We study evolutionary games on graphs. The individuals of a population occupy the vertices of the graph and interact with their neighbors to receive payoff. We consider finite population size, regular graphs, probabilistic death-birth updating and weak selection. There are two types of strategies, A and B, and a payoff matrix [(a, b),(c, d)]. The initial condition is given by an arbitrary configuration where each vertex is occupied by either A or B. The conjugate initial condition is obtained by swapping A and B. We ask: when is the fixation probability of A for the original configuration greater than the fixation probability of B for the conjugate configuration? The answer is a linear condition of the form σa+b > c+σd. We calculate σ for any initial condition. For large population size we obtain the well known result σ = (k + 1)/(k − 1), but now this result extends to any mixed initial condition. As a specific example we study evolution of cooperation. We calculate the critical benefit-to-cost ratio for natural selection to favor the fixation of cooperators for any initial condition. We obtain results that specify which initial conditions reduce and which initial conditions increase the critical benefit-to-cost ratio. Adding more cooperators to the initial condition does not necessarily favor cooperation. But strategic placing of cooperators in a network can enhance the takeover of cooperation.

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The concept of a selfdual connection on a fourdimensional Riemannian manifold is generalized to the 4ndimensional case of any quaternionic Khler manifold. The generalized selfdual connections are minima of a modified YangMills functional. It is shown that our definitions give a correct framework for a mapping theory of quaternionic Khler manifolds. The mapping theory is closely related to the construction of YangMills fields on such manifolds. Some monopolelike equations are discussed.

Technological innovations have revolutionized the process of scientific research and knowledge discovery. The availability of massive data and challenges from frontiers of research and development have reshaped statistical thinking, data analysis and theoretical studies. The challenges of high-dimensionality arise in diverse fields of sciences and the humanities, ranging from computational biology and health studies to financial engineering and risk management. In all of these fields, variable selection and feature extraction are crucial for knowledge discovery. We first give a comprehensive overview of statistical challenges with high dimensionality in these diverse disciplines. We then approach the problem of variable selection and feature extraction using a unified framework: penalized likelihood methods. Issues relevant to the choice of penalty functions are addressed. We demonstrate that for a host of statistical problems, as long as the dimensionality is not excessively large, we can estimate the model parameters as well as if the best model is known in advance. The persistence property in risk minimization is also addressed. The applicability of such a theory and method to diverse statistical problems is demonstrated. Other related problems with high-dimensionality are also discussed.