Abstract. The smoothing Newton method for solving a system of nonsmooth equations F (x) = 0, which may arise from the nonlinear complementarity problem, the variational inequality problem or other problems, can be regarded as a variant of the smoothing method. At the kth step, the nonsmooth function F is approximated by a smooth function f (·, ε k ), and the derivative of f (·, ε k ) at x k is used as the Newton iterative matrix. The merits of smoothing methods and smoothing Newton methods are global convergence and convenience in handling. In this paper, we show that the smoothing Newton method is also superlinearly convergent if F is semismooth at the solution and f satisfies a Jacobian consistency property. We show that most common smooth functions, such as the Gabriel-Moré function, have this property. As an application, we show that for box constrained variational inequalities if the involved function is P -uniform, the iteration sequence generated by the smoothing Newton method will converge to the unique solution of the problem globally and superlinearly (quadratically).
Objective
To determine the effect of vitamin D3 analogue (EB‐1089) on the growth and proliferation of a prostate cancer cell line (PC‐3).
Materials and methods
PC‐3 cells (104 cells per well) were plated into 24‐well tissue culture plates. After 24 h, the culture medium was replaced with one containing the vitamin D3 analogue EB‐1089; a control treatment using only replacement medium was conducted in parallel. Cell proliferation was measured by the incorporation of 3H‐thymidine 7 and 12 days after the addition of the vitamin D3 analogue. Cells were precipitated with 5% trichloroacetic acid and the radioactivity determined using a scintillation counter. Each experiment was performed at least five times.
Results
There was a significant dose‐dependent inhibition of cell growth after 7 and 12 days of treatment with EB‐1089, varying from 40 to 70% of the 3H‐thymidine incorporation by controls, respectively. The maximum inhibition occurred with 0.1 μmol/L EB‐1089 on day 7 and day 12 (both P<0.01). Longer incubation times appeared to have a greater effect when higher concentrations of EB‐1089 were used.
Conclusion
These in vitro studies have shown that the vitamin D3 analogue EB‐1089 can significantly reduce the growth rate of the prostate cancer cell line PC‐3. This would support the hypothesis that deficiency of vitamin D increases the risk of prostate cancer and further in vivo testing of vitamin D is warranted for its potential role in active therapy.
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