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The numerical range and numerical radius of some model operators are investigated. Among other things, a complete description of the numerical range of square-zero operators on Hilbert spaces, including the skew-symmetric Volterra operator (V0f)(x)=∫-xxf(t)dt on the Lebesgue space L2(-1,1),L2(-1,1), is also given. We study the effect of international business trips on the productivity of twelve Australian industries during 1991/2–2005/6.We find strong positive effects, particularly for departing, rather than incoming, international visits. This result supports viewing business visits as investments rather than expenditures. In an extensive category satisfying a mild chain condition, the arithmetic of multiplication and addition (cartesian product and coproduct) of objects is shown to be very close Cheap Louis Vuitton Wallet to that of natural numbers. Examples of such categories abound, e.g. in algebraic geometry. A definition of internal family of equations in the algebras for an internal type in Louis Vuitton Outlet Online an elementary topos with natural numbers object is given. The category of algebras which satisfy an internal family of equations is shown to be monadic over the base topos and internally cocomplete provided the topos in Grothendieck over a topos with the internal Axiom of Choice.