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Tumor necrosis factor-α (TNF-α), an adipokine, is produced in adipocytes, and the elevation of its levels has been linked to obesity and insulin resistance in some population. In this study the relationship between TNF-α promoter gene polymorphism and obesity in an Iranian population has been studied. Subjects were randomly selected from Tehran Cohort Lipid and Glucose Study. Adult participants placed in three groups according to their body mass index (BMI): BMI < 25, 25 ≤ BMI < 30, BMI ≥ 30 and under-18 subjects placed in two groups, under 85th percentile BMI and above 85th percentile. Finally, 244 persons were selected for G-308A and G-238A polymorphisms analysis. The FBS, high-density lipoprotein cholesterol (HDL-C), low-density lipoprotein cholesterol (LDL-C), triglyceride, cholesterol levels and blood pressure and HOMA of all subjects were measured. The polymorphism −308 and −238 were revealed by restriction fragment length polymorphism (RFLP; NCOI and MSPI) after the promoter site was amplified by PCR. The allele frequency of TNF-α polymorphism was in the Hardy–Weinberg equilibrium. There was no relation between BMI and the frequency of this allele. The fact that there is no association between G-308A and G-238A TNF-α promoter polymorphisms and obesity probably shows that it is not an important risk factor for obesity and consequently for cardiovascular disease.

Suppose T and S are bounded adjointable operators with close range between Hilbert C*-modules, then T S has closed range if and only if Ker(T ) + Ran(S) is an orthogonal summand, if and only if Ker(S * ) + Ran(T * ) is an orthogonal summand. Moreover, if the Dixmier (or minimal) angle between Ran(S) and Ker(T ) ∩ [Ker(T ) ∩ Ran(S)] ⊥ is positive and Ker(S * ) + Ran(T * ) is an orthogonal summand then T S has closed range. Introduction.The closeness of range of operators is an attractive and important problem which appears in operator theory, especially, in the theory of Fredholm operators and generalized inverses.In this paper we will investigate when the product of two operators with closed range again has closed range. This problem was first studied by Bouldin for bounded operators between Hilbert spaces in [3,4]. Indeed, for Hilbert space operators T, S whose ranges are closed, he proved that the range of T S is closed if and only if the Dixmier (or minimal) angle between Ran(S) and Ker(T ) ∩ [Ker(T ) ∩ Ran(S)] ⊥ is positive, where the Dixmier angle between subspaces M and N of a certain Hilbert space is the angle α 0 (M, N) in [0, π/2] whose cosine is defined by c 0 (M, N) = sup{ x, y : x ∈ M, x ≤ 1 , y ∈ N, y ≤ 1}.Nikaido [24,25] also gave topological characterizations of the problem for the Banach space operators. Recently (Dixmier and Friedrichs) angles between linear subspaces have been studied systematically by Deutsch [7], he also has reconsidered the closeness of range of the product of two operators with closed range. In this note we use C*-algebras techniques to reformulate some results of Bouldin and Deutsch in the framework of Hilbert C*-modules.Some further characterizations of modular operators with closed range are obtained.Hilbert C*-modules are essentially objects like Hilbert spaces, except that the inner product, instead of being complex-valued, takes its values in a C*-algebra. Since the geometry of these modules emerges from the C*-valued inner product, some basic properties of Hilbert 2000 Mathematics Subject Classification. Primary 47A05; Secondary 15A09, 46L08, 46L05.