Variational Principles and Conservation Conditions in Volterra's Ecology and in Urban Relative Dynamics

Contribution to the Metropolitan Study: 14 -- From an analytical viewpoint, Volterra's variational principles and their associated integrands in single and multiple species interaction under absolute growth conditions in the field of mathematical ecology are reconsidered and simplified. They are then compared with the conservation conditions found appropriate to hold in a class of dynamic problems of relative growth in urban analysis. The comparison assists in interpreting the integrands of geographical (spatial) associations as a "stationary effort fitness function" associated with a cumulative entropy measure of the relative urban dynamic spatial distributions. From a substantive viewpoint, the paper shows the theoretical conditions, which would result in all spatial activity to be concentrated into a single point, so that inferences can be made regarding the conditions under which the activity will disperse. It also demonstrates that assuming a particular problem formulation, in this case a relative dynamic framework in an inactive environment, will result in obtaining spatial competitive exclusion. This is demonstrated in a parsimonious manner

Period-doubling in discrete relative spatial dynamics and the Feigenbaum sequence

AbstractIt is shown in this paper that although the period-doubling Feigenbaum sequence and the associated universal numbers in discrete maps of the logistic type hold over parameters, their true nature have them holding over slopes of the corresponding Poincaré maps. This finding enables one to find these Feigenbaum slope sequences in more complex maps. Further, it is demonstrated by an example in discrete relative growth spatial dynamics that a Feigenbaum sequence does not hold over the bifurcation parameter

A 5-year study on the effect of hormone therapy, tibolone and raloxifene on vaginal bleeding and endometrial thickness

Objectives: To study the effect of standard and low-dose estrogen-progestin therapy (EPT), tibolone and raloxifene on the incidence of vaginal spotting/bleeding and endometrial thickness over a 5-year period. Methods: Seven hundred eighty-six postmenopausal women were studied in an open prospective design. Vaginal spotting/bleeding and endometrial thickness as assessed by transvaginal ultrasonography was compared between six categories of women over a 5-year period: three categories in women on continuous combined estrogen-progestin therapy, one category under tibolone, one category under raloxifene and one under no treatment. More specifically, women received tibolone 2.5 mg (N = 204), raloxifene HCl 60 mg (N = 137), conjugated equine estrogens 0.625 mg/medroxyprogesterone acetate 5 mg (N = 122), 17β-estradiol 2 mg/norethisterone acetate 1 mg (N = 58), 17β-estradiol 1 mg/norethisterone acetate 0.5 mg (N = 76) or no therapy (controls, N = 189). Women with suspected endometrial pathology were referred for hysteroscopy. Results: Bleeding/spotting incidence was highest among standard dose EPT users (conjugated equine estrogens 0.625 mg/medroxyprogesterone acetate 5 mg: 40.1%, 17β-estradiol 2 mg/norethisterone acetate 1 mg: 44.8%, p < 0.001 compared to controls). Low-dose EPT associated with lower incidence of spotting/bleeding (34.1%). The incidence under tibolone and raloxifene was 22.5% and 2.9%, respectively, while 3.2% of women not receiving therapy reported vaginal spotting/bleeding. Mean endometrial thickness was not significantly affected in any of the groups studied. The drop-out rate due to spotting/bleeding was higher in the two higher dose EPT regimens. After logistic regression analysis, age at baseline was the only significant predictor of subsequent spotting/bleeding (b = -0.25, S.E. = 0.09, p = 0.006), while menopausal age and pre-treatment serum FSH had marginal significance. Conclusions: EPT, tibolone and raloxifene do not appear to associate with significant changes in endometrial thickness in the majority of cases. The low-dose EPT regimen associated with a decreased incidence of unscheduled spotting/bleeding compared to the standard dose regimens. Tibolone expressed a favorable endometrial profile, as seen in its effect on unscheduled spotting/bleeding and mean endometrial thickness. Raloxifene associated with the lowest incidence in S/B and the lowest drop-out rate.s © 2005 Elsevier Ireland Ltd. All rights reserved