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Register a new userThe Relationship Between a Function, a Functions Inverse, and their Antiderivatives with an Emphasis in Finding Exact Roots with the Technique of Integration
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Judah Francis Unmuth-Yockey
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Using a new technique involving integration it is possible to find the exact roots of simple functions. In this case, simple functions are defined as smooth functions having an inverse, and that inverse having an antiderivative. This technique now makes it possible to find the exact roots of certain functions without the use of numerical or iterative methods.
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A concise analytical formula is developed for the inverse of an invertible 3 x 3 matrix using a telescoping method, and is generalized to larger square matrices. The formula is confirmed using randomly generated matrices in Matlab
We show how a metric space induces a linear functional (a mean) on real-valued functions with domains in that metric space. This immediately induces a relative measure on a collection of subsets of the underlying set.
The Riemann Xi-function Xi(t)=xi(1/2+it) is a particularly interesting member of a broad family of entire functions which can be expanded in terms of symmetrized Pochhammer polynomials depending on a certain scaling parameter beta. An entire function in this family can be expressed as a specific integral transform of a function A(x) to which can be associated a unique minimal beta-sequence beta(min,n)-> infinity as n-> infinity, having the property that the Pochhammer polynomial approximant Xi(n,t,beta(n)) of order n to the function Xi(t) has real roots only in t for all n and for all beta(n)>= beta(min,n). The importance of the minimal beta-sequence is related to the fact that its asymptotic properties may, by virtue of the Hurwitz theorem of complex analysis, allow for making inferences about the zeros of the limit function Xi(t) in case the approximants Xi(n,t,beta(n)) converge. The objective of the paper is to investigate numerically the properties, in particular the very large n properties, of the minimal beta-sequences for different choices of the function A(x) of compact support and of exponential decrease, including the Riemann case.
Let n be a non-null positive integer and $d(n)$ is the number of positive divisors of n, called the divisor function. Of course, $d(n) leq n$. $d(n) = 1$ if and only if $n = 1$. For $n > 2$ we have $d(n) geq 2$ and in this paper we try to find the smallest $k$ such that $d(d(...d(n)...)) = 2$ where the divisor function is applied $k$ times. At the end of the paper we make a conjecture based on some observations.
In these notes, we consider the problem of finding the logarithm or the square root of a real matrix. It is known that for every real n x n matrix, A, if no real eigenvalue of A is negative or zero, then A has a real logarithm, that is, there is a real matrix, X, such that e^X = A. Furthermore, if the eigenvalues, xi, of X satisfy the property -pi < Im(xi) < pi, then X is unique. It is also known that under the same condition every real n x n matrix, A, has a real square root, that is, there is a real matrix, X, such that X^2 = A. Moreover, if the eigenvalues, rho e^{i theta}, of X satisfy the condition -pi/2 < theta < pi/2, then X is unique. These theorems are the theoretical basis for various numerical methods for exponentiating a matrix or for computing its logarithm using a method known as scaling and squaring (resp. inverse scaling and squaring). Such methods play an important role in the log-Euclidean framework due to Arsigny, Fillard, Pennec and Ayache and its applications to medical imaging. Actually, there is a necessary and sufficient condition for a real matrix to have a real logarithm (or a real square root) but it is fairly subtle as it involves the parity of the number of Jordan blocks associated with negative eigenvalues. As far as I know, with the exception of Highams recent book, proofs of these results are scattered in the literature and it is not easy to locate them. Moreover, Highams excellent book assumes a certain level of background in linear algebra that readers interested in the topics of this paper may not possess so we feel that a more elementary presentation might be a valuable supplement to Higham. In these notes, I present a unified exposition of these results and give more direct proofs of some of them using the Real Jordan Form.
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