From c341b8c26ecb69891ed52b0bbdaa7326b71f8782 Mon Sep 17 00:00:00 2001 From: kolosovpetro Date: Sat, 21 Sep 2024 12:20:25 +0200 Subject: [PATCH 1/7] section files numbered --- CHANGELOG.md | 6 +++++ ...eBinomialTheoremAndDiscreteConvolution.pdf | Bin 421494 -> 421615 bytes ...eBinomialTheoremAndDiscreteConvolution.tex | 22 +++++++++--------- .../{abstract.tex => 01-abstract.tex} | 0 ... 02-definitions-notations-conventions.tex} | 0 ...x => 03-introduction-and-main-results.tex} | 0 ...=> 04-polynomial-p-and-its-properties.tex} | 0 ...eeb-polynomial-p-and-binomial-theorem.tex} | 0 ...al-p-in-terms-of-discrete-convolution.tex} | 0 ...mial-theorem-and-discrete-convolution.tex} | 0 ...=> 08-derivation-of-the-coefficient-a.tex} | 0 .../{conclusion.tex => 09-conclusion.tex} | 0 src/sections/acknowledgements.tex | 3 +-- 13 files changed, 18 insertions(+), 13 deletions(-) rename src/sections/{abstract.tex => 01-abstract.tex} (100%) rename src/sections/{definitions-notations-conventions.tex => 02-definitions-notations-conventions.tex} (100%) rename src/sections/{introduction.tex => 03-introduction-and-main-results.tex} (100%) rename src/sections/{polynomial-p-and-its-properties.tex => 04-polynomial-p-and-its-properties.tex} (100%) rename src/sections/{odd-binomial-theorem-as-partial-case-of-p.tex => 05-relation-betweeb-polynomial-p-and-binomial-theorem.tex} (100%) rename src/sections/{relation-between-p-and-convolution-of-polynomials.tex => 06-polynomial-p-in-terms-of-discrete-convolution.tex} (100%) rename src/sections/{relation-between-binomial-theorem-and-discrete-convolution.tex => 07-relation-between-binomial-theorem-and-discrete-convolution.tex} (100%) rename src/sections/{derivation-of-coefficients-a.tex => 08-derivation-of-the-coefficient-a.tex} (100%) rename src/sections/{conclusion.tex => 09-conclusion.tex} (100%) diff --git a/CHANGELOG.md b/CHANGELOG.md index ca7d31d..245a8f4 100644 --- a/CHANGELOG.md +++ b/CHANGELOG.md @@ -5,6 +5,12 @@ 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] - 2024-09-20 + +### Changed + +- Section files numbered + ## [1.0.0] - 2024-09-20 ### Changed diff --git a/out/OnTheBinomialTheoremAndDiscreteConvolution.pdf b/out/OnTheBinomialTheoremAndDiscreteConvolution.pdf index 26122cb937e3cf05d1ce7d13ee6ada59c52b4a14..cb48a5e6405685c501fa0703fc723ff4f9e8cf1e 100644 GIT binary patch delta 17245 zcmai52RN7Q_qX@VEXvAW-#sKdq$IK;A|)a#iHye%5h7oOhDst^c0{C58Ii1vD5DU` zD*xO2R&PDe@4Eh7@AvAx&i8)q``l-I&gY!x+tR?+*}<)a>9k3w7_5{0={T8>pm3-o zC=|+rGOxjarW9y0FeUZSq7nxLO7_vw({HY!Ea@<8DY0T+praw9lI+7}p>fN-XZi`e zwQCl0`i9zcYn=x;Y?FK99PX7}^O2eiN#sh4I#6&)`CiH7`}nu7 zwkB>dK3yD&I$puicS-r>Q6s_U$}fX5GD1(;4Pe(N&*WgkV()Zr%}cl`C!eipr6*gE zRnp4bz0!2YF7|`+`s&Q+RD+`Jw{W$_;?*Myj*&TAA-cD>Zw< zKJB&+*U^z%9xcy!+k@6l_H5ARmHB9COV1FAjgCF0Lb|i{h2Ui!hO8^`0xwyjNf({{ 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\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} diff --git a/src/sections/abstract.tex b/src/sections/01-abstract.tex similarity index 100% rename from src/sections/abstract.tex rename to src/sections/01-abstract.tex diff --git a/src/sections/definitions-notations-conventions.tex b/src/sections/02-definitions-notations-conventions.tex similarity index 100% rename from src/sections/definitions-notations-conventions.tex rename to src/sections/02-definitions-notations-conventions.tex diff --git a/src/sections/introduction.tex b/src/sections/03-introduction-and-main-results.tex similarity index 100% rename from src/sections/introduction.tex rename to src/sections/03-introduction-and-main-results.tex diff --git a/src/sections/polynomial-p-and-its-properties.tex b/src/sections/04-polynomial-p-and-its-properties.tex similarity index 100% rename from src/sections/polynomial-p-and-its-properties.tex rename to src/sections/04-polynomial-p-and-its-properties.tex diff --git a/src/sections/odd-binomial-theorem-as-partial-case-of-p.tex b/src/sections/05-relation-betweeb-polynomial-p-and-binomial-theorem.tex similarity index 100% rename from src/sections/odd-binomial-theorem-as-partial-case-of-p.tex rename to src/sections/05-relation-betweeb-polynomial-p-and-binomial-theorem.tex diff --git a/src/sections/relation-between-p-and-convolution-of-polynomials.tex b/src/sections/06-polynomial-p-in-terms-of-discrete-convolution.tex similarity index 100% rename from src/sections/relation-between-p-and-convolution-of-polynomials.tex rename to src/sections/06-polynomial-p-in-terms-of-discrete-convolution.tex diff --git a/src/sections/relation-between-binomial-theorem-and-discrete-convolution.tex b/src/sections/07-relation-between-binomial-theorem-and-discrete-convolution.tex similarity index 100% rename from src/sections/relation-between-binomial-theorem-and-discrete-convolution.tex rename to src/sections/07-relation-between-binomial-theorem-and-discrete-convolution.tex diff --git a/src/sections/derivation-of-coefficients-a.tex b/src/sections/08-derivation-of-the-coefficient-a.tex similarity index 100% rename from src/sections/derivation-of-coefficients-a.tex rename to src/sections/08-derivation-of-the-coefficient-a.tex diff --git a/src/sections/conclusion.tex b/src/sections/09-conclusion.tex similarity index 100% rename from src/sections/conclusion.tex rename to src/sections/09-conclusion.tex diff --git a/src/sections/acknowledgements.tex b/src/sections/acknowledgements.tex index ba0217e..40d6da6 100644 --- a/src/sections/acknowledgements.tex +++ b/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. 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4fa6856..2985306 100644 --- a/src/sections/03-introduction-and-main-results.tex +++ b/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 \\ From cfc49356b3a191cc8600b200a3b3ea55ab38eaac Mon Sep 17 00:00:00 2001 From: kolosovpetro Date: Sat, 21 Sep 2024 12:31:04 +0200 Subject: [PATCH 3/7] fix grammar section 4 --- src/sections/04-polynomial-p-and-its-properties.tex | 8 ++++---- 1 file changed, 4 insertions(+), 4 deletions(-) diff --git a/src/sections/04-polynomial-p-and-its-properties.tex b/src/sections/04-polynomial-p-and-its-properties.tex index 464b72d..9944315 100644 --- a/src/sections/04-polynomial-p-and-its-properties.tex +++ b/src/sections/04-polynomial-p-and-its-properties.tex @@ -1,10 +1,10 @@ \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 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 $x,k\in\mathbb{R}$, 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 + @@ -54,7 +54,7 @@ &=-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 $x,k\in\mathbb{R}$ From e32491444074a5216935d3ced32a1b4cfc47f700 Mon Sep 17 00:00:00 2001 From: kolosovpetro Date: Sat, 21 Sep 2024 12:34:25 +0200 Subject: [PATCH 4/7] fix grammar section 5 and 6 --- ...-relation-betweeb-polynomial-p-and-binomial-theorem.tex | 2 +- .../06-polynomial-p-in-terms-of-discrete-convolution.tex | 7 ++++--- 2 files changed, 5 insertions(+), 4 deletions(-) diff --git a/src/sections/05-relation-betweeb-polynomial-p-and-binomial-theorem.tex b/src/sections/05-relation-betweeb-polynomial-p-and-binomial-theorem.tex index ee174b8..d81d950 100644 --- a/src/sections/05-relation-betweeb-polynomial-p-and-binomial-theorem.tex +++ b/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} diff --git a/src/sections/06-polynomial-p-in-terms-of-discrete-convolution.tex b/src/sections/06-polynomial-p-in-terms-of-discrete-convolution.tex index e9b4a92..aaa9b8a 100644 --- a/src/sections/06-polynomial-p-in-terms-of-discrete-convolution.tex +++ b/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} From a51f4dd148a4698568c76fba5235ce613592e368 Mon Sep 17 00:00:00 2001 From: kolosovpetro Date: Sat, 21 Sep 2024 12:38:05 +0200 Subject: [PATCH 5/7] grammar fix section 8 --- ...eBinomialTheoremAndDiscreteConvolution.pdf | Bin 421681 -> 421711 bytes .../08-derivation-of-the-coefficient-a.tex | 4 ++-- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/out/OnTheBinomialTheoremAndDiscreteConvolution.pdf b/out/OnTheBinomialTheoremAndDiscreteConvolution.pdf index c0533d36be648557d7d1a869696eaefc3ebe56d5..7dca8a5c3f6af4b7bc3d2820fe692ca261a85c52 100644 GIT binary patch delta 24814 zcmV)PK()WI-Wku{8L+}m0W*^UK`DRTTFY|VMi$-kE3gDBLc8CONqJ#Aaz=?GWo*u5 zsvIw939>liLjy9FHD908XaEG@21&V?Dtk6209|z7d+zJphH3I4p~;V<=abQsm&PWR zIIU$exlS0f#0Z`kA&8SUnOr4r(~I+QCZ$Rze?RpXI(>C^K4$69;pX}2Dn{Uyjf!z<@MgzF#9-n@8yIyntuFD}miHRdk8c=gZ8S-F{ZHr3;I zlb=SkQuWD8VW1sxN^!5KwuFl$<4}{apr#=HjVCXqYYQ@oOM^8^nO|4)u{P<$a??hwZ2&uo9bo1(#J8kX#*k?eoVTW2TwwQs56Ce 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z?pBf_U?0-)K$0RU)C(lk45|#sME;9czjG1b-zpMf_3Ge#~3riCUM$&mzF zT_A&x+kXrGCt=};qA@J#u^vJ~4Ss9k-*5$FumNg+Ae<)>9GaLQgAD2T9z~EWD|>BKSLGdAcIj$ez1k0 zpfh)*eQ+Lt3&QLrg%CzsL_-l+A0}h4P6`VQ5~Y101IpgH;UYZ3agH#DY-?#`+D%?8QJD`{i%W58-QE1qDp(F?z z5R&OoAh3x=#{!8KV9Y=g1cr5(2N8=Q`_WPaKX;t0AZH+OBs!_V#^wi$Ay_*IdY|#H ztxZGI7ty(q7I;|xqRPPNfuI>|y?GR>Y52VXoxq@N@!vbP^yOy<8Xyy8&a?=KiC(Pm zAnGk>5omX&p_8Ch0y7>)erO4{Xy}H5fl5J_ORxmN)+#UnlO54a@%stPoQTrE#G#No zdQc%#(9O^O|AuC;s{$XQFTyl~xSAng2h{;a6&J&Bgv&&9S;Y`MU~t?z}E2a9*nKn9S}i< z%;|e`gx|e>pbYd&Om~q)^Ih?q8B5FM5i(0(Yv?!>VQnpv)|S>* Date: Sat, 21 Sep 2024 12:41:55 +0200 Subject: [PATCH 7/7] update changelog --- CHANGELOG.md | 3 ++- 1 file changed, 2 insertions(+), 1 deletion(-) diff --git a/CHANGELOG.md b/CHANGELOG.md index 245a8f4..f1d18c5 100644 --- a/CHANGELOG.md +++ b/CHANGELOG.md @@ -5,11 +5,12 @@ 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] - 2024-09-20 +## [1.0.1] - 2024-09-21 ### Changed - Section files numbered +- English grammar fixes ## [1.0.0] - 2024-09-20