The Fundamental Crossed Module of the Complement of a Knotted Surface

We prove that if $M$ is a CW-complex and $M^1$ is its 1-skeleton then the crossed module $\Pi_2(M,M^1)$ depends only on the homotopy type of $M$ as a space, up to free products, in the category of crossed modules, with $\Pi_2(D^2,S^1)$. From this it follows that, if $G$ is a finite crossed module and $M$ is finite, then the number of crossed module morphisms $\Pi_2(M,M^1) \to G$ can be re-scaled to a homotopy invariant $I_G(M)$, depending only on the homotopy 2-type of $M$. We describe an algorithm for calculating $\pi_2(M,M^{(1)})$ as a crossed module over $\pi_1(M^{(1)})$, in the case when $M$ is the complement of a knotted surface $\Sigma$ in $S^4$ and $M^{(1)}$ is the handlebody made from the 0- and 1-handles of a handle decomposition of $M$. Here $\Sigma$ is presented by a knot with bands. This in particular gives us a geometric method for calculating the algebraic 2-type of the complement of a knotted surface from a hyperbolic splitting of it. We prove in addition that the invariant $I_G$ yields a non-trivial invariant of knotted surfaces in $S^4$ with good properties with regards to explicit calculations.Comment: A perfected version will appear in Transactions of the American Mathematical Societ

On Yetter's Invariant and an Extension of the Dijkgraaf-Witten Invariant to Categorical Groups

We give an interpretation of Yetter's Invariant of manifolds $M$ in terms of the homotopy type of the function space $TOP(M,B(G))$, where $G$ is a crossed module and $B(G)$ is its classifying space. From this formulation, there follows that Yetter's invariant depends only on the homotopy type of $M$, and the weak homotopy type of the crossed module $G$. We use this interpretation to define a twisting of Yetter's Invariant by cohomology classes of crossed modules, defined as cohomology classes of their classifying spaces, in the form of a state sum invariant. In particular, we obtain an extension of the Dijkgraaf-Witten Invariant of manifolds to categorical groups. The straightforward extension to crossed complexes is also considered.Comment: 45 pages. Several improvement

Entrepreneurial attitude, geographical isolation and university students - some evidence from the Atlantic

Some regions like Island of Madeira show high levels of firm birth rate. But the entrepreneurial experience is quite different from the European reality given the high level of micro-business owing o subsistence reasons, as a response to the reduced opportunity costs, the lack of profitable employment options and the high levels of unemployment. As a consequence the high level of entrepreneurship is only partially related to high tech innovative firms, qualified employment growth and economic diversification. The majority of the new firms are linked with traditional sectors (restaurants, boutiques, personal services and civil construction. Firm creation is also a result of the EU integration and cohesion policies. On the other hand infra-structure development policies explain the increasing importance of the public administration in terms of employment and consequently the low levels of unemployment. Traditionally, in the islands, the government intervention in terms of employment, economic planning is considered excessive. The island economies have been able to benefit from large streams of international solidarity in terms of high external aid per capita especially due to their strategic relevance. But the global economic and political change associated with the globalisation put increasing pressure on the island forcing them to reformulate their economic, social and political options. International donors and institutions like World Bank stresses issues such as economic diversification, economic and social modernisation and macroeconomic policies focused o supply side effects and the development of economic growth determining factors. Due to the reduced levels of international aid, islands are obliged to diminish levels of government intervention connected to public employment and direct production activities and to enhance private initiatives and entrepreneurship. In what concerns the outermost regions the EU enlargement demands increasing levels of competitivity, financial autonomy, economic diversification and entrepreneurial attitude. Given the lack of studies in this geographical area, the on-going economic, social and cultural modernisation induced by the integration in The EU sphere and the widespread perception about the changing times, we intend in this study to give some answers to the following questions: .how is the entrepreneurial attitude affected by the historical record of high levels of government intervention and public employment? .what kind of impacts results from the perceived “island penalty”, in terms of propensity towards entrepreneurship? .What is the main obstacle to the entrepreneurial event? .Should one wish to create a firm, what is the probability of the stated preference is in the high tech sectors? What kind of support will be required and welcomed from the public institutions? Studies and academic studies in islands have some advantages. The agglomeration of institutions, populations, firms and social networks in a reduced geographical space enable us to capture in some detail a vast group of variables, relationships and cause effects linked to a specific subject. Islands societies have a large and cohesive social capital, and share a homogenous set of values and cultural attitudes, which facilitates experiences of collective action. To conduct an empirical test in order to find out the most influential variables in the entrepreneurship attitude we use logit equations. The sample is made up of local university students, theoretically the most apt in developing innovative firms. We investigate also the differences between economics and managements and humanities students in terms of entrepreneurship propensity. An important matter in isolated, peripheral and underdeveloped regions is the diffusion of innovations. Consequently, student’s sources of information and knowledge regarding the overall tendencies of profitable, innovative and fashionable entrepreneurial experiences must be identified. Therefore, this paper describes the changing and uncertain economic and political environment faced by islands societies. A contextualisation of the relationship between entrepreneurship, economic growth and insular penalty is stretched and lastly, we provide an empirical study related to the entrepreneurial attitude in an insular region: The Island of Madeira.

A stochastic spreadsheet model analysing investment options for the development of pasture on beef cattle farms : a dissertation submitted in partial fulfilment of the requirements for the degree of Master in Applied Science at Massey University

The decision to proceed with farm development to increase animal production is complex. Standalone personal computer software to study either the financial or physical aspects of farm development is available, but models which integrate these components and account for the risks associated with the investment are not. A stochastic spreadsheet (Microsoft Excel®) model was therefore developed to predict the profitability, feasibility and risk of pasture development for two case farms: one in southern Brazil and the other near Wanganui in New Zealand. Pasture was developed at different rates for each farm and the model was used to predict the associated physical and financial changes over-time and a probability distribution of the net present values (NPV) of the net operating profit after tax and before interest (NOPAT) relative to the status quo situation. The extra pasture was used solely for increasing beef cattle production. On the Brazilian case farm the development of 2,263 ha at two rates was studied. The continuation of the status quo had first degree stochastic dominance in terms of the NPV over both development rates; it was superior by about NZ$46.000 for the 200 ha/y option and ca. NZ$ 110.000 for the 500 ha/y option at a 16% discount rate. However, at a 6% discount rate the 500 ha/y development rate had first degree stochastic dominance in terms of the NPV over both the continuation of the status quo (by about NZ$960.000) and the 200 ha/y option (ca. NZ$ 120.000). This indicates that pasture development could proceed profitability if interest rates continue to fall in Brazil as predicted. For the New Zealand case farm the development of 247 ha at 50 ha/y had first degree stochastic dominance over the 25 ha/y (ca. NZ$24,000) and continuation of the status quo (ca. NZ$ 208.000) at a 6% discount rate. Pasture development should therefore continue. Stochastic analysis of the pasture development investment options gave a better insight into the likely outcomes for a project, and provides the farmer with more information for making a decision on whether, and how. to proceed with farm development. The model could easily be adapted for studying farm development with respect to other types of livestock enterprises Keywords: pastures, development, risk, feasibility, profitability, model

Categorifying the sl(2,C)sl(2,C) Knizhnik-Zamolodchikov Connection via an Infinitesimal 2-Yang-Baxter Operator in the String Lie-2-Algebra

We construct a flat (and fake-flat) 2-connection in the configuration space of $n$ indistinguishable particles in the complex plane, which categorifies the $sl(2,C)$-Knizhnik-Zamolodchikov connection obtained from the adjoint representation of $sl(2,C)$. This will be done by considering the adjoint categorical representation of the string Lie 2-algebra and the notion of an infinitesimal 2-Yang-Baxter operator in a differential crossed module. Specifically, we find an infinitesimal 2-Yang-Baxter operator in the string Lie 2-algebra, proving that any (strict) categorical representation of the string Lie-2-algebra, in a chain-complex of vector spaces, yields a flat and (fake flat) 2-connection in the configuration space, categorifying the $sl(2,C)$-Knizhnik-Zamolodchikov connection. We will give very detailed explanation of all concepts involved, in particular discussing the relevant theory of 2-connections and their two dimensional holonomy, in the specific case of 2-groups derived from chain complexes of vector spaces.Comment: The main result was considerably sharpened. Title, abstract and introduction updated. 50 page

Invariants of Welded Virtual Knots Via Crossed Module Invariants of Knotted Surfaces

We define an invariant of welded virtual knots from each finite crossed module by considering crossed module invariants of ribbon knotted surfaces which are naturally associated with them. We elucidate that the invariants obtained are non trivial by calculating explicit examples. We define welded virtual graphs and consider invariants of them defined in a similar way.Comment: New results. A perfected version will appear in Compositio Mathematic