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Decomposition of Transition Kernels Via Tridiagonal Multivariate Integral Kernel Enhanced Multivariance Products Representation (TMIKEMPR)
Last modified: 2020-02-02
Abstract
In today's conditions, technological developments and diversity of the problems of human life make the investigation of multivariable functions more remarkable. As the multivariance of the functions increase, it becomes necessary to classify and minimize the process complexity. Kernels can be defined as the multivariate functions which include two different type independent variables. Even though kernels or kernel functions come to mind as symmetrical mathematical objects, this is not the most general case. Nonsymmetric function can also be considered as kernels especially in the cases of integral operators (or summation operators) which are not self-
adjoint. We consider the kernel in most general meaning. We use it as a multivariate function denoted by K(x,y) where x and y stand for two N1 and N2 element set of scalar independent variables here. In this work we are going to focus on the cases where N1 is not equal to N2. The cases where N1=N2 has been investigated in a separate work where the symmetry K(x,y) = K(y,x)) has also been considered. Hence, the cases where N1 = N2 without symmetry can also be investigated by using the method we develop in this work. As long as N1 is not equal to N2 the corresponding kernel maps from one function space to another whose dimensions are different. This corresponds in fact to a transition from one space to another. Even in the case where N1 =N2 but symmetry does not exist, the action represented by this kernel is a transition because the domain and space of images are different. By following the terminology we have used in the companion of this paper, ys can be called “operand or operation” coordinates while xs can be considered as “image variables”. Even in this work, our purpose is again to decompose a given kernel function (like the purpose in the companion paper) in such a way that it can be expressed as a (finite or denumerable infinite) linear combination of functions each of which is a binary product of two functions depending on only either x or y. One way to this end is, unlikely to the companion paper, the singular value decomposition based representation. However, we prefer to develop a new TKEMPR-like algorithm which is based on an appropriately defined four component partially bivariate EMPR expansion. This is done in such a way that the scalar independent variables are replaced by two set of scalars, x and y. Then this new bivariate EMPR is constructed through the same philosophy of usual bivariate EMPR by replacing the singlefold integrations by N-tuple integrations. The remaining is exactly in the same way of tridiagonalization used in TKEMPR. The difference from the companion paper comes from the nonexistence of symmetry.
The authors are very grateful to Professor Metin Demiralp for his invaluable supervising supports to this work during the studies.
adjoint. We consider the kernel in most general meaning. We use it as a multivariate function denoted by K(x,y) where x and y stand for two N1 and N2 element set of scalar independent variables here. In this work we are going to focus on the cases where N1 is not equal to N2. The cases where N1=N2 has been investigated in a separate work where the symmetry K(x,y) = K(y,x)) has also been considered. Hence, the cases where N1 = N2 without symmetry can also be investigated by using the method we develop in this work. As long as N1 is not equal to N2 the corresponding kernel maps from one function space to another whose dimensions are different. This corresponds in fact to a transition from one space to another. Even in the case where N1 =N2 but symmetry does not exist, the action represented by this kernel is a transition because the domain and space of images are different. By following the terminology we have used in the companion of this paper, ys can be called “operand or operation” coordinates while xs can be considered as “image variables”. Even in this work, our purpose is again to decompose a given kernel function (like the purpose in the companion paper) in such a way that it can be expressed as a (finite or denumerable infinite) linear combination of functions each of which is a binary product of two functions depending on only either x or y. One way to this end is, unlikely to the companion paper, the singular value decomposition based representation. However, we prefer to develop a new TKEMPR-like algorithm which is based on an appropriately defined four component partially bivariate EMPR expansion. This is done in such a way that the scalar independent variables are replaced by two set of scalars, x and y. Then this new bivariate EMPR is constructed through the same philosophy of usual bivariate EMPR by replacing the singlefold integrations by N-tuple integrations. The remaining is exactly in the same way of tridiagonalization used in TKEMPR. The difference from the companion paper comes from the nonexistence of symmetry.
The authors are very grateful to Professor Metin Demiralp for his invaluable supervising supports to this work during the studies.