Ercan Gürvit

Evaluation of Univariate Integrals via Fluctuationlessness Theorem

A recently developed Fluctuation Method is used in approximating the remainder term of the integral of the Taylor expansion. This provides us with a new numerical integration method.

Univariate Function Decomposition via Tridiagonal Vector Enhanced Multivariance Products Representation (TVEMPR)

This work is devoted to the decomposition of a univariate function by using very recently developed Tridiagonal Vector Enhanced Multivariance Products Representation (TVEMPR). To this end the target function is expressed as a bilinear form over the power vector of the independent variable and the functions coefficient vector. Both vectors are composed of denumerable infinite number of elements. The power vector of the independent variable is decomposed via Tridiagonal Vector Enhanced Multivariance Products Representation. The core matrix of the decomposition contains a 2×2 type left uppermost block as the only nonzero agent. Then the bilinear form, and therefore the function can be expressed thoroughly to get a decomposition as a linear combination of certain functions which are in fact derived from the original target function. This is the simplest case. Some other but complicated cases which start with multi outer products are left to future works. The support vectors have been chosen as proportional to certain power vectors of some given parameters to proceed from rather simplicity.

Taylor Series Expansion with the Fluctuation Freely Approximated Remainder Over Highly Oscillatory Basis Functions

A new formulation is developed here to approximate highly oscillatory functions by applying the Fluctuationlessness Theorem to the remainder term of the Taylor polynomial. To this end a trigonometric basis set is utilized. Because of the limitation of space in this extended abstract the implementation of results are left to the presentation.

Evaluation of Multivariate Integrals via Fluctuationlessness Theorem and Taylor’s Remainder

A recently developed Fluctuationlessness Method is used in approximating the multiple remainder terms of the integral of the multivariate Taylor expansion. This provides us with a new numerical integration method for multivariate functions.

Fluctuationless Univariate Integration Through Taylor Expansion with Remainder by Using Oscillatory Function Basis Sets

This work uses a recently developed fluctuation free matrix representation method in approximating the integral of the Taylor expansion remainder term. The basis set used for the matrix representation contains common factors of sine and cosine functions with the same frequencies and the same origin. This provides a new numerical univariate integration method to us such that the approximation quality can be controlled by the number of the expansion terms in the Taylor expansion of the integrand and by the dimension of the subspace over which the matrix representations are built. The number of oscillations in the basis set is also another quality control agent and may help to get better approximants in the case of high oscillations. Due to the limitation of space in this extended abstract results of the implementations are left to the presentation.

Multi Nodalset Fluctuation Free Approximation in Taylor Remainder’s Evaluation

The matrix representation of a univariate function is equal to the image of the independent variable matrix representation under that function at the no fluctuation limit. In recent studies this fact is extended in such a way that the matrix representation of a univariate function can be expressed as a linear combination of the same function with two different matrix arguments. This idea makes us think for more than two matrices whose images under the target function are combined to get better approximation. This paper focuses on the application of this approximation method on the integral representation of the remainder term of the Taylor series expansion. In this work the basic conceptual background is given. Some illustrative implementations will be given at the relevant conference presentation.

Multi Nodalset Fluctuation Free Integration in Taylor Remainder’s Evaluation

The matrix representation of a univariate function is equal to the image of the independent variable matrix representation under that function at the no fluctuation limit. In recent studies of BEBBYT group this fact is extended in such a way that the matrix representation of a univariate function can be expressed as a linear combination of the same function with two different matrix arguments each of which characterizes a deviation from the matrix representation of the independent variable when all fluctuations except the very first few are ignored. This idea urges us to search for more than two matrices whose images under the target function are combined to get better approximation. This paper focuses on the application of this approximation method on the integral representation of the Taylor series expansion. Here the basic conceptual background is given. Some illustrative implementations will be given at the relevant conference presentation.

Fluctuation studies in the infinite interval matrix representations of operator products and their decompositions

In this work a study on finite dimensional matrix approximations to products of quantum mechanical operators is conducted. It is emphasized that the matrix representation of the product of two operators is equal to the product of the matrix representation of each of the operators when all the fluctuation terms are ignored. The calculation of the elements of the matrices corresponding to the matrix representation of various operators, based on three terms recursive relation is defined. Finally it is shown that the approximation quality depends on the choice of higher values of n, namely the dimension of Hilbert space.

Taylor Formula with Remainder Term Evaluated under a Weight Factor at Fluctuationlessness Limit

Taylor expansion of a weighted function is taken into consideration. Fluctuationlessness theorem is applied to the remainder term in integral form and finally an expression which provides an approximation method to a family of non‐analytic functions is obtained.