Projections in a normed linear space and a generalization of the pseudo-inverse

The concept of a "projection function" in a finite-dimensional real or complex normed linear space H (the function P_M which carries every element into the closest element of a given subspace M) is set forth and examined.

If dim M = dim H - 1, then P_M is linear. If P_N is linear for all k-dimensional subspaces N, where 1 ≤ k < dim M, then P_M is linear.

The projective bound Q, defined to be the supremum of the operator norm of P_M for all subspaces, is in the range 1 ≤ Q < 2, and these limits are the best possible. For norms with Q = 1, P_M is always linear, and a characterization of those norms is given.

If H also has an inner product (defined independently of the norm), so that a dual norm can be defined, then when P_M is linear its adjoint P_M^H is the projection on (kernel P_M)^⊥ by the dual norm. The projective bounds of a norm and its dual are equal.

The notion of a pseudo-inverse F⁺ of a linear transformation F is extended to non-Euclidean norms. The distance from F to the set of linear transformations G of lower rank (in the sense of the operator norm ∥F - G∥) is c/∥F⁺∥, where c = 1 if the range of F fills its space, and 1 ≤ c < Q otherwise. The norms on both domain and range spaces have Q = 1 if and only if (F⁺)⁺ = F for every F. This condition is also sufficient to prove that we have (F⁺)^H = (F^H)⁺, where the latter pseudo-inverse is taken using dual norms.

In all results, the real and complex cases are handled in a completely parallel fashion.

The Spanish Pension System: Overview and Long-Term Projections

In the middle of the so-called pension crisis, the ageing of population and the sharp decrease in affiliations to Social Security are threatening the well-functioning of the Spanish public pension system. The purpose of this paper is to present the main challenges to be faced by the Spanish pension system, as well as to shed light on the main determinants that will condition the evolution of pension expenditure over GDP along the following decades. This quantitative analysis, which considers the 2011 and 2013 pension reforms, uses the latest data on the Spanish demographic, labor market and institutional factors in order to project the evolution of the system from the expenditure side up to 2060. For the purpose of analyzing the dynamics and the underlying drivers of pension expenditure, the so-called aggregate accounting approach will be utilized. The alternative scenarios proposed allow for exploring the robustness of the results within the baseline scenario, which constitutes the reference point of projections in the exercise. The analysis concludes that, by the end of the projected period, Spain will count on a similar public pension expenditure ratio as the current one.

Fish demand and supply projections

It has been predicted that the global demand for fish for human consumption will increase by more than 50% over the next 15 years. The FAO has projected that the increase in supply will originate primarily from marine fisheries, aquaculture and to a lesser extent from inland fisheries, but with a commensurate price increase. However, there are constraints to increased production in both marine and inland fisheries, such as overfishing, overexploitation limited potential increase and environmental degradation due to industrialization. The author sees aquaculture as having the greatest potential for future expansion. Aquaculture practices vary depending on culture, environment, society amd sources of fish. Inputs are generally low-cost, ecologically efficient and the majority of aquaculture ventures are small-scale and family operated. In the future, advances in technology, genetic improvement of cultured species, improvement in nutrition, disease management, reproduction control and environmental management are expected along with opportunities for complimentary activities with agriculture, industrial and wastewater linkages. The main constraints to aquaculture are from reduced access to suitable land and good quality water due to pollution and habitat degradation. Aquaculture itself carries minimal potential for aquatic pollution. State participation in fisheries production has not proven to be the best way to promote the fisheries sector. The role of governments is increasingly seen as creating an environment for economic sectors to make an optimum contribution, through support in areas such as infrastructure, research, training and extension and a legal framework. The author feels that a holistic approach integrating the natural and social sciences is called for when fisheries policy is being examined.

Building social and ecological resilience to climate change in Roviana, Solomon Islands: PASAP country activity for Solomon Islands: Brief review: climate change trends and projections for Solomon Islands

As part of the Australian Government’s International Climate Change Adaptation Initiative (ICCAI), the Pacific Adaptation Strategy Assistance Program (PASAP) aims to enhance the capacity of partner countries to assess key vulnerabilities and risks, formulate adaptation strategies and plans, mainstream adaptation into decision-making, and inform robust longterm national planning and decision-making in partner countries. The Department of Climate Change and Energy Efficiency contracted University of Queensland (UQ) and University of California, Santa Barbara (UCSB) to lead the project: “Building social and ecological resilience to climate change in Roviana, Solomon Islands” (2010-2012). Under this project The WorldFish Center was subcontracted to undertake outputs 5 and 6 of Objective three: (5) Review of climate change evidence and projections for the study area and (6) Vulnerability and adaptation assessment for the study area. This report addresses the first of these and comprises a desktop review of climate change evidence and projections for the study area.