Contraction of electroweak model and neutrino

The electroweak model, which lepton sector correspond to the contracted gauge group SU(2; j) × U(1), j ?? 0, whereas boson and quark sectors are standard one, is suggested. The field space of the model is fibered under contraction in such a way that neutrino fields are in the fiber and all other fields are in the base. Properties of the fibered field space are understood in context of semi-Riemannian geometry. This model describes in a natural manner why neutrinos so rarely interact with matter, as well as why neutrino cross section increase with the energy. Dimensionfull parameter of the model is interpreted as neutrino energy. Dimensionless contraction parameter j at low energy is connected with the Fermi constant of weak interactions and is approximated as j ² ?? 10^?5.     

The structure of optimal solutions for harvesting a renewable resource

We consider the problem of optimal harvesting of a renewable resource whose dynamics are governed by logistic growth and whose payoff is proportional to the harvest. We consider both the case of a finite and an infinite time horizon and analyse the structure of the optimal solutions and their dependence on the parameters of the model. We show that the optimal policy can only have one of three structures: (1) maximal harvesting effort until the resource is depleted, (2) zero harvesting during an initial time interval followed by a subsequent switch to maximal harvesting effort, or (3) a singular solution, which corresponds to an intermediate level of harvesting, accompanied by the most rapid approach path. All three scenarios emerge, with minor variations, with finite and infinite time horizons, depending on the particular combination of parameters of the system. We characterize the conditions under which the singular solution is optimal and present suggestions for designing an optimal and sustainable harvesting strategy. Recommendations for Resource Managers :

• We have rigorously explored a standard optimal harvesting model and its steady states.
• We show that three different types of solutions may emerge: (i) maximal harvesting eventually leading to a complete depletion of the stock; (ii) maximal harvesting with a potential period of idleness leading to a positive stock; (iii) an initial phase of either no or full harvesting followed by a period of intermediate harvesting intensity leading to a positive stock (singular solution).
• With some modifications, similar results hold for a finite planning horizon.
• Which of these three scenarios emerges in the finite horizon case depends not only on the parameter values but also on the length of the planning horizon.