In this paper we reprove three classical results on compact groups from Hofmann - Mostert book [8] . Notably we get a very short proof of Borel one point theorem [8, p.310].

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We introduce a new multiple weights class and generalize the Adams inequality to the multilinear fractional integral operator on weighted Morrey spaces. We also investigate the boundedness of the multilinear fractional integral operator from weighted Morrey spaces to BMO or Lipschitz spaces.

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We introduce a notion of a matched pair of locally compact groupoids and construct several C∗-algebras from a matched pair of locally compact groupoids without assuming the existence of quasi-invariant measures on the unit space. We also show that there exist natural representations of the above C∗-algebras when there exists an invariant measure.

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We introduce a notion of rank for C∗-algebras (or Banach algebras), which is viewed as a replacement of the topological stable rank of Rieffel. We study its basic properties and close relation with the stable rank. We also consider a replacement of the connected stable rank of Rieffel.

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In this paper, we establish a simple yet effective Taylor series expansion method of approximating the solutions of nonlinear Hammerstein equations. The method lends itself to numerical computations which can be done in parallel. Numerical examples are provided to demonstrate the effectiveness of the current method.

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An overlapped wavelet method is proposed to detect the number and locations of the hidden periodicities in two dimensionally indexed random fields, by checking if the empirical wavelet coefficients of periodogram have significantly large absolute values across fine scale levels. The magnitudes of the amplitudes are also estimated using wavelet coefficients of smaller scale levels than that for the detection of periodicities. The strong consistency of the estimators is established. Some numerical examples are given to test the performance of our method.

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In the present paper, a sequential decision problem on a Markov process is set up which takes into account a partial maintenance, and observe some monotonic properties of an optimal policy. We develop an optimal maintenance policy for products. During their life cycle, a condition of this item deteriorates, and a state of an item goes from state to state according to a Markovian transition rule based on the stochastic convexity. The decision-maker decides a level of repair with cost which varies with this level. This problem is how much to expend to maintain this item to minimize the total expected cost. A dynamic programming formulation implies a recursive equation about expected cost obtainable under optimal policy.

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Emphasis is placed on the valuation of plain vanilla option when the price process of underlying asset is described by the stochastic Verhulst-Gompertz Equation with network externality effects in a complete market. The method is based on the change of measure, Girsanov theorem and martingale valuation techinique. The application to an exchange option is made attempt and the valuation formula for this option like the Black-Scholes one is derived. A simple relationship similar to the put-call-parity between the exchange call option and the put option is provided. Our results will be useful to analyze and to hedge the price evolution at a sudden rise or crash of stock and commodity markets.

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