In this paper the complex-valued bes 528 t linear unbiased estimator of an unknown constant mean of white noise was derived the ordinary least-squares estimator of an unknown constant mean of random field (arithmetic mean) charged by an imaginary error.

Understanding how spatial configurations of economic activity emerge is important when formulating spatial planning and economic policy. A simple model was proposed by Simon, who assumed that firms grow at a rate proportional to their size, and that new divisions of firms with certain probabilities relocate to other firms or to new centres of economic activity. Simon's model produces realistic results in the sense that the sizes of economic centres follow a Zipf distribution, which is also observed in reality. It lacks realism in the sense that mechanisms such as cluster formation, congestion (defined as an overly high density of the same activities) and dependence on the spatial distribution of external parties (clients, labour markets) are ignored. <br />The present paper proposed an extension of the Simon model that includes both centripetal and centrifugal forces. Centripetal forces are included in the sense that firm divisions are more likely to settle in locations that offer a higher accessibility to other firms. Centrifugal forces are represented by an aversion of a too high density of activities in the potential location. The model is implemented as an agent-based simulation model in a simplified spatial setting. By running both the Simon model and the extended model, comparisons are made with respect to their effects on spatial configurations. To this end a series of metrics are used, including the rank-size distribution and indices of the degree of clustering and concentration.

This Chapter is written for the Festschrift celebrating the 70th birthday of the distinguished economist Duncan Foley from the New School for Social Research in New York. This Chapter reviews applications of statistical physics methods, such as the principle of entropy maximization, to the probability distributions of money, income, and global energy consumption per capita. The exponential probability distribution of wages, predicted by the statistical equilibrium theory of a labor market developed by Foley in 1996, is supported by empirical data on income distribution in the USA for the majority (about 97%) of population. In addition, the upper tail of income distribution (about 3% of population) follows a power law and expands dramatically during financial bubbles, which results in a significant increase of the overall income inequality. A mathematical analysis of the empirical data clearly demonstrates the two-class structure of a society, as pointed out Karl Marx and recently highlighted by the Occupy Movement. Empirical data for the energy consumption per capita around the world are close to an exponential distribution, which can be also explained by the entropy maximization principle.

We investigate the structure of the profit landscape obtained from the most basic, fluctuation based, trading strategy applied for the daily stock price data. The strategy is parameterized 9c5 by only two variables, p and q. Stocks are sold and bought if the log return is bigger than p and less than -q, respectively. Repetition of this simple strategy for a long time gives the profit defined in the underlying two-dimensional parameter space of p and q. It is revealed that the local maxima in the profit landscape are spread in the form of a fractal structure. The fractal structure implies that successful strategies are not localized to any region of the profit landscape and are neither spaced evenly throughout the profit landscape, which makes the optimization notoriously hard and hypersensitive for partial or limited information. The concrete implication of this property is demonstrated by showing that optimization of one stock for future values or other stocks renders worse profit than a strategy that ignores fluctuations, i.e., a long-term buy-and-hold strategy.

We introduce a new threshold model of social networks, in which the nodes influenced by their neighbours can adopt one out of several alternatives. We characterize social networks for which adoption of a product by the whole network is possible (respectively necessary) and the ones for which a unique outcome is guaranteed. These characterizations directly yield polynomial time algorithms that allow us to determine whether a given social network satisfies one of the above properties. <br />We also study algorithmic questions for networks without unique outcomes. We show that the problem of determining whether a final network exists in which all nodes adopted some product is NP-complete. In turn, the problems of determining whether a given node adopts some (respectively, a given) product in some (respectively, all) network(s) are either co-NP complete or can be solved in polynomial time. <br />Further, we show that the problem of computing the minimum possible spread of a product is NP-hard to approximate with an approximation ratio better than $\Omega(n)$, in contrast to the maximum spread, which is efficiently computable. Finally, we clarify that some of the above problems can be solved in polynomial time when there are only two products.