Non-orientable case

Remark 4. The above proof can be easily adopted to the case of a separatrix loop on a general two-dimensional surface, regardless whether it is orientable or non-orientable. In both cases the map will have the form (13.2.9). Note, however, that if a small neighborhood of the separatrix loop is homeomorphic to an annulus, then A>0 and if a neighborhood of f is a Mobius band, then A < 0 (the latter corresponds, obviously, to the non-orientable case). In the case > 0, the Andronov-Leontovich theorem holds without changes. 

Note that in the orientable case (A > 0), the separatrix loop at /x = 0 is a limit trajectory (o -limit if > 1, or a-limit if i/ < 1) for nearby trajectories from the inner neighborhood U. On the contrary, in the non-orientable case... 

Molecular orientation and the testing direction strongly influence the observed modulus of a polymer sample. Fibers are typically highly oriented and exhibit much higher modulus values than non-oriented samples prepared from the same polymer. In the case of films, we typically observe anisotropy a film exhibits a range of modulus values depending upon the testing direction. 

Here is a fixed integer, the Euler characteristic that marks the particular topology of the surface on which the polyhedron is embedded. However, in order to describe the topology completely, one also has to specify the orientability of the surface [3]. A surface is orientable if there is no walk on the surface that would take you from the outside to the inside. Such is the case of a sphere with handles. Otherwise, it is non-orientable. This is the case of a sphere with crosscaps. Based on this orientability, the infinite class of surfaces can be divided into two subclasses ... 

The special case rA = rB where all vertices are equidistant. The problem is therefore depicted by the complete graph Ki0 on ten vertices. This graph has genus seven and can therefore be embedded on the non-orientable surface with seven crosscaps. 

Morphology Some polymers, like PETP, are spun in a nearly amorphous state or show a low degree of crystallinity. In other polymers, such as nylon, the undrawn material is already semi-crystalline. In the latter case the impact of extension energy must be sufficient to (partly) "melt" the folded chain blocks (lamellae) in all cases non-oriented material has to be converted into oriented crystalline material. In order to obtain high-tenacity yarns, the draw ratio must be high enough to transform a fraction of the chains in more or less extended state. 

One aspect of combined interactions which has proved useful is the apphcation of an external magnetic field to non-oriented polycrystalline absorbers with i ->i y-transitions. In the zero-field case the quadrupole doublet is symmetrical and the sign of e qQ is not determined. Collins [31] used a first-order perturbation treatment to show that an applied field would cause the transition from the 1 state of Fe to split into a doublet, and the... 

In the case of low Si steels (< 1 wt% Si), the last two annealing steps are applied by the user after lamination punching (semi-finished sheet). Table 4.3-15 lists the specifications, including all relevant properties for non-oriented magnetic steel sheet. 

Functionalized protonic acids such as ( ) camphor sulfonic acid (HCSA) and dodecyl benzene sulfonic acid (HDBSA) are used as primary dopants while m-cresol, xylene, etc. are used as the secondary dopants. The free-standing non-oriented PANI-CSA films cast from the above solvents have conductivities in the range 100—400 S cm , a value two orders of magnitude larger than that of non-oriented films of emeraldine base doped with aqueous HCl. MacDiarmid et al. [34] found that the conductivity of POT-CSA is 0.2 S cm in w-cresol and 2 S cm in chloroform whereas that of PANI-CSA is 140 S cm in w-cresol and 0.16 S cm in chloroform. These differences can be attributed as due to the interaction of the non-polar covalent methyl groups in POT with less polar chloroform, whereas in the case of RANI, the polar backbone can interact more readily with the polar /n-cresol. 

In this case, the film was even more ordered. When the deposition rate and the film thickness were increased, a non-oriented phase appeared in the EPR spectra. The contents of the non-oriented phase in the investigated samples varied from 0 up to 80%. The first actually obtained film contained a significant amount of the disordered phase, but later we found a mode of deposition to completely remove it. Thin films had the best ordering, and in this aspect the copper dipivaloil methanate differ from the copper phthalocyanine. 

In the same section we give the bifurcation diagrams for the codimension two case with a first zero saddle value and a non-zero first separatrix value (the second term of the Dulac sequence) at the bifurcation point. Leontovich s method is based on the construction of a Poincare map, which allows one to consider homoclinic loops on non-orientable two-dimensional surfaces as well, where a small-neighborhood of the separatrix loop may be a Mobius band. Here, we discuss the bifurcation diagrams for both cases. 

The original proof of the Andronov-Leontovich theorem assumes that the system is defined on the plane. We choose here a somewhat different approach which can be easily adopted to the case of systems defined on non-orientable two-dimensional surfaces as well. 

Let us consider next the case where — 1 < A < 0 which corresponds to a separatrix loop F on a non-orientable surface (the case 4 < — 1 follows similarly by a reversion of time). A neighborhood of V is then a Mobius band whose median is F. The Poincare map in this case also has the form (13.3.8) with the function satisfying estimates (13.3.7). However, now we need more smoothness. So we assume that the system is at least C -smooth, i.e. r > 4 in (13.3.7). 

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