MATH-55

CHANGELOG.md

@@ -5,6 +5,13 @@ All notable changes to this project will be documented in this file.
 The format is based on [Keep a Changelog](https://keepachangelog.com/en/1.0.0/),
 and this project adheres to [Semantic Versioning v2.0.0](https://semver.org/spec/v2.0.0.html).
 
+## [1.0.1] - 2024-09-21
+
+### Changed
+
+- Section files numbered
+- English grammar fixes
+
 ## [1.0.0] - 2024-09-20
 
 ### Changed

src/OnTheBinomialTheoremAndDiscreteConvolution.tex

@@ -79,49 +79,49 @@
 }
 \begin{document}
     \begin{abstract}
-        \input{sections/abstract}
+        \input{sections/01-abstract}
     \end{abstract}
     \maketitle
     \tableofcontents
 
 
     \section{Definitions, notations and conventions} \label{sec:definitions-notations-and-conventions}
-    \input{sections/definitions-notations-conventions}
+    \input{sections/02-definitions-notations-conventions}
     \clearpage
 
 
     \section{Introduction and main results} \label{sec:introduction}
-    \input{sections/introduction}
+    \input{sections/03-introduction-and-main-results}
 
 
     \section{Polynomial \texorpdfstring{$\polynomialP{m}{b}{x}$}{P[m,b,x]} and its properties}
     \label{sec:polynomial-p-and-its-properties}
-    \input{sections/polynomial-p-and-its-properties}
+    \input{sections/04-polynomial-p-and-its-properties}
 
 
-    \section{Polynomial \texorpdfstring{$\polynomialP{m}{b}{x}$}{P[m,b,x]} in terms of Binomial theorem}
+    \section{Relation between the polynomial \texorpdfstring{$\polynomialP{m}{b}{x}$}{P[m,b,x]} and Binomial theorem}
     \label{sec:odd-binomial-expansion-as-partial-case-of-polynomial-p}
-    \input{sections/odd-binomial-theorem-as-partial-case-of-p}
+    \input{sections/05-relation-betweeb-polynomial-p-and-binomial-theorem}
 
 
     \section{Polynomial \texorpdfstring{$\polynomialP{m}{b}{x}$}{P[m,b,x]} in terms of Discrete convolution}
     \label{sec:relation-between-p-and-convolution-of-polynomials}
-    \input{sections/relation-between-p-and-convolution-of-polynomials}
+    \input{sections/06-polynomial-p-in-terms-of-discrete-convolution}
 
 
     \section{Relation between Binomial theorem and Discrete convolution}
     \label{sec:relation-between-binomial-theorem-and-discrete-convolution}
-    \input{sections/relation-between-binomial-theorem-and-discrete-convolution}
+    \input{sections/07-relation-between-binomial-theorem-and-discrete-convolution}
 
 
-    \section{Derivation of coefficient \texorpdfstring{$\coeffA{m}{r}$}{A[m,r]}}
+    \section{Derivation of the coefficient \texorpdfstring{$\coeffA{m}{r}$}{A[m,r]}}
     \label{sec:derivation-of-coefficients-a}
-    \input{sections/derivation-of-coefficients-a}
+    \input{sections/08-derivation-of-the-coefficient-a}
 
 
     \section{Conclusion}
     \label{sec:conclusion}
-    \input{sections/conclusion}
+    \input{sections/09-conclusion}
 
 
     \section{Acknowledgements}

src/sections/introduction.tex → src/sections/03-introduction-and-main-results.tex

@@ -2,18 +2,18 @@
 \begin{align*}
     \polynomialP{m}{b}{x} = \sum_{k=0}^{b-1} \sum_{r=0}^{m} \coeffA{m}{r} k^r(x-k)^r
 \end{align*}
-where $\coeffA{m}{r}$ is real coefficient.
+where $\coeffA{m}{r}$ is a real coefficient.
 By means of Lemma~\ref{lemma_polynomial_p_and_odd_power},
 the polynomial $\polynomialP{m}{b}{x}$ has the following relation with Binomial theorem~\cite{AbraSteg72}
 \begin{align*}
     \polynomialP{m}{x+y}{x+y} = \sum_{r=0}^{2m+1} \binom{2m+1}{r} x^{2m+1-r} y^r
 \end{align*}
-From the other hand, polynomial $\polynomialP{m}{b}{x}$ might be expressed in terms of discrete convolution
+On the other hand, polynomial $\polynomialP{m}{b}{x}$ might be expressed in terms of discrete convolution
 of polynomial $n^j$
 \begin{align*}
     \polynomialP{m}{x+1}{x} = \sum_{r=0}^{m} \coeffA{m}{r} \convPower{n}{r}{x}, \quad n\geq 0
 \end{align*}
-It is of first necessity to notice that  $n^r$ of discrete convolution $\convPower{n}{r}{x}$ evaluated at $x$
+It is important to notice that  $n^r$ of discrete convolution $\convPower{n}{r}{x}$ evaluated at $x$
 is implicit piecewise-defined polynomial such as
 \begin{equation*}
     n^{r} =
@@ -34,8 +34,8 @@
     =
     -1 + \sum_{r=0}^{2m+1} \binom{2m+1}{r} x^{2m+1-r} y^r, \quad n > 0
 \end{equation*}
-Also, the following generalizations for multinomial case are discussed,
-see Corollaries~\ref{cor_mult_exp_and_macaulay_conv},~\ref{cor_mult_exp_and_macaulay_conv_strict}
+Additionally, the following generalizations for the multinomial case are discussed in
+corollaries~\ref{cor_mult_exp_and_macaulay_conv} and ~\ref{cor_mult_exp_and_macaulay_conv_strict}
 \begin{gather*}
     \sum_{r=0}^{m} \coeffA{m}{r} \convPower{n}{r}{\multifoldSum{t}} =
     1 + \sum_{\multifoldSum[k]{t}=2m+1} \binom{2m+1}{k_1, k_2,\ldots, k_t} \prod_{\ell=1}^{t} x_\ell^{k_\ell},
@@ -50,12 +50,13 @@
 \begin{equation*}
     x^{2m+1} = \sum_{r=0}^{m} \coeffA{m}{r} \sum_{k=0}^{x-1} k^r (x-k)^r
 \end{equation*}
-From the other prospective, the theorem~\ref{thm_odd_power_by_macaulays_convolution_strict} concludes as follows
+From another perspective, the theorem~\ref{thm_odd_power_by_macaulays_convolution_strict} concludes as follows
 \begin{equation*}
     x^{2m+1} = \sum_{r=0}^{m} \coeffA{m}{r} \sum_{k=1}^{x} k^r (x-k)^r
 \end{equation*}
 In its explicit form an identity $x^{2m+1} = \sum_{r=0}^{m} \coeffA{m}{r} \sum_{k=0}^{x-1} k^r (x-k)^r$ looks like
-as follows
+as follows.
+For example,
 \begin{align*}
     x^3 &= \sum_{k=1}^{x} 6k (x-k) + 1 \\
     x^5 &= \sum_{k=1}^{x} 30k^2 (x-k)^2 + 1 \\

src/sections/polynomial-p-and-its-properties.tex → src/sections/04-polynomial-p-and-its-properties.tex

@@ -1,10 +1,11 @@
 \label{sec:polynomial-p-and-their-properties}
-We continue our mathematical journey from short overview of polynomial $\polynomialL{m}{x}{k}$ that is
-essential part of polynomial $\polynomialP{m}{b}{x}$ since that
+We continue our mathematical journey from the short overview
+of polynomial $\polynomialL{m}{x}{k}$ which is
+an essential part of polynomial $\polynomialP{m}{b}{x}$ since that
 $\polynomialP{m}{b}{x} = \sum_{k=0}^{b-1} \polynomialL{m}{x}{k}$.
-Polynomial $\polynomialL{m}{x}{k}, \; m\in\mathbb{N}$ is polynomial of degree $2m$ in $x,k\in\mathbb{R}$,
+Polynomial $\polynomialL{m}{x}{k}$ is a polynomial of degree $2m$ in $x,k\in\mathbb{R}$,
 see definition~\eqref{eq:def_polynomial_l}.
-In explicit form the polynomial $\polynomialL{m}{x}{k}$ is as follows
+In its explicit form the polynomial $\polynomialL{m}{x}{k}$ is as follows
 \begin{equation*}
     \polynomialL{m}{x}{k} =
     \coeffA{m}{m} k^m(x-k)^m +
@@ -13,7 +14,7 @@
     \coeffA{m}{0}
 \end{equation*}
 where $\coeffA{m}{r}$ are real coefficients defined by~\eqref{eq:def_coeff_a}.
-Coefficients $\coeffA{m}{r}$ are nonzero only for $r$ within the interval $r \in \{m\} \cup \left[0,\frac{m-1}{2}\right]$.
+Coefficients $\coeffA{m}{r}$ are nonzero for $r$ only within the range $r \in \{m\} \cup \left[0,\frac{m-1}{2}\right]$.
 For example,
 \begin{table}[H]
     \setlength\extrarowheight{-6pt}
@@ -54,15 +55,15 @@
         &=-140 k^6+420 k^5 x-420 k^4 x^2+140 k^3 x^3+14 k^2-14 k x+1
     \end{split}
 \end{equation*}
-It is worth to notice that $\polynomialL{m}{x}{k}$ is symmetrical over $x$
+It is important to notice that $\polynomialL{m}{x}{k}$ is symmetric over $x$
 \begin{ppty}
     \label{ppty_symmetry_of_polynomial_l}
     For every $x,k\in\mathbb{R}$
     \begin{equation*}
         \polynomialL{m}{x}{k} = \polynomialL{m}{x}{x-k}
     \end{equation*}
 \end{ppty}
-This might be seen in the following tables
+This might be seen from the following tables
 \begin{table}[H]
     \setlength\extrarowheight{-6pt}
     \begin{tabular}{c|cccccccc}

src/sections/odd-binomial-theorem-as-partial-case-of-p.tex → src/sections/05-relation-betweeb-polynomial-p-and-binomial-theorem.tex

@@ -5,7 +5,7 @@
         \polynomialP{m}{x+y}{x+y} = \sum_{r=0}^{2m+1} \binom{2m+1}{r} x^{2m+1-r} y^r
     \end{equation*}
 \end{lem}
-By Lemma~\ref{lemma_polynomial_p_and_odd_power} and equation~\eqref{eq:p_all_forms} the following
+By means of lemma~\ref{lemma_polynomial_p_and_odd_power} and equation~\eqref{eq:p_all_forms} the following
 polynomial identities straightforward
 \begin{equation*}
     x^{2m+1}

src/sections/relation-between-p-and-convolution-of-polynomials.tex → src/sections/06-polynomial-p-in-terms-of-discrete-convolution.tex

@@ -1,7 +1,7 @@
 In this section we discuss the relation between $\polynomialP{m}{b}{x}$ and discrete convolution of
 polynomials.
 To show that $\polynomialP{m}{b}{x}$ involves the discrete convolution of polynomial $n^r$
-let's remind the definition of $\polynomialP{m}{b}{x}$
+recall the definition of the polynomial $\polynomialP{m}{b}{x}$
 \begin{equation*}
     \polynomialP{m}{b}{x} = \sum_{k=0}^{b-1} \sum_{r=0}^{m} \coeffA{m}{r} k^r (x-k)^r
     = \sum_{r=0}^{m} \coeffA{m}{r} \sum_{k=0}^{b-1} k^r (x-k)^r
@@ -11,7 +11,8 @@
 (f \ast f)[n]
     = \sum_{k} f(k) f(n-k)
 \end{equation*}
-General formula of discrete convolution for polynomials $f(n) = n^j, \; n\geq a \in \mathbb{R}$ may be derived immediately
+General formula of discrete convolution for polynomials $f(n) = n^j, \; n\geq a \in \mathbb{R}$
+can be derived immediately
 \begin{equation*}
     \begin{split}
         \convPower{n}{j}{x}
@@ -30,7 +31,7 @@
     \]
 \end{lem}
 It is of first importance to keep in mind that  $n^r$ of discrete convolution $\convPower{n}{r}{x}$ evaluated at $x$
-is implicit piecewise-defined polynomial such as
+is an implicit piecewise-defined polynomial such as
 \begin{equation*}
     n^{r} =
     \begin{cases}

src/sections/derivation-of-coefficients-a.tex → src/sections/08-derivation-of-the-coefficient-a.tex

@@ -43,7 +43,7 @@
         &-\underbrace{\sum_{j} \binom{r}{j} \frac{(-1)^j}{r+j+1} \bernoulli{r+j+1} n^{r-j}}_{(\diamond)}
     \end{split}
 \end{equation*}
-Hence, introducing $\ell=2r+1-s$ to $(\star)$ and $\ell=r-j$ to $(\diamond)$, we get
+Hence, by introducing $\ell=2r+1-s$ into $(\star)$ and $\ell=r-j$ into $(\diamond)$, we get
 \begin{equation*}
     \begin{split}
         \sum_{k=0}^{n-1} k^r (n-k)^r
@@ -73,7 +73,7 @@
 \end{equation}
 Taking the coefficient of $n^{2r+1}$ for $r=m$ in~\eqref{eq:proof2} we get $\coeffA{m}{m} = (2m+1) \binom{2m}{m}$.
 Since that $\text{odd } \ell \leq r$ in explicit form is $2j + 1 \leq r$, it follows that $j \leq \frac{m-1}{2}$,
-where $j$ is iterator.
+where $j$ is an iterator.
 Therefore, taking the coefficient of $n^{2j+1}$ for an integer $j$ in the range $\frac{m}{2} \leq j \leq m$,
 we get $\coeffA{m}{j} = 0$.
 Taking the coefficient of $n^{2d+1}$ for $d$ in the range $m/4 \leq d < m/2$ we get

src/sections/acknowledgements.tex

@@ -1,4 +1,3 @@
 I'd like to thank to Dr. Max Alekseyev for sufficient help in the derivation of the real coefficients $\coeffA{m}{r}$.
 Also, I'd like to thank to OEIS editors Michel Marcus, Peter Luschny, Jon E. Schoenfield and others
-for their useful volunteer work and for useful comments and during the
-work on OEIS sequences related to this manuscript.
+for their useful volunteer work and for useful comments during the work on OEIS sequences related to this manuscript.