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zZ*am?_=j4FZ{!!2l9CC(fgx$a8u^7J$ELtvpk4i6ftr#2-L-75Krtu~^xKP~d-ezl zWO%+uPnu`A9zoq1sjKGyo#IwIXOsrM`8y*sDm^nhJtnph9Dw0^rjeUluK_`g{tv1j BYS#b& diff --git a/src/sections/abstract.tex b/src/sections/abstract.tex index 3d43b0c..88b7f05 100644 --- a/src/sections/abstract.tex +++ b/src/sections/abstract.tex @@ -7,7 +7,6 @@ For every $t\in\mathbb{T}_1$ and $x,b\in \Lambda^2$ \[ \frac{\Delta t^{2m+1}}{\Delta t} = -% \frac{\partial P(m,b,x)}{\Delta x} \bigg |_{x = t, \; b = \sigma(t)} + \frac{\partial P(m,b,x)}{\Delta x} (m, \sigma(t), t) + \frac{\partial P(m,b,x)}{\Delta b} (m, t, t) \] diff --git a/src/sections/introduction.tex b/src/sections/introduction.tex index f0e590e..fd40cb0 100644 --- a/src/sections/introduction.tex +++ b/src/sections/introduction.tex @@ -16,4 +16,4 @@ In context of Computer Science, namely object oriented programming paradigm, the time scale calculus may be thought as unified interface of derivative operator. Furthermore, the idea of time-scale calculus was slightly extended -in~\cite{bayour2017truly,benkhettou2016conformable,caputo2009time,martins2009calculus}. \ No newline at end of file +in~\cite{bayour2017truly,benkhettou2016conformable,caputo2009time,martins2009calculus}. diff --git a/src/sections/main-results.tex b/src/sections/main-results.tex index b36a653..e096b11 100644 --- a/src/sections/main-results.tex +++ b/src/sections/main-results.tex @@ -1,31 +1,31 @@ -Time scale derivative of the polynomial $t^{2m+1}$ may be expressed as follows +Timescale derivative of the polynomial $t^{2m+1}$ may be expressed as follows \begin{thm} \label{main_theorem} - Let $\polynomialP{m}{b}{x}$ be a $2m+1-$degree integer-valued polynomial defined by~\eqref{eq:polynomial_p}. - Let be a two-dimensional time scale - $\Lambda^2 = \mathbb{T}_1 \times \mathbb{T}_2 = \{t=(x, b) \colon \; x\in\mathbb{T}_1, \; b\in\mathbb{T}_2 \}$. - Let be $\mathbb{T}_1 = \mathbb{T}_2$. - For every $t\in\mathbb{T}_1, \; x,b\in \Lambda^2, \; m\in\mathbb{N}, \; m = const$ + Let $P(m,b,x)$ be a $2m+1$-degree polynomial in $x,b$. + Let be a two-dimensional timescale + $\Lambda^2 = \mathbb{T}_1 \times \mathbb{T}_2 = \{t=(x, b) \colon \; x\in\mathbb{T}_1, \; b\in\mathbb{T}_2 \}$ + such that $\mathbb{T}_1 = \mathbb{T}_2$. + For every $t\in\mathbb{T}_1$ and $x,b\in \Lambda^2$ \[ - (t^{2m+1})^{\Delta} = - \frac{\partial \polynomialP{m}{b}{x}}{\Delta x} \bigg |_{x = t, \; b = \sigma(t)}+ - \frac{\partial \polynomialP{m}{b}{x}}{\Delta b} \bigg |_{x = t, \; b = t}, + \frac{\Delta t^{2m+1}}{\Delta t} = + \frac{\partial P(m,b,x)}{\Delta x} (m, \sigma(t), t) + + \frac{\partial P(m,b,x)}{\Delta b} (m, t, t) \] where \begin{itemize} \setlength\itemsep{1em} - \item $\sigma(t) > t$ is forward jump operator, + \item $\sigma(t) > t$ -- is forward jump operator - \item $\frac{\partial \polynomialP{m}{b}{x}}{\Delta x} \bigg |_{x = t, \; b = \sigma(t)}$ + \item $\frac{\partial \polynomialP{m}{b}{x}}{\Delta x} (m, \sigma(t), t)$ -- is the value of the partial derivative on time scales of - $\polynomialP{m}{b}{x}$ with respect to the variable $x$, evaluated at $x = t, \; b = \sigma(t)$, + $\polynomialP{m}{b}{x}$ with respect to the variable $x$ evaluated in point $x = t, \; b = \sigma(t)$ - \item $\frac{\partial \polynomialP{m}{b}{x}}{\Delta b} \bigg |_{x = t, \; b = t}$ + \item $\frac{\partial \polynomialP{m}{b}{x}}{\Delta b} (m, t, t)$ -- is the value of the partial derivative on time scales of - $\polynomialP{m}{b}{x}$ with respect to the variable $b$, evaluated at $x = t, \; b = t$. + $\polynomialP{m}{b}{x}$ with respect to the variable $b$, evaluated at $x = t, \; b = t$ \end{itemize} \end{thm} -In other words, theorem ~\ref{main_theorem} says +In simpler words, the theorem ~\ref{main_theorem} says \begin{center} \begin{quotation} For every odd-exponent polynomial $t^{2m+1}$, its derivative on time scales equals to the sum @@ -51,4 +51,4 @@ ^{\Delta}_{t} = \frac{\partial}{\Delta x} \left( \sum_{k=0}^{b-1} \sum_{r=0}^{m} \coeffA{m}{r} k^r(x-k)^r \right) \Bigg |_{x = t, \; b = \sigma(t)} + \frac{\partial}{\Delta b} \left( \sum_{k=0}^{b-1} \sum_{r=0}^{m} \coeffA{m}{r} k^r(x-k)^r \right) \Bigg |_{x = t, \; b = t} -\end{align*} \ No newline at end of file +\end{align*}