In this paper, we study the weak convergence of Mann’s type iteration procedure and the existence of nonexpansive retractions for commutative semigroups in Banach spaces which satisfy Opial’s condition. Further, we introduce an implicit iteration procedure for nonexpansive semigroups and then prove a strong convergence theorem for the nonexpansive semigroups in general Banach spaces.

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In this paper, we introduce new properties of topological spaces defined by stars of coverings, which are called strict star-Lindel¨ofness and strict starcompactness. Following the fundamental studies on star covering properties by E.K.van Douwen, G.M.Reed, A.W.Roscoe and I.J.Tree ([1]), there have been several related studies (see M.V.Matveev [7], [8], [9], for survey). Star-Lindel¨of spaces have many nice properties. However, star-Lindel¨ofness is not preserved by closed subspaces ([7]). We define strict star-Lindel¨ofness to modify this defect and still so as to keep possible properties of star-Lindel¨ofness. Furthermore, we investigate relationships among these covering properties and give various examples.

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In this paper, we introduce the generalized R − KKM mapping and the new class R_ψN− KKM(X, Y ), and we get some fixed point theorems, matching theorems, coincidence theorems, and minimax inequalities on L-convex spaces.

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We address an optimal portfolio selection problem of maximizing so we call CVaR (Conditional Value-at-Risk) based Sharpe ratio of return rate of portfolio, which is defined as the ratio of the expected excess return to CVaR. The Sharpe ratio defined as the ratio of expected excess return to standard deviation, the most common traditional performance measure, takes standard deviation as a risk measure, however, its has been received a lot of criticisms. In our CVaR based Sharpe ratio, the standard deviation is replaced by CVaR, which is a remarkable coherent risk measure which overcomes essential defects of standard deviation. Although our new performance measure is expected to enlarge the applicable area of practical investment problems for which the original Sharpe ratio is not suitable, however, we should device effective computational methods to solve optimal portfolio selection problems with very large number of investment opportunities. In order to deal with rather complicated non-concave objective function, which comes from the introduction of CVaR, in this paper, we propose a Genetic Algorithm (GA) approach. The paper briefly reviews the literature in the area of application of Genetic Algorithms to financial problems, and then details the development of Genetic Algorithm for portfolio selection. In order to evaluate CVaR for each portfolio, by utilizing the results of Rockafellar and Uryasev (2000), we introduce an auxiliary decision variable to obtain a tractable concave maximization problem. Furthermore, if we estimate or approximate required expected values by sampling methods or historical data, we can reduce this concave maximization problem to an LP (Linear Programming) problem. Therefore, our problem could be solved by GA which incorporates LP for evaluating values of CVaR. Numerical experiments from real Japanese financial data are conducted to test our approach to conclude that, by suitable choice of probability parameter of three evolution operation, we could effectively solve optimal selection problems with practical sizes.

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In this paper we consider a large deviations problem for a discrete-time polling system consisting of two-parallel queues and a single server. The arrival process of each queue is a superposition of traffic streams generated by a number of mutually independent and identical Markovian on/off sources, and the single server serves the two queues according to the so-called Bernoulli service schedule. Using the large deviations techniques, we derive the upper and lower bounds of the probability that the queue length of each queue exceeds a certain level (i.e., the buffer overflow probability). These results have important implications for traffic management of high-speed communication networks such as call admission control and bandwidth allocation.

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The most familiar parlor games High-Hand-Wins poker, Hi-Lo poker and La Relance poker are discussed, under the situation where the players’ hands are delivered by extremely non-uniform distributions. It is shown that if players’ hands are distantly (closely) distributed from (to) the opponent’s one, then they behave cautiously (boldly), when the amount of bet is not so small.

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In this paper we introduce the notion of a quadratic BF-algebra, and obtain that quadratic BF-algebras, quadratic Q-algebras, BG -algebras and B-algebras are equivalent notions on a field X with |X| ≥ 3, and hence every quadratic BF-algebra is a BCI-algebra.

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In our paper “L²-boundedness of Marcinkiewicz integrals along surfaces with variable kernels”, we have found that Theorems 3 and 5 are not correct. Here, we reformulate Theorems 3 and 5, so that L²-boundedness holds as in Theorems 1 and 4.

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