ON STRICTLY CONVEX AND STRICTLY 2-CONVEX 2-NORMED SPACES

In this paper a new duality mapping is defined, and it is our object to show that there is a similarity among these three types of characterizations of a strictly convex 2-normed space. This enables us to obtain more new results along each of two types of characterizations. We shall also investigate a strictly 2-convex 2-normed space in terms of the above two different types.

Let X be a 2-normed space throughout this paper.If x,y,z X are nonzero vectors, we denote by V(x), V(x, y) and V(x, y,z) the linear manifolds of X generated by x, x and y, x, y and z, respectively.STRICTLY CONVEX 2-NORMED SPACES.
Recall from [5] thatXis said to be strictly convex if ]x + y,z -IIx,zll -II y,zll x for z V (x,y) implies x y.In this section we shall give several characterizations of this space in terms of 2-semi-inner products and duality mappings.But first we need the following lemma which is essential to our consequent theorems, and which is a portion of Theorem 1 in 11] plus three new statements (8), (9), and (10).
The concept of 2-semi-inner product defined by Siddiqui and Rizvi [14] is 2-dimensional analogue of that of the usual semi-inner product in functional analysis.A 2-semi-inner product is a mapping [.,. .]onX X X into real numbers such that (i) [x +x',y Iz]-[x,y Iz]+[x',y Iz];, (ii) [ax, y z] a [x,y ]z] for any real a; (iii) [x,x z] 0; [x,x ]z] 0 if and only if x and z arc linearly dependent; and (iv) I[x,y Every 2-nor,med space can be made into a 2-semi-inner product space, and the norm is given by Ilx,yll -- [x,x [yf [14].
Let us define another type of duality mapping as follows: DEFINITION.Let l'(x,z) be the same as l(x,z) which has the following additional properties: (i) IIx,zll y,zll if and only if II/11 I111 for z q V(x,y),/" .l(x,z)and g lO,,z); and (ii) IIx,zll llx,,ll if and only if ll/l llhll forxqV(z,w), J'.l(x,z)andh l(x,w).
In a similar manner we can prove the following analogous result.

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