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Author: John Liaperdos (ioannis.liaperdos@gmail.com)
Last update: April 28, 2015
Description: Provides an example of a PhD Thesis for Universidad de Antioquia
using the teipel-thesis-en pdfLaTeX class.
Character encoding: UTF-8
use the "modern" or "classic" option to switch between
a modern or classic font, respectively.
add/remove the "hyperref" option to enable/disable hyperlinks:
(remember to remove auxiliary files after adding/removing
the "hyperref" option).
add/remove the "printer" option to typeset a printer-friendly
(grayscale)/color version of the thesis.
use the "watermark" option to indicate a draft copy of the thesis
use the "histinit" option to enable "historiated initials".
(If used, all chapter initials declared by the \InitialCharacter{}
macro are enlarged. If ommitted, arguments of \InitialCharacter{}
are typeset as normal text.)
use the "plain" option to disable tikz graphics in title page
and part/chapter headers (might help to avoid compilation timeouts).
Note that "plain" disables CD label and CD cover creation.
use the "noindex" option to (hopefully) avoid compilation timeouts
when compiling online (disables index generation - note that "\indexGR",
"\index" invocations need not be removed when toggling this option).
use the "frontispiece" option to typeset a frontispiece at the back of the cover page
use the `letter' option for a US letter page layout, instead of A4
\documentclass[modern,hyperref,watermark,histinit,frontispiece,plain]{teipel-thesis-en}

Calculating the value of Ck ∈ {1, ∞} class of smoothness real-valued function's derivative in point of R+ in radius of convergence of its Taylor polynomial (or series), applying an analog of Newton's binomial theorem and q-difference operator. (P,q)-power difference introduced in section 5. Additionally, by means of Newton's interpolation formula, the discrete analog of Taylor series, interpolation using q-difference and p,q-power difference is shown.

In this paper, we derive and prove, by means of Binomial theorem and Faulhaber's formula, the following identity
between $m$-order polynomials in \(T\)
\(\sum_{k=1}^{\ell}\sum_{j=0}^m A_{m,j}k^j(T-k)^j=\sum_{k=0}^{m}(-1)^{m-k}U_m(\ell,k)\cdot T^k=T^{2m+1}, \ \ell=T\in\mathbb{N}.\)

The main aim of this paper to establish the relations between forward, backward and central finite (divided) differences (that is discrete analog of the derivative) and partial & ordinary high-order derivatives of the polynomials.

Kolosov Petro

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