\documentclass[10pt,letterpaper]{article} \usepackage[top=0.85in,left=2.79in,footskip=0.79in,marginparwidth=2in]{geometry} \usepackage{amsmath, amsthm, amscd, amsfonts, amssymb, graphicx, tikz, color} \usepackage[bookmarksnumbered, colorlinks, plainpages]{hyperref} % use Unicode characters - try changing the option if you run into troubles with special characters (e.g. umlauts) \usepackage[utf8]{inputenc} % clean citations \usepackage{cite} % hyperref makes references clicky. use \url{www.example.com} or \href{www.example.com}{description} to add a clicky url \usepackage{nameref,hyperref} % line numbers \usepackage[right]{lineno} % improves typesetting in LaTeX \usepackage{microtype} \DisableLigatures[f]{encoding = *, family = * } % text layout - change as needed \raggedright \setlength{\parindent}{0.5cm} \textwidth 5.25in \textheight 8.79in % Remove % for double line spacing %\usepackage{setspace} %\doublespacing % use adjustwidth environment to exceed text width (see examples in text) \usepackage{changepage} % adjust caption style \usepackage[aboveskip=1pt,labelfont=bf,labelsep=period,singlelinecheck=off]{caption} % remove brackets from references \makeatletter \renewcommand{\@biblabel}[1]{\quad#1.} \makeatother % headrule, footrule and page numbers \usepackage{lastpage,fancyhdr,graphicx} \usepackage{epstopdf} \pagestyle{myheadings} \pagestyle{fancy} \fancyhf{} \rfoot{\thepage/\pageref{LastPage}} \renewcommand{\footrule}{\hrule height 2pt \vspace{2mm}} \fancyheadoffset[L]{2.25in} \fancyfootoffset[L]{2.25in} % use \textcolor{color}{text} for colored text (e.g. highlight to-do areas) \usepackage{color} % define custom colors (this one is for figure captions) \definecolor{Gray}{gray}{.25} % this is required to include graphics \usepackage{graphicx} % use if you want to put caption to the side of the figure - see example in text \usepackage{sidecap} \usepackage{leftidx} % use for have text wrap around figures \usepackage{wrapfig} \usepackage[pscoord]{eso-pic} \usepackage[fulladjust]{marginnote} \reversemarginpar \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{proposition}[theorem]{Proposition} \newtheorem{corollary}[theorem]{Corollary} \theoremstyle{definition} \newtheorem{definition}[theorem]{Definition} \newtheorem{example}[theorem]{Example} \newtheorem{xca}[theorem]{Exercise} \theoremstyle{remark} \newtheorem{remark}[theorem]{Remark} \theoremstyle{observation} \newtheorem{observation}[theorem]{Observation} \theoremstyle{question} \newtheorem{question}[theorem]{Question} \theoremstyle{problem} \newtheorem{problem}[theorem]{Problem} \numberwithin{equation}{section} \usepackage{fancyhdr} \pagestyle{fancy} \fancyhf{} \fancyhead[LE,RO]{Henry Garrett · Independent Researcher · Department of Mathematics · DrHenryGarrett@gmail.com · Manhattan, NY, USA } \fancyfoot[LE,RO]{Henry Garrett · Independent Researcher · Department of Mathematics · DrHenryGarrett@gmail.com · Manhattan, NY, USA } % document begins here \begin{document} \vspace*{0.35in} \linenumbers % title goes here: \begin{flushleft} {\Large \textbf\newline{ New Ideas In Recognition of Cancer And Neutrosophic SuperHyperGraph By Eulerian-Path-Cut As Hyper Eulogy-Path-Cut On Super EULA-Path-Cut } } \newline \newline Henry Garrett · Independent Researcher · Department of Mathematics · DrHenryGarrett@gmail.com · Manhattan, NY, USA % authors go here: \end{flushleft} \section{ABSTRACT} In this scientific research, some extreme notions and Neutrosophic notions are defined on the family of SuperHyperGraphs and Neutrosophic SuperHyperGraphs. Some well-known classes are used in this scientific research. A basic familiarity with Neutrosophic SuperHyper Eulerian-Path-Cut theory, SuperHyperGraphs, and Neutrosophic SuperHyperGraphs theory are proposed. \\ \vspace{4mm} \textbf{Keywords:} Neutrosophic SuperHyperGraph, SuperHyperEulerian-Path-Cut, Cancer's Neutrosophic Recognition \\ \textbf{AMS Subject Classification:} 05C17, 05C22, 05E45 \section{Neutrosophic Preliminaries Of This Scientific Research On the Redeemed Ways} In this section, the basic material in this scientific research, is referred to [Single Valued Neutrosophic Set](\textbf{Ref.}\cite{HG38},Definition 2.2,p.2), [Neutrosophic Set](\textbf{Ref.}\cite{HG38},Definition 2.1,p.1), [Neutrosophic SuperHyperGraph (NSHG)](\textbf{Ref.}\cite{HG38},Definition 2.5,p.2), [Characterization of the Neutrosophic SuperHyperGraph (NSHG)](\textbf{Ref.}\cite{HG38},Definition 2.7,p.3), [t-norm](\textbf{Ref.}\cite{HG38}, Definition 2.7, p.3), and [Characterization of the Neutrosophic SuperHyperGraph (NSHG)](\textbf{Ref.}\cite{HG38},Definition 2.7,p.3), [Neutrosophic Strength of the Neutrosophic SuperHyperPaths] (\textbf{Ref.}\cite{HG38},Definition 5.3,p.7), and [Different Neutrosophic Types of Neutrosophic SuperHyperEdges (NSHE)] (\textbf{Ref.}\cite{HG38},Definition 5.4,p.7). Also, the new ideas and their clarifications are addressed to \textbf{Ref.}\cite{HG38}. \begin{definition}(Different Neutrosophic Types of Neutrosophic SuperHyperEulerian-Path-Cut).\\ Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider a Neutrosophic SuperHyperSet $V'=\{V_1,V_2,\ldots,V_s\}$ and $E'=\{E_1,E_2,\ldots,E_z\}.$ Then either $V'$ or $E'$ is called \begin{itemize} \item[$(i)$] \textbf{Neutrosophic e-SuperHyperEulerian-Path-Cut} if the following expression is called \textbf{Neutrosophic e-SuperHyperEulerian-Path-Cut criteria} holds \begin{eqnarray*} && \forall E_a\in P: P~\text{is} \\&& \text{a SuperHyperPath and it has} \\&& \text{the all number of SuperHyperEdges}; \end{eqnarray*} \item[$(ii)$] \textbf{Neutrosophic re-SuperHyperEulerian-Path-Cut} if the following expression is called \textbf{Neutrosophic re-SuperHyperEulerian-Path-Cut criteria} holds \begin{eqnarray*} && \forall E_a\in P: P~\text{is} \\&& \text{a SuperHyperPath and it has} \\&& \text{the all number of SuperHyperEdges}; \end{eqnarray*} and $|E_i|_{\text{NEUTROSOPIC CARDINALITY}}=|E_j|_{\text{NEUTROSOPIC CARDINALITY}};$ \item[$(iii)$] \textbf{Neutrosophic v-SuperHyperEulerian-Path-Cut} if the following expression is called \textbf{Neutrosophic v-SuperHyperEulerian-Path-Cut criteria} holds \begin{eqnarray*} && \forall V_a\in P: P~\text{is} \\&& \text{a SuperHyperPath and it has} \\&& \text{the all number of SuperHyperEdges}; \end{eqnarray*} \item[$(iv)$] \textbf{Neutrosophic rv-SuperHyperEulerian-Path-Cut} if the following expression is called \textbf{Neutrosophic v-SuperHyperEulerian-Path-Cut criteria} holds \begin{eqnarray*} && \forall V_a\in P: P~\text{is} \\&& \text{a SuperHyperPath and it has} \\&& \text{the all number of SuperHyperEdges}; \end{eqnarray*} and $|V_i|_{\text{NEUTROSOPIC CARDINALITY}}=|V_j|_{\text{NEUTROSOPIC CARDINALITY}};$ \item[$(v)$] \textbf{Neutrosophic SuperHyperEulerian-Path-Cut} if it's either of Neutrosophic e-SuperHyperEulerian-Path-Cut, Neutrosophic re-SuperHyperEulerian-Path-Cut, Neutrosophic v-SuperHyperEulerian-Path-Cut, and Neutrosophic rv-SuperHyperEulerian-Path-Cut. \end{itemize} \end{definition} \begin{definition}((Neutrosophic) SuperHyperEulerian-Path-Cut).\\ Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider a Neutrosophic SuperHyperEdge (NSHE) $E=\{V_1,V_2,\ldots,V_s\}.$ Then $E$ is called \begin{itemize} \item[$(i)$] an \textbf{Extreme SuperHyperEulerian-Path-Cut} if it's either of Neutrosophic e-SuperHyperEulerian-Path-Cut, Neutrosophic re-SuperHyperEulerian-Path-Cut, Neutrosophic v-SuperHyperEulerian-Path-Cut, and Neutrosophic rv-SuperHyperEulerian-Path-Cut and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperEdges in the conseNeighborive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperEulerian-Path-Cut; \item[$(ii)$] a \textbf{Neutrosophic SuperHyperEulerian-Path-Cut} if it's either of Neutrosophic e-SuperHyperEulerian-Path-Cut, Neutrosophic re-SuperHyperEulerian-Path-Cut, Neutrosophic v-SuperHyperEulerian-Path-Cut, and Neutrosophic rv-SuperHyperEulerian-Path-Cut and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperEdges of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperEulerian-Path-Cut; \item[$(iii)$] an \textbf{Extreme SuperHyperEulerian-Path-Cut SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyperEulerian-Path-Cut, Neutrosophic re-SuperHyperEulerian-Path-Cut, Neutrosophic v-SuperHyperEulerian-Path-Cut, and Neutrosophic rv-SuperHyperEulerian-Path-Cut and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperEdges of an Extreme SuperHyperSet $S$ of high Extreme cardinality conseNeighborive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperEulerian-Path-Cut; and the Extreme power is corresponded to its Extreme coefficient; \item[$(iv)$] a \textbf{Neutrosophic SuperHyperEulerian-Path-Cut SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyperEulerian-Path-Cut, Neutrosophic re-SuperHyperEulerian-Path-Cut, Neutrosophic v-SuperHyperEulerian-Path-Cut, and Neutrosophic rv-SuperHyperEulerian-Path-Cut and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the Neutrosophic SuperHyperPolynomial contains the Neutrosophic coefficients defined as the Neutrosophic number of the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperEdges of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperEulerian-Path-Cut; and the Neutrosophic power is corresponded to its Neutrosophic coefficient; \item[$(v)$] an \textbf{Extreme V-SuperHyperEulerian-Path-Cut} if it's either of Neutrosophic e-SuperHyperEulerian-Path-Cut, Neutrosophic re-SuperHyperEulerian-Path-Cut, Neutrosophic v-SuperHyperEulerian-Path-Cut, and Neutrosophic rv-SuperHyperEulerian-Path-Cut and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the maximum Extreme cardinality of an Extreme SuperHyperSet $S$ of high Extreme cardinality of the Extreme SuperHyperVertices in the conseNeighborive Extreme sequence of Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperEulerian-Path-Cut; \item[$(vi)$] a \textbf{Neutrosophic V-SuperHyperEulerian-Path-Cut} if it's either of Neutrosophic e-SuperHyperEulerian-Path-Cut, Neutrosophic re-SuperHyperEulerian-Path-Cut, Neutrosophic v-SuperHyperEulerian-Path-Cut, and Neutrosophic rv-SuperHyperEulerian-Path-Cut and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperVertices of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperEulerian-Path-Cut; \item[$(vii)$] an \textbf{Extreme V-SuperHyperEulerian-Path-Cut SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyperEulerian-Path-Cut, Neutrosophic re-SuperHyperEulerian-Path-Cut, Neutrosophic v-SuperHyperEulerian-Path-Cut, and Neutrosophic rv-SuperHyperEulerian-Path-Cut and $\mathcal{C}(NSHG)$ for an Extreme SuperHyperGraph $NSHG:(V,E)$ is the Extreme SuperHyperPolynomial contains the Extreme coefficients defined as the Extreme number of the maximum Extreme cardinality of the Extreme SuperHyperVertices of an Extreme SuperHyperSet $S$ of high Extreme cardinality conseNeighborive Extreme SuperHyperEdges and Extreme SuperHyperVertices such that they form the Extreme SuperHyperEulerian-Path-Cut; and the Extreme power is corresponded to its Extreme coefficient; \item[$(viii)$] a \textbf{Neutrosophic SuperHyperEulerian-Path-Cut SuperHyperPolynomial} if it's either of Neutrosophic e-SuperHyperEulerian-Path-Cut, Neutrosophic re-SuperHyperEulerian-Path-Cut, Neutrosophic v-SuperHyperEulerian-Path-Cut, and Neutrosophic rv-SuperHyperEulerian-Path-Cut and $\mathcal{C}(NSHG)$ for a Neutrosophic SuperHyperGraph $NSHG:(V,E)$ is the Neutrosophic SuperHyperPolynomial contains the Neutrosophic coefficients defined as the Neutrosophic number of the maximum Neutrosophic cardinality of the Neutrosophic SuperHyperVertices of a Neutrosophic SuperHyperSet $S$ of high Neutrosophic cardinality conseNeighborive Neutrosophic SuperHyperEdges and Neutrosophic SuperHyperVertices such that they form the Neutrosophic SuperHyperEulerian-Path-Cut; and the Neutrosophic power is corresponded to its Neutrosophic coefficient. \end{itemize} \end{definition} \section{ Neutrosophic SuperHyperEulerian-Path-Cut But As The Extensions Excerpt From Dense And Super Forms} \begin{definition}(Neutrosophic event).\\ Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability Eulerian-Path-Cut. Any Neutrosophic k-subset of $A$ of $V$ is called \textbf{Neutrosophic k-event} and if $k=2,$ then Neutrosophic subset of $A$ of $V$ is called \textbf{Neutrosophic event}. The following expression is called \textbf{Neutrosophic probability} of $A.$ \begin{eqnarray} E(A)=\sum_{a\in A}E(a). \end{eqnarray} \end{definition} \begin{definition}(Neutrosophic Independent).\\ Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability Eulerian-Path-Cut. $s$ Neutrosophic k-events $A_i,~i\in I$ is called \textbf{Neutrosophic s-independent} if the following expression is called \textbf{Neutrosophic s-independent criteria} \begin{eqnarray*} E(\cap_{i\in I}A_i)=\prod_{i\in I}P(A_i). \end{eqnarray*} And if $s=2,$ then Neutrosophic k-events of $A$ and $B$ is called \textbf{Neutrosophic independent}. The following expression is called \textbf{Neutrosophic independent criteria} \begin{eqnarray} E(A\cap B)=P(A)P(B). \end{eqnarray} \end{definition} \begin{definition}(Neutrosophic Variable).\\ Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability Eulerian-Path-Cut. Any k-function Eulerian-Path-Cut like $E$ is called \textbf{Neutrosophic k-Variable}. If $k=2$, then any 2-function Eulerian-Path-Cut like $E$ is called \textbf{Neutrosophic Variable}. \end{definition} The notion of independent on Neutrosophic Variable is likewise. \begin{definition}(Neutrosophic Expectation).\\ Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability Eulerian-Path-Cut. A Neutrosophic k-Variable $E$ has a number is called \textbf{Neutrosophic Expectation} if the following expression is called \textbf{Neutrosophic Expectation criteria} \begin{eqnarray*} Ex(E)=\sum_{\alpha \in V}E(\alpha)P(\alpha). \end{eqnarray*} \end{definition} \begin{definition}(Neutrosophic Crossing).\\ Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability Eulerian-Path-Cut. A Neutrosophic number is called \textbf{Neutrosophic Crossing} if the following expression is called \textbf{Neutrosophic Crossing criteria} \begin{eqnarray*} Cr(S)=\min\{\text{Number of Crossing in a Plane Embedding of } S\}. \end{eqnarray*} \end{definition} \begin{lemma} Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability Eulerian-Path-Cut. Let $m$ and $n$ propose special Eulerian-Path-Cut. Then with $m\geq 4n,$ \end{lemma} \begin{proof} Consider a planar embedding $G$ of $G$ with $cr(G)$ crossings. Let $S$ be a Neutrosophic random k-subset of $V$ obtained by choosing each SuperHyperVertex of $G$ Neutrosophic independently with probability Eulerian-Path-Cut $p := 4n/m,$ and set $H:=G[S]$ and $H:=G[S].$ \\ Define random variables $X, Y, Z$ on $V$ as follows: $X$ is the Neutrosophic number of SuperHyperVertices, $Y$ the Neutrosophic number of SuperHyperEdges, and $Z$ the Neutrosophic number of crossings of $H.$ The trivial bound noted above, when applied to $H,$ yields the inequality $Z \geq cr(H) \geq Y-3X.$ By linearity of Neutrosophic Expectation, $$E(Z) \geq E(Y )-3E(X).$$ Now $E(X) = pn,~E(Y ) = p^2m$ (each SuperHyperEdge having some SuperHyperEnds) and $E(Z) = p^4cr(G)$ (each crossing being defined by some SuperHyperVertices). Hence $$p^4cr(G) \geq p^2m-3pn.$$ Dividing both sides by $p^4,$ we have: \begin{eqnarray*} cr(G)\geq \frac{pm-3n}{p^3}=\frac{n}{{(4n/m)}^3}=\frac{1}{64}{m^3}{n^2}. \end{eqnarray*} \end{proof} \begin{theorem} Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability Eulerian-Path-Cut. Let $P$ be a SuperHyperSet of $n$ points in the plane, and let $l$ be the Neutrosophic number of SuperHyperLines in the plane passing through at least $k+1$ of these points, where $1 \leq k \leq 2 \sqrt{2n}.$ Then $l<32n^2/k^3.$ \end{theorem} \begin{proof} Form a Neutrosophic SuperHyperGraph $G$ with SuperHyperVertex SuperHyperSet $P$ whose SuperHyperEdge are the segments between conseNeighborive points on the SuperHyperLines which pass through at least $k+1$ points of $P.$ This Neutrosophic SuperHyperGraph has at least $kl$ SuperHyperEdges and Neutrosophic crossing at most $l$ choose two. Thus either $kl < 4n,$ in which case $l < 4n/k \leq32n^2/k^3,$ or $l^2/2 > \text{l choose 2} \geq cr(G) \geq {(kl)}^3/64n^2$ by the Neutrosophic Crossing Lemma, and again $l < 32n^2/k^3.$ \end{proof} \begin{theorem} Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability Eulerian-Path-Cut. Let $P$ be a SuperHyperSet of $n$ points in the plane, and let $k$ be the number of pairs of points of $P$ at unit SuperHyperDistance. Then $k < 5n^{4/3}.$ \end{theorem} \begin{proof} Assume a Neutrosophic SuperHyperGraph (NSHG) $S$ is a pair $S=(V,E).$ Consider $S=(V,E)$ is a probability Eulerian-Path-Cut. Draw a SuperHyperUnit SuperHyperCircle around each SuperHyperPoint of $P.$ Let $n_i$ be the Neutrosophic number of these SuperHyperCircles passing through exactly $i$ points