres, where we use pseudo-Liouville operators. The theory can be formulated in an equivalent manner by introducing modified cumulant distributions, which are closely related to the amplitudes. The modified cumulants have the same spatial asymptotic properties as ordinary cumulants, but have superior short-time and small-distance behavior.Work supported by the National Science Foundation.