Dimensional dual hyperovals in projective spaces and c.AGast-geometries
The two subjects of the title are studied, first independently and then by making them interact. Many questions arise from this interaction.In our intrinsic study of the first subject, we construct a new family of examples and define the notions of quotient and universality. For the second subject, we restrict the class c.AG* of circular extensions of dual affine spaces under the two geometrical conditions (LL) and (T) by showing that, apart from some extreme cases, every such rank three geometry can be erected into a rank 4 geometry. In particular, c.AG*n(q)-geometries satisfying (LL) and (T) with n ≥ 3 must have q even. We then deduce similar results for the first subject.
On some d-dimensional dual hyperovals in PG(d(d+3)/2, 2)
Let d>=3. Let H be a d+1-dimensional vector space over GF(2) and {e"0,...,e"d} be a specified basis of H. We define Supp(t)@?{e"t"""1,...,e"t"""l}, a subset of a specified base for a non-zero vector t=e"t"""1+...+e"t"""l of H, and Supp(0)@[email protected]?. We also define J(t)@?Supp(t) if |Supp(t)| is odd, and J(t)@?Supp(t)@?{0} if |Supp(t)| is even. For s,[email protected]?H, let {a(s,t)} be elements of [email protected]?([email protected]?H) which satisfy the following conditions: (1) a(s,s)=(0,0), (2) a(s,t)=a(t,s), (3) a(s,t) (0,0) if s t, (4) a(s,t)=a(s^',t^') if and only if {s,t}={s^',t^'}, (5) {a(s,t)|[email protected]?H} is a vector space over GF(2), (6) {a(s,t)|s,[email protected]?H} generate [email protected]?([email protected]?H). Then, it is known that [email protected]?{X(s)|[email protected]?H}, where X(s)@?{a(s,t)|[email protected][email protected]?{s}}, is a dual hyperoval in PG(d(d+3)/2,2)=([email protected]?([email protected]?H))@?{(0,0)}. In this note, we assume that, for s,[email protected]?H, there exists some x"s","t in GF(2) such that a(s,t) satisfies the following equation: a(s,t)[email protected][email protected]?J(t)a(s,w)+x"s","t(a(s,0)+a(s,e"0)). Then, we prove that the dual hyperoval constructed by {a(s,t)} is isomorphic to either the Huybrechts' dual hyperoval, or the Buratti and Del Fra's dual hyperoval.
Implementation of Tourism Satellite Account: Assessing the Contribution of Tourism to the Croatian Economy
Compiling the indicators of tourism consumption and the output of tourism industries, Tourism Satellite Account (TSA) provides a framework for internationally harmonized measurements of direct macroeconomic contribution of tourism. Ivandic and Marusic provide a detailed methodological approach for the transfer of TSA theoretical concepts to the assessment of tourism economic contribution on the case of the Croatia. Giving that the TSA compilation is still characterized by a surprisingly wide range of ‘styles’ used, the chapter contributes to a better understanding of the issues related to measurement of direct tourism impacts. It also gives a set of baseline criteria for resource allocation within tourism policy and the further evaluation of its effectiveness.