On the Convergence of Lacunary Walsh-Fourier Series

We study the Walsh-Fourier series of S_{n_j}f, along a lacunary subsequence of integers {n_j}. Under a suitable integrability condition, we show that the sequence converges to f a.e. Integral condition is only slightly larger than what the sharp integrability condition would be, by a result of Konyagin. The condition is: f is in L loglog L (logloglog L). The method of proof uses four ingredients, (1) analysis on the Walsh Phase Plane, (2) the new multi-frequency Calderon-Zygmund Decomposition of Nazarov-Oberlin-Thiele, (3) a classical inequality of Zygmund, giving an improvement in the Hausdorff-Young inequality for lacunary subsequences of integers, and (4) the extrapolation method of Carro-Martin, which generalizes the work of Antonov and Arias-de-Reyna.Comment: 18 pages. v2: Several typos corrected. Final version of the paper, accepted to LM

Weighted bounds for variational Walsh-Fourier series

For 1<p<infty, and weight w in A_p, and function f in L^p(w), we show that the r-variation of the Walsh-Fourier sums are finite, for r sufficiently large as function of w. (That r is a function of w is necessary.) This strengthens a result of Hunt-Young and is a weighted extension of a variation norm Carleson theorem of Oberlin-Seeger-Tao-Thiele-Wright. The proof uses phase plane analysis and a weighted extension of a variational inequality of Lepingle.Comment: v2: A major revision of v1, which only considered the pointwise convergence, not the finer variational informatio

DEMAND FOR DIFFERENTIATED VEGETABLES

To obtain a healthier diet, Americans need to consume not only more vegetables, but also a healthier mix of vegetables. Household demands for eight categories of vegetables are investigated, using ACNielsen's Homescan data. A maximum simulated likelihood estimation procedure results in elasticity estimates which are somewhat larger than those obtained from both time-series and cross-section data in the literature. Even these larger elasticities are not large enough to bridge the dietary consumption gap without, and possibly even with, substantial price or food expenditure subsidies. Furthermore, Homescan data do indicate some significant differences in preferences for types of vegetables by household characteristics, such as race and ethnicity. This information could be used in designing more effective public interventions for boosting vegetable consumption in the United States.Food Consumption/Nutrition/Food Safety,

Conjunctival blue nevus

The authors report a case of a conjunctival blue nevus and review the literature pertaining to these pigmented lesions in this location, describing clinical and histological report of a patient with a blue nevus of the palpebral conjunctiva with a literature review. A 64-year-old white female was evaluated for a darkening pigmented lesion of the left lower palpebral conjunctiva. Examination revealed a 3 mm x 6mm blue-black lesion with sharply demarcated edges and an irregular border. Histopa - thology showed plump spindle-shaped, pigmented melanocytic cells revealing a branching network of dendritic processes with small, elongated, and hyperchromatic nuclei consistent with a common blue nevus. No recurrence was noted at 9-month follow-up. Blue nevi of the conjunctiva are lesions that have a low risk for malignant transformation but can appear clinically similar to primary acquired melanosis or melanoma. Blue nevi of the conjunctiva are rare and represent 0.5%-3.0% of pigmented conjunctival lesions. There was one reported case in a literature search of a malignant melanoma arising from a conjunctival cellular blue nevus. Treatment is complete wide excisional biopsy

Oncocytoma of the upper conjunctival fornix

Oncocytomas are tumors characterized by large, eosinophilic epithelial cells with abundant mitochondria that form ductular or glandular spaces. While these tumors have been described in other organs, those of the ocular adnexa occur infrequently, with the caruncle being the most common site of involvement. Conjunctival oncocytomas are extremely rare and are believed to arise from the ductal elements of the lacrimal gland proper and the accessory lacrimal glands of the conjunctiva. We describe the clinical and histological features of a case of an oncocytoma presenting as an atypically located superior fornix mass in a 78-yearold female, with a review of the literature