© Hindawi Publishing Corp. SEVEN-DIMENSIONAL CONSIDERATIONS OF EINSTEIN’S CONNECTION. III. THE SEVEN-DIMENSIONAL EINSTEIN’S CONNECTION

The purpose of the present paper is to prove a necessary and sufficient condition for a unique Einstein’s connection to exist in 7-g-UFT and to display a surveyable tensorial representation of seven-dimensional Einstein’s connection in terms of the unified field tensor, employing the powerful recurrence relations of the third kind obtained in earlier papers. All considerations in this paper are restricted to the first and second classes of X7, since the case of the third class, the simplest case, was already studied by many authors.

Einstein's connection in a generalized Riemannian manifold X n has been investigated by many authors for lower-dimensional cases n = 2,...,6.In a series of papers, we obtain a surveyable tensorial representation of seven-dimensional Einstein's connection in terms of unified field tensor, with main emphasis on the derivation of powerful and useful recurrence relations which hold in X 7 .
In [2], which we denote by I in the present paper, we gave a brief survey of Einstein's unified field theory and derived the recurrence relations of the first kind which hold in a general X n .In [1], which we denote by II in the present paper, we derived a powerful recurrence relations of the second and third kinds which hold in seven-dimensional Einstein's generalized Riemannian manifold X 7 .These relations will be used in the present paper to find a tensorial representation of seven-dimensional Einstein's connection.All considerations in this paper are based on the results and symbolism of I and II.Whenever necessary, they will be quoted in the present paper.
In the following theorem, we prove a necessary and sufficient condition for a unique Einstein's connection to exist in 7-g-UFT.
Theorem 1 (for the first and second class).A necessary and sufficient condition for the existence and uniqueness of the solution of [1, (2.5) or (2.26)] in 7-g-UFT is given by: For the first class, where ( For the second class with the second category, For the second class with the first category, Proof.For the first class, the symmetric scalars M xyz defined by [1, (2.27)] take values as in Table 1, in virtue of [2, (3a)].
It may be easily verified that the product of 3 factors in the first row of Table 1 is g given by [2, (1)], that of the 4 factors in the second row is 1−3K 2 + 9K 4 − 27K 6 , that of the 7 factors in the third row is 1 2 , and that of the 8 factors in the fourth row is 1 − 2K 4 − 8K 6 + (K 4 ) 2 − 4K 2 K 6 .After a lengthy calculation, we obtain the product of the 12 factors in the fifth row as where D, E, and F are given by (2).Therefore, our assertion follows in virtue of [1, (2.28)].
The proof of the second class may be obtained easily from (1) and (4) by simply substituting the corresponding conditions of each case.
In the following three theorems, we establish a linear system containing the torsion tensor S = S ωλµ of the Einstein's connection, employing the powerful recurrence relations of the third kind obtained in [2, Theorem 5].
At least one of x, y, z is 7 and no two take 1, 2 nor 3, 4 nor 5, 6 No two are equal and no two take 1, 2 nor 3, 4 nor 5, 6 Two are equal and one of three is most 7 Theorem 2 (for the first class of 7-g-UFT).The system of [1, (2.26) or (2.5)] is reduced to the following linear system of 66 equations:

S ,
given by where

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