The mean-variance model from the inverse of the variance-covariance matrix

In this paper we obtain the main results of the Markowitz mean-variance model from the inverse of the covariance matrix, following a shorter and mathematically rigorous path. We also obtain the equilibrium expression of Sharpes capital asset pricing model (CAPM).capm, beta, portfolio composition

Managing induced tourism image: Relational patterns and the life cycle

The tourism image is an element that conditions the competitiveness of tourism destinations by making them stand out in the minds of tourists. In this context, marketers of tourism destinations endeavour to create an induced image based on their identity and distinctive characteristics.A number of authors have also recognized the complexity of tourism destinations and the need for coordination and cooperation among all tourism agents, in order to supply a satisfactory tourist product and be competitive in the tourism market. Therefore, tourism agents at the destination need to develop and integrate strategic marketing plans.The aim of this paper is to determine how cities of similar cultures use their resources with the purpose of developing a distinctive induced tourism image to attract tourists and the extent of coordination and cooperation among the various tourism agents of a destination in the process of induced image creation.In order to accomplish these aims, a comparative analysis of the induced image of two cultural cities is presented, Girona (Spain) and Perpignan (France). The induced image is assessed through the content analysis of promotional brochures and the extent of cooperation with in-depth interviews of the main tourism agents of these destinations.Despite the similarities of both cities in terms of tourism resources, results show the use of different attributes to configure the induced image of each destination, as well as a different configuration of the network of tourism agents that participate in the process of induced image creation

The degree-number of vertices problem in Manhattan networks

Generally speaking, the aim of this work is to study the problem (Delta,N) (or the degree-number of vertices problem) for the case of a Manhattan digraph. A digraph is a network formed by vertices and directed edges called arcs (in the case of graphs the edges have no direction). The diameter of a graph is the minimum distance that exists between two of the farthest vertices. In the diameter of a digraph we must take into account that arcs have direction. A double-step digraph consists of N vertices and a set of arcs of the form (i,i+a) and (i,i+b), with a and b positive integers called 'steps', that is, there are connections from vertex i to vertices i+a and i+b (operations are modulo N). This digraph is denoted by G(N;a,b). A double-step graph G(N;+-a,+-b) consists of N vertices, but the edges are of the form (i,i+-a) and (i,i+-b), with steps a and b (positive integers), therefore, there are connections from vertex i to vertices i+a, i-a, i+b and i-b (mod N). In a Manhattan digraph, the arcs have directions like the ones of the streets and avenues of Manhattan (or l'Eixample in Barcelona), that is, if an arc goes to the right, the 'next one' goes to the left and if an arc goes upwards, the 'next one' goes downwards. The (Delta,N) problem consists in finding the minimum diameter of a graph or digraph given the number of vertices N and the maximum degree Delta. As this problem has been solved for the case of double-step graphs G(N;+-a,+-b), we expand these graphs transforming every vertex into a directed cycle of order 4 and every edge into two arcs in opposite directions, so that we obtain a Manhattan digraph M. In this work we find the relation between the steps of the double-step graph G(N;+-a,+-b) and the ones of the Manhattan digraph M. Moreover, we made a program that calculates the diameter of the so-called New Amsterdam digraph NA, related to the Manhattan digraph M, from the parameters of the original graph G(N;+-a,+-b).Català: En termes generals, l’objectiu d’aquest treball és estudiar el problema (o problema grau-nombre de vèrtexs) per al cas del digraf Manhattan. Un digraf és una xarxa constituïda per vèrtexs i per arestes dirigides anomenades arcs (en el cas de grafs, les arestes no tenen direcció). El diàmetre d’un graf és la mínima distància possible que hi ha entre dos dels vèrtexs més allunyats entre si. En el diàmetre d’un digraf hem de tenir en compte que els arcs tenen direcció. Un digraf de doble pas consta de vèrtexs i un conjunt d'arcs de la forma i , amb i enters positius anomenats “passos", és a dir, que existeixen enllaços des del vèrtex cap els vèrtexs i (les operacions s'han d'entendre sempre mòdul ). Aquest digraf es denota . Un graf de doble pas també consta de vèrtexs, però les arestes són de la forma i , amb passos i (enters positius), per tant, existeixen enllaços des del vèrtex cap els vèrtexs i (mod ) . En un digraf Manhattan els arcs tenen les direccions com les dels carrers i les avingudes de Manhattan (o de l'Eixample de Barcelona), és a dir, si un arc va cap a la dreta, el "següent" va cap a l'esquerra i si un arc va cap a dalt, el "següent" va cap a baix. El problema consisteix a trobar el diàmetre mínim d'un graf o digraf fixats el nombre de vèrtexs i el grau . Com que aquest problema ha estat resolt per al cas de grafs de doble pas , hem expandit aquests grafs transformant cada vèrtex en un cicle dirigit de 4 vèrtexs i cada aresta en dos arcs de sentits oposats, de manera que obtenim un digraf Manhattan . En aquest treball trobem la relació entre els passos del graf de doble pas i els del digraf Manhattan . A més, hem fet un programa que calcula el diàmetre del digraf anomenat New Amsterdam , que està relacionat amb el Manhattan , a partir dels paràmetres del graf original