Conclusion and Remarks

3 Remarks and Conclusions. - Based on the above we have the following remarks  

For all the problems the exponentially fitted and trigonometrically fitted methods are much more efficient than the other methods with the same basic characteristics for the same cost. 

From exponentially fitted and trigonometrically fitted methods the most efficient are the linear symmetric multistep methods. This is because these methods have two additional properties [symmetry (i.e. symplecticness) and non-empty interval of periodicity] to the dissipative methods and one additional property (non-empty interval of periodicity) to the symplectic methods. 

As a conclusion we can say that symmetric (non-dissipative) methods are more efficient because they have non-empty interval of periodicity and because they are symplectie (in the case of linear symmetric multistep methods). We note also that symmetric linear multistep methods are very simple in programming and have very low computational cost (only one function evaluation per step). 

The microenvironment may also influence apoptosis. Exposure of isolated thymocytes to TPA plus ionomycin for 24 hours enhanced apoptosis. On the other hand, when ffiymocytes were cultured in intact lobes, a 24 hour TPA plus ionomycin exposure only marginally induced apoptosis. Therefore, it appears that removing thymocytes from their thymic microenvironment makes the cells more susceptible to certain stimuli, possibly by altering their physiological status. (Moore et al., 1992). Viral infection may also alter apoptosis. Epstein-Barr virus infected human Burkitt s lymphoma cells were particularly sensitive to treatment with PdBu (42% apoptosis at 72 hours), whereas its virus free counterpart displayed only 12% apoptosis (Ishii and Gobe, 1993). 

Apoptosis can be induced or prevented through different pathways (TNFa-induced, Fas-induced, TRAIL-induced). In lymphokine-activated killer cells Fas-mediated cytotoxicity could be dissociated from perforin-mediated cytotoxicity by their different requirement of TPA-sensitive PKC 

Staurosporine, UCN-01, and UCN-02 (a weak PKC inhibitor) induced a concentration- and time-dependent increase in apoptosis, whereas neither CGP 41251, RO 31-8220, nor GF 109203X induced apoptosis in immature rat thymocytes. In the human cell line BM13674 the specific inhibition of PKA gave rise to significantly increased levels of apoptosis at postirradiation compared to values after radiation e3q osure only. Calphostin C which caused 68% inhibition of PKC activity in irradiated cells, did not alter the level of radiation-induced apoptosis (Findik et al., 1995). 

The induction or prevention of apoptosis may depend, in addition to or independent of PKC, on calcium, pH, pS3, cyclin-dependent kinases, cyclin- 

Overexpression of one PKC isoenzyme may lead to altered expression and activity of one or more of the other PKC isoenzymes. For certain effects several PKC isoenzymes maybe involved. For example, the combined effects of PKCX, E and are essential for the transcriptional activation of c-fos by oncogenic H-ras (Kampfer et aL, 1998). PKCX. and participate in the ras-mediated reorganization of the F-actin cytoskeleton (Oberall et al., 1999). PKC can control the phosphorylation and activation of PKC5 (Ziegler et al., 1999). It may be that not the levels of PKCs, but the levels of the dephospho-rylated forms of PKCs are important for apoptosis (Whelan and Parker, 1998). 

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Molecules in magnetic fields 4. Final remarks and conclusions Acknowledgements References... 

Remarks and Conclusions. Based on the results presented in the Figure 17 we can concluded the following ... 

Remarks and Conclusion. - (I) Resonance Problem For the resonance problem the most accurate methods are the methods derived by Simos and Williams (Case IV, Case V and Case VI of the family),20 the method derived by Simos,24 the hybrid sixth algebraic order methods derived by Thomas and Simos (Case IV and Case V of the family),25 the new phase fitted method of algebraic order eight developed in this critical review in Section 2.3.1, the eighth algebraic order exponentially-fitted method derived by Simos,30 the eighth... 

Remarks and Conclusion. - The most accurate fourth order P-stable method with constant coefficients is the P-stable method proposed by Simos66 with phase-lag of order sixteen. The most accurate sixth order method with constant coefficients is the modification of the family of sixth order methods of Simos69 with phase-lag of order twenty-two developed in this review. The most accurate eighth order method with constant coefficients is the eighth order method of Simos77 with phase-lag of order eighteen and interval of periodicity... 

Remarks and Conclusion. - The most efficient variable-step method for the solution of coupled differential equations arising from the Schrodinger equation is the variable-step Bessel and Neumann fitted method of Simos.44 Efficient variable-step method for the solution of the above problem is also the variable-step Bessel and Neumann fitted method of Simos 43 Efficient methods also are the new exponentially-fitted variable step method which is produced based on the new exponentially-fitted tenth algebraic order method developed in Section 2.1.6 and the method of Simos,30 the exponentially-fitted variable-step method developed by Thomas and Simos,25 the new variabe-step method which is developed using the modification of the family of... 

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