We consider the single layer shallow water equations on a bounded domain M ⊂ R 2 forced by a multiplicative white noise, and obtain the existence and uniqueness of a maximal pathwise solution for a short period of time. The proof relies on the Skorohod representation theorem, the Gyöngy-Krylov theorem, stopping time arguments, and isotropic estimates.
Let F q be a finite field of q elements where q is a large odd prime power andWe prove bounds on the number of incidences between a point set P and a Q-sphere set S, denoted by I(P, S), as the following.We prove this estimate by studying the spectra of directed graphs. We also give a version of this estimate over finite rings Z q where q is an odd integer. As a consequence of the above bounds, we give an estimate for the pinned distance problem. In Sections 4 and 5, we prove a bound on the number of incidences between a random point set and a random Q-sphere set in F d q . We also study the finite field analogues of some combinatorial geometry problems, namely, the number of generalized isosceles triangles, and the existence of a large subset without repeated generalized distances.
Vietnam has pilot-tested a payment for forest environmental services (PFES) program in an effort to restore and protect forest areas, some of which have been severely degraded by the excessive cutting of trees by small-scale farmers planting annual crops on steep, sloping lands. The pilot program implemented in southern Vietnam seems to be successful, yet the program in northern Vietnam has not produced the desired rates of planting and maintaining forest areas. The reasons for these mixed results include differences in socio-economic characteristics and also the production and marketing opportunities available to rural households in the project areas. To gain insight regarding program participation, we examine the household-level opportunity costs of planting and maintaining small Household opportunity costs of protecting and developing forest lands 903 plots of forest trees in northern Vietnam. We find that small-scale farmers in Hoa Binh Province, with limited financial resources, prefer the annual revenue stream provided by crops such as maize and cassava, rather than waiting for 7 years to obtain revenue from a forest planting. Farmers in Son La Province, with limited access to markets, prefer annual crops because they are not able to sell bamboo shoots and other forest products harvested from their small plots. In both provinces, the payments offered for planting and maintaining forest trees are smaller than the opportunity costs of planting and harvesting annual crops. Thus, most households likely would choose not to participate in the PFES program, at current payment rates, if given the opportunity to decline.Keywords: Bamboo, cassava, feasibility, maize, payment rate, PES, PFES, smallholders Acknowledgement: We appreciate the helpful comments of two reviewers, who encouraged us to clarify the discussion and enhance our empirical analysis. We appreciate also the financial support provided by the Federal Ministry for Economic Cooperation and Development, Germany, through the project, "Opportunities for economic incentives to promote sustainable land and water management on the sloping lands of South and Southeast Asia."
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