© Hindawi Publishing Corp. ON THE CONVOLUTION PRODUCT OF THE DISTRIBUTIONAL KERNEL Kα,β,γ,ν

We introduce a distributional kernel Kα,β,γ,ν which is related to the operator ⊕ k iterated k 
times and defined by ⊕ k=[(∑r=1p∂2/∂xr2)4−(∑j=p


Introduction. The operator ⊕ k can be factorized in the form
where p + q = n is the dimension of the space R n , i = √ −1, and k is a nonnegative integer.The operator 2 is first introduced by Kananthai [2] and named the Diamond operator denoted by (1.2) We denote the operators L 1 and L 2 by (1.3) Thus (1.1) can be written by where R H α (u), R e β (v), S γ (w), and T ν (z) are defined by (2.2), (2.4), (2.6), and (2.7), respectively.We defined the distributional kernel K α,β,γ,ν by Since the functions R H α (u), R e β (v), S γ (w), and T ν (z) are all tempered distributions and the supports of R H α (u) and R e β (v) are compact (see [2, pages 30-31] and [1, pages 152-153]), then the convolution on the right-hand side of (1.5) exists and also is a tempered distributions.Thus K α,β,γ,ν is well defined and also is a tempered distribution.

Preliminaries
Definition 2.1.Let x = (x 1 ,x 2 ,...,x n ) ∈ R n and write Denote by Γ + = {x ∈ R n : x 1 > 0 and u > 0} the interior of forward cone and Γ + denote its closure.For any complex number α, we define the function where the constant K n (α) is given by the formula For any complex number β, define the function where (2.5) For any complex numbers γ and ν, define where β+β where R e β and R β are given by (2.2).

Proof. See [5, page 20].
Lemma 2.5 (the convolution product of R H α (x)).The convolution product is given by (i) where R H α and R H α are defined by (2.1) with p even, (2.10) The proof of this lemma is given by Téllez [6, pages 121-123].
Lemma 2.6 (the convolutions product of S γ (w) and T ν (z)).The convolutions product is given by T ν+ν where S γ and T ν are defined by (2.6) and (2.7), respectively.

Proof. (i) Now
where ϕ ∈ Ᏸ the space of infinitely differentiable function with compact supports.We have ω .. , and

Call for Papers
As a multidisciplinary field, financial engineering is becoming increasingly important in today's economic and financial world, especially in areas such as portfolio management, asset valuation and prediction, fraud detection, and credit risk management.For example, in a credit risk context, the recently approved Basel II guidelines advise financial institutions to build comprehensible credit risk models in order to optimize their capital allocation policy.Computational methods are being intensively studied and applied to improve the quality of the financial decisions that need to be made.Until now, computational methods and models are central to the analysis of economic and financial decisions.
However, more and more researchers have found that the financial environment is not ruled by mathematical distributions or statistical models.In such situations, some attempts have also been made to develop financial engineering models using intelligent computing approaches.For example, an artificial neural network (ANN) is a nonparametric estimation technique which does not make any distributional assumptions regarding the underlying asset.Instead, ANN approach develops a model using sets of unknown parameters and lets the optimization routine seek the best fitting parameters to obtain the desired results.The main aim of this special issue is not to merely illustrate the superior performance of a new intelligent computational method, but also to demonstrate how it can be used effectively in a financial engineering environment to improve and facilitate financial decision making.In this sense, the submissions should especially address how the results of estimated computational models (e.g., ANN, support vector machines, evolutionary algorithm, and fuzzy models) can be used to develop intelligent, easy-to-use, and/or comprehensible computational systems (e.g., decision support systems, agent-based system, and web-based systems) This special issue will include (but not be limited to) the following topics: • Computational methods: artificial intelligence, neural networks, evolutionary algorithms, fuzzy inference, hybrid learning, ensemble learning, cooperative learning, multiagent learning

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Application fields: asset valuation and prediction, asset allocation and portfolio selection, bankruptcy prediction, fraud detection, credit risk management • Implementation aspects: decision support systems, expert systems, information systems, intelligent agents, web service, monitoring, deployment, implementation