Weighted ergodic theorems for Banach-Kantorovich lattice Lp(∇^,μ^)L_{p}(\hat{\nabla},\hat{\mu})

In the present paper we prove weighted ergodic theorems and multiparameter weighted ergodic theorems for positive contractions acting on $L_p(\hat{\nabla},\hat{\mu})$. Our main tool is the use of methods of measurable bundles of Banach-Kantorovich lattices.Comment: 11 page

An unusual presentation of anetoderma: a case report

BACKGROUND: Anetoderma is a benign condition with focal loss of dermal elastic tissue resulting in localized areas of flaccid or herniated saclike skin. Currently, anetoderma is classified as either primary (idiopathic), or secondary anetoderma (which is associated with a variety of skin conditions, penicillamine use, or neonatal prematurity). Lesions appear on the upper arms, trunk, and thighs. CASE PRESENTATION: We report a 14-year-old boy, which was noticed to have had multiple, white, non-pruritic areas on the acral sites of upper and lower extremities for two years. In physical examination, the patient had normal mental development. Skin lesions consisted of scattered, white to skin-colored papules, less than 1 cm in diameter, and with central protrusion, with distribution on dorsal part of the index finger, forearms, distal portion of thighs and calves. Lesions were detected neither on the trunk nor the proximal areas of extremities. There are no sensory changes associated with the lesions. Otherwise, his general health was good. He did not have any medication consumption history. Family history was negative. Laboratory examinations were within normal limits. Skin biopsy from one of his lesions was done, that confirmed the diagnosis of anetoderma. CONCLUSIONS: In summary, we report a case of anetoderma on unusual sites of the skin. We could not find similar reports of anetoderma developing on distal extremities without involvement of the upper trunk and proximal arms, in the medical literature