Mean Value Theorem Of Calculus Research Articles

Correction and Supplement to Approach for a Proof of Riemann Hypothesis by Second Mean-Value Theorem

From the theorem 1 formulated in [1], a set of functions of measure zero within the set of all corresponding functions has to be excluded. These are the cases where the Omega functions Ω(u) are piece-wise constant on intervals of equal length and non-increasing due to application of second mean-value theorem or, correspondingly, where for the Xi functions Ξ(z) the functions Ξ(y)y are periodic functions on the imaginary axis y with z=x+iy. This does not touch the results for the Omega function to the Riemann hypothesis by application of the second mean-value theorem of calculus and the majority of other Omega functions in the suppositions, but makes their derivation correct. The corresponding calculations together with a short recapitulation of the main steps to the basic equations for the restrictions of the mean-value functions and the application to piece-wise constant Omega functions (staircase functions) are represented.

Approach to a Proof of the Riemann Hypothesis by the Second Mean-Value Theorem of Calculus

By the second mean-value theorem of calculus (Gauss-Bonnet theorem) we prove that the class of functions with an integral representation of the form with a real-valued function which is non-increasing and decreases in infinity more rapidly than any exponential functions , possesses zeros only on the imaginary axis. The Riemann zeta function as it is known can be related to an entire function with the same non-trivial zeros as . Then after a trivial argument displacement we relate it to a function with a representation of the form where is rapidly decreasing in infinity and satisfies all requirements necessary for the given proof of the position of its zeros on the imaginary axis z=iy by the second mean-value theorem. Besides this theorem we apply the Cauchy-Riemann differential equation in an integrated operator form derived in the Appendix B. All this means that we prove a theorem for zeros of on the imaginary axis z=iy for a whole class of function which includes in this way the proof of the Riemann hypothesis. This whole class includes, in particular, also the modified Bessel functions for which it is known that their zeros lie on the imaginary axis and which affirms our conclusions that we intend to publish at another place. In the same way a class of almost-periodic functions to piece-wise constant non-increasing functions belong also to this case. At the end we give shortly an equivalent way of a more formal description of the obtained results using the Mellin transform of functions with its variable substituted by an operator.

Optimal fuzzy rules cover extrema

A fuzzy system approximates a function by covering the graph of the function with fuzzy rule patches and averaging patches that overlap. But the number of rules grows exponentially with the total number of input and output variables. the best rules cover the extrema or bumps in the function—they patch the bumps. For mean-squared approximation this follows from the mean value theorem of calculus. Optimal rules can help reduce the computational burden. to find them we can find or learn the zeroes of the derivative map and then center input fuzzy sets at these points. Neural systems can then both tune these rules and add rules to improve the function approximation. © 1995 John Wiley & Sons, Inc.