The field of ocean surface waves can be expressed as a superposition of infinite number of wavetrains which have different wave-lengths and directions of propagation. These wavetrains, not just propagating independently, interact with each other through the nonlinearity of the free surface boundary conditions, thus producing a physical situation often called "wave turbulence" or "weak turbulence". Owing to the recent advances in computing facilities as well as numerical methods, large-scale direct numerical simulations of water wave turbulence based on the primitive governing equations are now becoming possible. Among many other applications, this advance would make it possible to assess the validity of those basic assumptions and hypotheses on which the traditional statistical theories have been build. Considering the current situation as above, a review on numerical methods which can be used for large-scale direct numerical simulations of water wave turbulence will be given. Some applications of those methods are also presented briefly.

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This paper surveys recent developments in numerical methods for option pricing, focusing on the approaches based on fast integral transforms. Under the Black-Scholes framework, the pricing of discrete path dependent options and Bermudan options can be reduced to evaluation of a series of convolutions of the Gaussian distribution and a known function. These convolutions can be computed efficiently using the double-exponential integration formula and fast integral transforms. The resulting algorithms have computational complexity of O (N) at each time step, where N is the number of sample points at each step, and the error decreases exponentially with N. Thus these algorithms can be shown to be faster and more accurate than any other existing algorithms. Extensions of this approach to a wider variety of options including weather derivatives and options under jump-diffusion models are also discussed.

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The paper deals with the Freak Wave with its mathematical basis. First, the concept of Freak Waves in the actual ocean is described from the beginning of their research up to present state of the arts. Non-linear wave theory in particular NLS equation is briefly explained with its important solutions including envelope soliton and the breather (SFB). The latter solution is surmised to be a possible mechanism of the occurrence of Freak Waves. It-rises from ordinary wave train caused by modulational instability well known after Benjamin-Feir. I deduced this most representative example within the technique of elementary mathematics herein, however most powerful and complex method of Inverse Scattering Technique (IST) revealed the decomposition of Freak Waves from ordinary quasi linear waves by Osborne et al.. For the practical application of the theory in the field of ocean engineering and naval architecture, the in-situ data acquisition and/or real time observation in the open ocean are indispensable to evade from this tremendous disaster. A crucial problem remains in this research is mentioned with the proposals in future.

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A Newly developed optimal control theory based on energy equation is introduced. The point of this theory is to utilize the self solving mechanism which is included in closed loop systems as a general characteristic. For this-purpose, the energy equation of controlled system is focused on. In this method, the functional consists of total energy of controlled system, performance function and energy function. This energy function is described by energy flow which is transmitted from controller to controlled system. Being shown as a first-order expression concerning control variables, the functional is able to induce the optimal system equation from the condition which minimizes the functional. Then optimal control is realized by putting the optimal equation into a closed loop system without solving the equation. This method is useful for nonlinear mechanical control systems because real time control can be realized by this method. Appling this theory to concrete problems, it is clarified that this theory is extremely appropriate for the optimal control of the systems with complex dispersion structures and the nonholonomic systems whose criteria functions are described by complex forms.

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