The

perturbative schemes generally built on the interaction representation yields a time ordered exponential [13,17] of terms ordered by the number of discrete interactions in the terms.

This equation can then be used to prove

perturbative renormalizability in QFT.

The amount of this information is dependent in case (i) on the ratio of small parameters in the

perturbative solution, which characterize strain intensities in the material caused by the propagating wave and predeformation, and in case (ii) on the ratio of the small parameter that characterizes strain intensity in the material caused by the propagating wave to the small parameter that describes the weak variation of material properties.

The couplings are such that they determine the local notion of causality and it is not clear when or how well a

perturbative scheme, which is generally built on free fields solutions, will work in the many body case.

Foundations of quantum chromodynamics; an introduction to

perturbative methods in gauge theories, 3d ed.

From useful algorithms for slowly convergent series to physical predictions based on divergent

perturbative expansions.

i[tau]]], it proves possible to develop a

perturbative analysis and estimate the value of [[phi].

Such an understanding is one of the goals of the rich field of hadron physics which has opened up between "traditional" nuclear physics and high energy collider physics and which tests the

perturbative limit of QCD.

If we, just for the moment, tentatively reintroduce the

perturbative running of coupling "constants" (renormalization group) we obtain m(graviton) [right arrow] [infinity] as r [right arrow] 0 implying that (quantum) gravity gets a dynamical cutoff for small separations, as an increasingly more massive quantum is harder to exchange, effectively making the interaction of gravity disappear in that limit, perhaps showing a way out of the ultraviolet divergencies of quantum gravity in a way reminiscent of how massive vector bosons cured the Fermi theory.

Among the topics are nearby cycles and periodicy in cyclic homology, the Gauss-Bonnet theorem for the noncommutative two torus, zeta phenomenology, absolute modular forms, the transcendence of values of transcendental functions at algebraic points, and the Hopf algebraic structure of

perturbative quantum gauge theories.

She has organized the discussion into three sections that address existing approaches for solving combinatorial problems in constraint programming, including exact approaches, heuristic approaches,

perturbative heuristic approaches, and constructive heuristic approaches; the features of ant colony optimization (based on the behavior of ant colonies in selecting shortest paths) and their applications to balancing diversification (seeking a good sampling of the search space) and intensification (seeking to guide the search towards the best combinations); applications to solving continuous problems, dynamic problems, and multi-objective optimization problems; and applications to constraint satisfaction problems.

A

perturbative solution for the spin wave function in the coordinate system with the magnetic field in the z direction yields an asymptotic form for the depolarization, D = ([pi]/6)exp[(4/3)-([pi][lambda]/2)], where the adiabaticity parameter [lambda] is