invariant

Continuous and Discontinuous Piecewise-Smooth One-Dimensional Maps: Invariant Sets and Bifurcation Structures

In this work our aim is to construct the soluble potentials within the framework of the Schrodinger invariant [I.sub.S](x) through solving the differential equation of the transformation z(x) directly and then to obtain its more general solutions instead of considering several special cases for the parameters [[alpha].sub.1], [[beta].sub.1], and [[gamma].sub.1].

Over the past twenty years, many interesting results were got for quadratic systems; the authors in [17,18] proved that quadratic systems with a pair of straight lines or an invariant hyperbola, ellipse, can have no limit cycles other than the possible ellipse itself.

Up to now, the researches on discrete models are still focused on the dynamical behaviors (including stability, periodic solutions, bifurcations, chaos, and chaotic control; see[1-6]), and most of the scholars have studied the Neimark-Sacker bifurcation, a codimension-1 bifurcation, which has shown one invariant closed curve bifurcating from an equilibrium point.

Model 1 is an unconstrained model where there is a configural invariance (participants of different groups conceptualize the constructs in the same way); model 2 holds all factor loadings are equal across groups; model 3 postulates factor loading and factor variances and covariances are equal; finally, model 4, hypothesizes factor loadings, factor variances and covariances and error variances and covariances are invariant across the groups.

Porti ([8]) has investigated the Reidemeister torsion of M associated with the adjoint representation Ado[Hol.sub.M] of its holonomy representation [Hol.sub.M] : [[pi].sub.1](M) [right arrow] PSL(2, C), and then Yamaguchi showed in [13] a relationship between the Porti's Reidemeister torsion and the twisted Alexander invariant explicitly.

spaces and gave some sufficient conditions for existence and non existence of an invariant metric with homogeneous geodesics on a homogeneous space of a compact Lie group G.

where [F.sup.[theta]] = {F [member of] F | [[theta].sup.-1]](F) = F} is the [sigma]-field of invariant sets (under [theta]).

An example of an affine invariant is illustrated in Figure 6.

[28] presented a novel illumination invariant method, namely, multiscale principal contour direction (MPCD).

In Section 2 invariant structure of canonical forms and their Lie algebraic properties are constructed.

For a QTAG-module M, the [sigma]th-Ulm invariant of M, [f.sub.M] ([sigma]) is the cardinal number g(Soc([H.sub.[sigma]](M))/Soc([H.sub.[sigma]+1] (M))) [1].