The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein.

The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. While the advice and information in this book are believed to be true and accurate at the date of publication, neither the authors nor the editors nor the publisher can accept any legal responsibility for any errors or omissions that may be made. The publisher makes no warranty, express or implied, with respect to the material contained herein.

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We consider standard fermion doublets and Dirac fermions within the framework of a radiative mechanism of mass generation. The mass matrix represents a solution of a multi-gap equation. Taking such a solution, we compute the mass matrix of the upper (lower) members of the doublets with respect to the basis where the lower (upper) members have a mass matrix that is flavor-diagonal. If this form lies in the vicinity of its diagonalized mass matrix, the mixing will be called "natural." Naturalness relations between the products of mixing angles and ratios of flavor masses result. They are of substantial predictive power in both the case of the quarks and the leptons.