work in progress

src/AStudyOnDynamicEquations.tex

@@ -21,7 +21,7 @@
 \newcommand \bernoulli [2][B] {{#1}\sb{#2}}
 \newcommand \curvePower [2]{\{#1\}\sp{#2}}
 \newcommand \coeffA [3][A] {{\mathbf{#1}} \sb{#2,#3}}
-\newcommand \polynomialP [4][P]{{\mathbf{#1}}\sp{#2} \sb{#3}(#4)}
+\newcommand \polynomialP [4][P] {#1 ( #2, #3, #4 )}
 
 % ordinary derivatives
 \newcommand \derivative [2] {\frac{d}{d #2} #1}                              % 1 - function; 2 - variable;
@@ -139,6 +139,7 @@
     \bibliography{AStudyOnDynamicEquations}
     \noindent \textbf{Version:} \input{sections/version}
 
+
     \section{Addendum 1: Mathematica scripts} \label{sec:mathematica_scripts}
     \input{sections/mathematica-scripts}
 

src/sections/abstract.tex

@@ -1,14 +1,14 @@
-Let $\mathbf{P}_b^m(x)$ be a $2m+1-$degree integer-valued polynomial in $x,b$.
-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 $\mathbb{T}_1 = \mathbb{T}_2$.
+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 $\mathbb{T}_1 = \mathbb{T}_2$.
 In this manuscript we derive and discuss the following partial dynamic equation on time scales.
-For every $t\in\mathbb{T}_1, \; x,b\in \Lambda^2, \; m = const, \; m\in\mathbb{N}$
+For every $t\in\mathbb{T}_1$ and $x,b\in \Lambda^2$
 \[
     (t^{2m+1})^{\Delta} =
-    \frac{\partial \mathbf{P}_b^m(x)}{\Delta x} \bigg |_{x = t, \; b = \sigma(t)}+
-    \frac{\partial \mathbf{P}_b^m(x)}{\Delta b}\bigg |_{x = t, \; b = t},
+    \frac{\partial P(m,b,x)}{\Delta x} \bigg |_{x = t, \; b = \sigma(t)} +
+    \frac{\partial P(m,b,x)}{\Delta b}\bigg |_{x = t, \; b = t}
 \]
-where $\sigma(t) > t$ is forward jump operator.
+such that $\sigma(t) > t$ is forward jump operator.
 In addition, we discuss various derivative operators in context of partial cases of above equation,
-we show finite difference, classical derivative, $q-$derivative, $q-$power derivative on behalf of it.
+we show finite difference, classical derivative, $q-$derivative, $q-$power derivative on behalf of it.