P. Hall's classical equality for the number of conjugacy classes in p-groups yields k(G) ,(3=2) log2 jGj when G is nilpotent. Using only Hall's theorem, this is the best one can do when jGj = 2. Using a result of G. J. Sherman, we improve the constant 3=2 to 5=3, which is best possible across all nilpotent groups and to 15=8 when G is nilpotent and jGj n 6= 8,16. These results are then used to prove that k(G) > log3 (jGj) when G=N is nilpotent, under natural conditions on N E G. Also, when G 0 is nilpotent of class c, we prove that k(G) ,(log jGj) t when jGj is large enough, depending only on (c,t).

There are many studies discussing the correlation between air pollution and human health hazards. Yet, in Tehran there is not a survey using time series methodology. Thus, we conducted a study based on time series data on the topic in Tehran, Iran. Mean levels of NO, NO2, NOX, CO, 03, SO2 and PM10 (particulate matters smaller than 10micrometer in diameter) were measured in one station of Tehran's Air Quality Control Corporation and were used as main independent variables. Mean temperature, mean humidity, day of the week, month and season were considered as potential confounders and deaths in people older than 64 years in Tehran was the dependent variable. All the variables were measured during Mar. 1998 to Dec. 1999. Concentrations of air pollutants were different between seasons and so were the means of daily deaths. Out of main independent variables, SO2, CO and PM10 showed statistically significant relations with the dependent variable (P<0.05). After controling for confounders, there was 3.4%, 2.6% and 3.36% increase in death rates, respectively, for each interquartile ascending (increase from 25th centile to 75th centile) in association to the mentioned pollutant centile concentration. No autocorrelation between residuals was observed (r= -0.059). The study showed that meteorological variables can confound the relation between air pollution and rate of deaths per day.  

Curriculum as the backbone of the educational system is one of the important factors in creating fundamental changes in the higher education system; The purpose of this research is to explain the effectiveness of entrepreneurship education model from the Islamic point of view and  the Leader, so that it can be used to examine the curriculum of architecture education in terms of the realization of entrepreneurship components in students. The method of the current research is a mixed exploratory one, which in the qualitative part, by examining the views of the Leader in his articles, books and speeches, descriptive propositions of his thought regarding the effectiveness of entrepreneurship education were extracted with the implication research method. In the quantitative part, the data was analyzed using structural equation method and using SPSS20 and smart PLS software. The findings showed that the model of entrepreneurship from the Leader'spoint of view can be divided into four main pillars: vision, knowledge, skill and business. And each of these main pillars overlaps with the model of effectiveness of entrepreneurship education in architecture curriculum. In this way, architecture higher education centers can, in addition to creating entrepreneurship curricula in the field of architecture, by identifying the potentials in the curricula of this field, strengthen the strengths and eliminate the weaknesses in them. This model, while providing the conditions for the realization of entrepreneurship education in academic fields, provides the context for the implementation of a local model in the light of Islamic orders.

In this paper, we study the notion of solvable L-subgroup of an L-group and provide its level subset characterization and this justifies the suitability of this extension. Throughout this work, we have used normality of an L-subgroup of an L-group in the sense of Wu rather than Liu.

This paper presents characterizations of M-fuzzifying matroids by means of two kinds of fuzzy operators, called M-fuzzifying derived operators and m-fuzzifying difference derived operators.