نشریه ریاضی و جامعه
سال هشتم شماره 4 (زمستان 1402)

In this paper, we introduce the Markovian regime-switching regression model, which is a graphical model based on the hidden Markov model. This model can be viewed as a clustered regression model, in which a Markov process models the transition from one cluster to another. These clusters are indeed the hidden states of the process, in the hidden Markov model, which are assumed to be a Markov process of order one. Besides, other assumptions of the hidden Markov model are assumed in this model, while the emission distribution is assumed to be the conditional distribution of the response given the covariates and the states. As an application of this model, the problem of prediction of PM 2.5 pollution in Tehran's air based on temperature and pressure during 2015-2017 using the Markovian regime-switching non-parametric additive transitive model, is considered and studied. Furthermore, the package hhsmm in R software, is introduced as a powerful tool for modeling the stated model.

State-switching models are models in which the distribution of a sequence of observations (usually during a time interval) is controlled by a sequence of hidden states, such that the conditional distribution of observations given each state is different from that given others. Hidden Markov and semi-Markov Models [27] are the most common instances of state-switching models, in which the hidden state is a Markov or semi-Markov process. Some other models, including the regime-switching models or Kalman-Filter model, are in this category. Various applications of such models are introduced by the researchers including, speech recognition [12], cognitive learning [24], brain performance modeling [15], modeling environmental processes [4, 5, 6], sequential analysis, reliability theory [7], biological analysis [8, 9, 27], and many other applications.

A hidden Markov model is constructed by the following items:   (1) Transition Probability Matrix $\pmb{\Gamma} =(\gamma_{ij})$, where\begin{equation*}\gamma_{ij}=\Pr(S_{t+1}=j|S_t=i),   i,j=1,\ldots,J,\end{equation*}such that\begin{equation*}\sum_{i=1}^{J}{\gamma_{ij}}=1,   j=1,\ldots,J.\end{equation*}   (2) Initial State Probability $ \pmb{\delta}=(\delta_j) $, where\begin{equation*}\delta_j=\Pr(S_1=j),    j=1,\ldots,J;    \sum_{j=1}^{J}{\delta_j}=1.\end{equation*}   (3) Observation distributions $ f_1(y),\ldots,f_J(y) $, where$$ f_j(y)=\Pr(Y_t=y | S_t=j);   j=1,\ldots,J,$$which are also called state-dependent distribution or emission distribution. When $ y_t $ is a continuous random variable, $ f_j(y) $ is a probability density function, which is usually a normal distribution or mixture of normal distributions.The regime-switching regression model is introduced by [14] as follows:(2.1)                       $$ y_{t} = x_{t}^T \beta_{s_t} + \sigma_{s_t}\epsilon_t,$$in which $\{y_t\}$ is the sequence of responses, $\{x_t\}$ is the sequence of covariates, $\{\epsilon_t\}$ are sequence of (usually) i.i.d. normally distributed errors with zero mean and a variance equal to 1, and $\beta_{s_t}$ and $\sigma_{s_t}$ are the regression coefficients and the standard deviation of errors at state $s_t$, respectively. A generalization of the model (2.1) to the the additive regime-switching regression model is introduced by [20] as follows:(2.2)   $$y_{t} = \mu_{s_t} + \sum_{j=1}^p f_{j,s_t}(x_{j,t}) + \sigma_{s_t}\epsilon_t,$$Letting $x_t = (y_{t-\ell},\ldots,y_{t-L}, z_{t-\ell},\ldots,z_{t-L})$, for lags $L > \ell \geq 1$ in (2.2), the non-parametric additive transitive regime-switching regression model is obtained. All models in this paper and all necessary tools for modeling, initialization, fitting, and prediction of these models are included in the R package hhsmm, which can be downloaded from https://cran.r-project.org/package=hhsmm. The reader is also referred to [3] for more information and examples about hhsmm package.

The data set for this paper is obtained from two sources. The AQI data set (PM2.5 values) are obtained from https://airnow.tehran.ir/, while the air temperature and pressure are obtained from Iran meteorological organization. Figure 3, shows the time-series plots of this data set. To visualize the additive regime-switching regression model, we first consider only the temperature as the covariate in the model. Figure 4, presents the prediction of PM2.5 in Tehran city air using a nonparametric additive regime-switching regression model, only based on the air temperature, in each of the four hidden states. The points in each state are colored by different colors and the curve of the prediction is drawn with the same color. As one can see from this figure, the predictive curves are fairly fitted to the points in each state. As a competitor model, we consider the single-state additive regression model. Figure 5, presents the result of the comparison of prediction precision of PM2.5 in Tehran city air, using two fitted models: the regime-switching regression model with four hidden states and the single state non-parametric additive regression model. The mean squared error in each model is presented in each plot. One can see that the non-parametric additive regime-switching regression model with four hidden states performs better than the single-state non-parametric additive regression model. Another introduced model is the additive transitive regime-switching regression model. Figure 6, shows the result of the prediction of PM2.5 in Tehran city air using a transitive regime switching non-parametric additive regression model with a 1-day lag. The mean squared error of the model is presented in the plot. One can see that this model performs better than the two other competitors. Finally, the additive transitive regime-switching model is used for the prediction of the future values of PM2.5. Figure 7, presents the out-of-sample prediction of PM2.5 in Tehran city air using a transitive regime switching non-parametric additive regression model with 10 days lag. The mean squared error of the model is equal to 99.3. The result of the prediction is satisfactory.

In this article, we have tried to introduce one of the most important topics in the subject of dynamical systems, namely the Melnikov function, in simple language. Melnikov function is one of the tools that can express the effect of the small perturbation on the homoclinic orbits. This issue is also related to breaking a homoclinic orbit under the effect of some small perturbations. When a small perturbation occurs in a dynamical system, some dynamical behaviors of the system may change. Here, we try to explain how to compute the formula of this function fluently. Therefore, in the first part, we will introduce some preliminary concepts and properties, and then in the second part, we will describe the construction of the Melnikov function. Finally, by stating the results of the fundamental matrix solutions and then using them, we will construct the Melnikov function near a homoclinic or hetroclinic orbit. 

One of the fundamental issues that have recently attracted the attention of researchers in the field of theoretical research is the issue of dynamical systems. The phenomenon of touching stable and unstable manifolds is an example of behavior in a dynamical system that results in the existence of special solutions. If the stable and unstable curves of an equilibrium point touch each other, then the solution is homoclinic, and if the stable curves of one equilibrium point intersect with the unstable curve of another equilibrium point, then we will have a heteroclinic solution. Investigating the existence of such solutions for a specific dynamical system has always been one of the important issues, and therefore many people have done good research in this field, for example, some examples of such research can be found in [8, 4,5] observed. In addition, many researchers have tried to solve certain cases of Hilbert's 16th problem using the Melnikov function. As an example, we can refer to [2, 5, 6]. The subject we will discuss in this article is related to the breaking of a homoclinic orbit. When a perturbation occurs in a dynamic system, some dynamical behaviors of the system may change; One of these changes can be the breaking of the homoclinic orbit in the phase space. One of the tools that can express the effect of small perturbations on the homoclinic orbit for us is the Melnikov function, which in this article we try to explain how to make fluently. To continue our discussion more easily, it is necessary to familiarize the reader with a series of basic concepts, so in the second part we will introduce some preliminary concepts, and then in the third part we will describe the construction of the Melnikov function.

We consider the differential equation $ \dot{x} = f(x,\mu) $ where $ x \in \mathbb{R}^2 $, $ \mu \in \mathbb{R} $ and $f$ is at least $C^2$. Suppose there are two hyperbolic saddles $ p_{-}(\mu) $ and $ p_{+}(\mu) $. Suppose that for $ \mu = 0 $ there is a solution of this equation in the form of $ x_{\ast}(t) $ such that\begin{align*}\lim_{t\rightarrow - \infty} x_{\ast}(t) = p_{-}(0),~~~~~~ \lim_{t\rightarrow + \infty} x_{\ast}(t) = p_{+}(0) \end{align*} Our goal is to check that when $ \mu $ changes, how this connection (that is, the saddle connection between $p_{-}(t)$ and $p_{+}(t)$) is broken.\\We put $ x_{\ast}(0) = x_{0} $. The velocity vector $ x_{\ast}(t) $ at $ t = 0 $ is:\begin{align*}\dot{x}_{\ast}(0) = f(x_{0},0) = (f_{1}(x_{0},0),f_{2}(x_{0},0 )).\end{align*}If we put\begin{align*}u_{0} = \frac{1}{\Vert f(x_{0},0)\Vert^{2}}(-f_{2}(x_{0},0) , f_{1 }(x_{0},0)), \end{align*}then obviously, the vector $ u_{0} $ is perpendicular to $ f(x_{0},0) $. We consider a line segment $ \Sigma $ passing through $ x_{0} $ and in the direction of $ u_{0} $. Therefore, $ \Sigma $ with the formula $ x = x_{0} + \xi u_{0} $ where $ \vert \xi \vert < \alpha $ for some $ \alpha>0$. Suppose $ x_{-}(t,\mu) $ is a solution of $ \dot{x} = f(x,\mu) $ such that$$1)~~x_{-}(0,\mu) \in \Sigma,$$$$2)~~\lim_{t\rightarrow -\infty}x_{-}(t,\mu) = p_{-}(\mu),$$$$3)~~x_{-}(t,0) = x_{\ast}(t).$$This answer is on the unstable manifold $ p_{-}(\mu) $.Similarly, suppose that $ x_{+}(t,\mu) $ is a solution of $ \dot{x} = f(x,\mu) $ such that$$1)~~ x_{+}(0,\mu) \in \Sigma,$$$$2)~~\lim_{t\rightarrow +\infty}x_{+}(t,\mu) = p_{+}(\mu),$$$$3)~~x_{+}(t,0) = x_{\ast}(t).$$This solution is also placed inside the stable manifold $ p_{+}(\mu) $ (pay attention to the Figures 1 and 2).  We define the separation function as follows:\begin{align*}s(\mu) = \xi_{-}(\mu) - \xi_{+}(\mu). \end{align*}We put $ \psi_{0} = (-f_{2}(x_{0},0),f_{1}(x_{0},0)) $ then $\dot{x}_{*}(t)=(\dot{x}_{*1}(t),\dot{x}_{*2}(t))$ is a solution of the equation $\dot{v}(t)=A(t)v$ and so,$$ \psi(t) = exp\left(-\int_{0}^{t} div f(x_{\ast}(s),0)ds\right)\left(-\dot{x}_{\ast2}(t), \dot{x}_{\ast1}(t)\right)$$is a solution of the adjoint equation $\dot{w}=-wA(t)$, with the initial condition$$\psi(0)=(-\dot{x}_{*2}(0),\dot{x}_{*1}(0))=(-f_2(x_0,0),f_1(x_0,0))=\psi_0,$$Since we have $\psi_0u_0=1,$ it follows that:\begin{align*}\frac{d \xi_{\pm}}{d\mu}(0) = \psi_{0}\frac{d \xi_{\pm}}{d\mu}(0)u_{0} = \psi_{0}\frac{\partial x_{\pm}}{\partial \mu}(0,0).\end{align*}Also $ \frac{\partial x_{\pm}}{\partial \mu}(t,0) $ is an answer of the following equation:\begin{align}\label{b50}\dot{v} = D_{x}f(x_{\ast}(t),0)v + \frac{\partial f}{\partial \mu}(x_{\ast}(t) ,0).\end{align}Let's assume that the state transition matrix of the homogeneous linear system $ \dot{v} = D_{x}f(x_{\ast}(t),0) v $ is $ \phi(t,s) $. Now, according to the parameter change formula, we have:\begin{align*}\frac{\partial x_{+}}{\partial\mu}(0,0)=\phi(0,T)\frac{\partial x_{+}}{\partial\mu}(T, 0)-\int_{0}^{T}\phi(0,s)\frac{\partial f}{\partial\mu}(x_{\ast}(s) ds, 0)\end{align*}And\begin{align*}\frac{\partial x_{-}}{\partial\mu}(0,0)=\phi(0,-T)\frac{\partial x_{-}}{ \partial\mu}(-T, 0)-\int_{-T}^{0}\phi(0,s)\frac{\partial f}{\partial\mu}(x_{\ast} (s), 0) ds\end{align*}So\begin{eqnarray}\label{b51}\frac{d \xi_{-}}{d \mu}(0) = \psi_0 \frac{\partial x_{-}}{\partial\mu}(0,0) &=& \psi (0)(\phi(0,-T)\frac{\partial x_{-}}{\partial \mu}(-T,0) + \int_{-T}^{0}\phi( 0,s)\frac{\partial f}{\partial\mu}(x_{\ast}(s),0)ds)\\&=& \psi(-T)\frac{\partial x_{-}}{\partial \mu}(-T,0) + \int_{-T}^{0}\psi(s) \frac{\partial f}{\partial \mu}(x_{\ast}(s),0)ds. \end{eqnarray}Now, if $t\rightarrow\infty$ then:\begin{align*}\lim_{T\rightarrow +\infty}\psi(-T)\frac{\partial x_{-}}{\partial\mu}(-T,0) = 0. \end{align*}So\begin{align*}\frac{d \xi_{-}}{d \mu}(0) = \int_{-\infty}^{0} \psi(s)\frac{\partial f}{\partial \mu}( x_{\ast}(s),0)ds. \end{align*}Similarly\begin{align*}\frac{d \xi_{+}}{d \mu}(0) = \int_{0}^{+\infty} \psi(s)\frac{\partial f}{\partial \mu }(x_{\ast}(s),0)ds. \end{align*}So\begin{align*}s'(0) = \xi'_{-}(0) - \xi'_{+}(0) = \int_{-\infty}^{\infty} \psi(s)\frac {\partial f}{\partial \mu}(x_{\ast}(s),0).ds \end{align*} The recent integral is called the Melnikov integral if the function $f$ is dependent on $t$, that is, the right side of the differential equation is $f(x,\mu, t)$, then the value of Melnikov integral will also depend on $t$.

This article presents a unique perspective on constructing the Melnikov integral, different from what is typically found in textbooks. The problem's assumptions are presented in a way that accounts for both homoclinic and heteroclinic states. By following the method outlined in the article, interested readers can calculate Melnikov's integral formula in the time-dependent mode by considering the time-dependent perturbation on the system.

The Catalan numbers are ubiquitous in counting problems which is one of the primary reasons for its popularity. From various sources like books and Wikipedia we see that in combinatorial mathematics. The Catalan numbers form is a sequence of natural numbers that occur in various counting problems, often involving recursively-defined objects such as polygon triangulation, balanced parentheses, mountain ranges, diagonal avoiding paths and binary tree. Belgian mathematician Eugene Charles Catalan discovered Catalan numbers in 1838, while studying well-formed sequences of parentheses. They are named after the Belgian mathematician Eugene Charles Catalan. Although they are named after Catalan, they were not first discovered by him. These numbers appear in a variety of disguises, we are so used to having them around, it is perhaps hard to imagine a time when they were either unknown, or known but obscure and underappreciated. The organization of this paper is as follows. The first we encountered a number of occurrences of the CBC and Catalan numbers. Then, we study to understand the connections between these numbers and well-known structures of Catalan numbers like dyck paths, binary trees, permutations, partitions and etc. We also discuss some algebraic interpretations and additional aspects of Catalan numbers.

For positive integers $n\geq k$, the \textit{Binomial coefficient} ${n\choose k}=\frac{n!}{k!(n-k)!}$ appears in Khayyam Triangle which is one of the most important object to solving and proposing combinatorics problems. Using properties of binomial coefficients, we can write\begin{eqnarray*}{2n\choose n}&=&{2n-1\choose n-1}+{2n-1\choose n}\\&=&{2n-1\choose n}+{2n-1\choose n}\\&=&2{2n-1\choose n},\end{eqnarray*}where show these numbers are even. The \textit{CBC}s $\binom{2n}{n}$ are centrally located in even numbered rows in Pascal’s triangle, as Figure 1 shows. It was conjectured by \textit{Paul Erd\H{o}s} that the CBC numbers are squarefree for all $n>4$. This conjecturewas proved for every sufficiently large $n$ [4,16]. These numbers appears in several importance sequences like \textit{Catalan, Narayana, Motzikin and etc.} This is one of the reasons for researchers to having special attention to these numbers. The $n$-Catalan numbers defined as $C_n=\frac{1}{n+1}{2n\choose n}$, with initial value $C_0=1$. To date, nearly 400 articles and problems have appeared on Catalan numbers. \textit{Richard P. Stanley} of Massachusetts Institute of Technology has listed over $270$ occurrences of Catalan numbers in his EnumerativeCombinatorics, vol. 2, and another seventy on his Web site Catalan Addendum [17,18]. The CBC numbers are given as sequence $A000108 $ in the On-Line Encyclopedia of Integer Sequences [20]. Using generating functions, It can be employed to derive the following explicit formula . To this end, consider the generating function $C(x)=\sum_{n=0}^{\infty} C_nx^n=C_0+C_1x+C_2x^2+\cdots$. Using some initial values of Catalan numbers, we can write\begin{align*}\label{GF}xC^2(x)&=(1+x+2x^2+5x^3+14x^4+42x^5+\cdots)^2\\&=1+2x+5x^2+14x^3+\cdots =C(x)-1.\end{align*}Then\begin{eqnarray}4x^2C^2(x)=4xC(x)-4x+1-1\end{eqnarray}Solving for $C(x)=\frac{1-\sqrt{1-4x}}{2x}$. In this paper, we introduce and prove some structures of Catalan numbers. The set family like $\{\S_i\}_{n=1}^{\infty}$ is called structures of Catalan numbers if it can define the bijection $F: \S_n\longrightarrow C_n$ between these family and Catalan numbers. In the Section 1, we shall only study Dyck paths, and provide several bijective proofs like polygon triangulation, balanced parentheses, mountain ranges, that they are counted by Catalan numbers. We first introduce Dyck and binary paths, and then state our main theorem in Section 2.In Section 3, we states structures of Catalan numbers and Nonintersecting Chords. Let we have $2n$ persons with labaled $v_1; v_2;\ldots;v_{2n}$ such that located at the boundary of a circle with equal distancebetween every two adjacent persons. Let $R_n$ be the set of different ways to pair the $2n$persons with edges (as straight lines) that do no intersect. Then, $|R_n| = |C_n|$. In Section 4, we find structures of Catalan numbers and triangulated polygons. Section 5 devoted to structures of Catalan numbers and words. A \textit{plane tree} is a rooted tree with an ordering specified for the children of eachvertex. In the Section 6, we proved $C_n$ counts the number of plane trees with $n+1$ vertices and also, states several new structures of Catalan numbers and trees. Sections 7 and 8 devoted to structures of Catalan numbers between permutations and partitions., respectability.

There are many equivalent ways to define Catalan numbers. The main results of this paper,focus on combinatorial interpretations of Catalannumbers. Some of these structures of Catalan numbers which is discuss in this paper are1) Dyck paths of length $n$ with two steps $(0,1)$ and $(1,0)$.2) binary parenthesizations of a string of $n+1$ letters.3) The set of binary trees with n nodes.4) Plane trees with n internal nodes, all of degree 2.5) Paths from $(0,0)$ to $(2n,0)$ with steps $(1,−1)$ and $(1, 1)$, never falling below the $x$-axis.6) Noncrossing pairs of sequences of $n+1$ steps $(1, 0)$ and $(0, 1)$, which only intersect at start and end.7) In Stanley, this is described as non-intersecting arcs joining $n$ pairs of points in the plane. Our preferred version of this representation is to think of it as partitions of $2n$ into blocks of size 2.8) $n$ nonintersecting chords joining $2n$ points on a circle.9) Ways of drawing $n +1$ points on a line with arcs connecting them such that the arcs do not pass below the line, the arcs are noncrossing, all the arcs at a given node exit in the same direction (left or right), and the graph thus formed is a tree.10) Partitions of an $(n + 2)$-gon into triangles.11) Noncrossing partitions of $[n]$.12) Noncrossing Murasaki diagrams with $n$ vertical bars.13) Nonnesting partitions of $[n]$.14) Permutations of the multiset $\{1^2, 2^2,\ldots,n^2\}$ such that the first occurrences of each number appear in increasing order, and there is no subsequenceof the form $abab$.15) $321$-avoiding permutations of $[n]$.16) Permutations w of $[2n]$ with $n$ cycles of length two such that the product $(1, 2,\ldots,2n)w$ has $n$ cycles.17) Permutations of $[n]$ that can be stack sorted.18) Binary parenthesizations of a string of $n + 1$ letters.19) The numbers of the sequences $(a_1,\ldots,a_n)$ such that $1\leq a_1\leq \cdots \leq a_n$ and $a_i\leq i$.20) Standard Young Tableux of shape $(n, n)$.

We have studied some nice structures of the Catalan numbers in this paper. We make frequent references to Stanley’s list of Catalan representations [19]. These can be found in exercise 6.19, where each of the representations discussed is given as a part of the exercise. So, the first, we encountered a number of occurrences of the CBC $\binom{2n}{n}$ and Catalan numbers. Then, we study to understand the connections between these numbers and well-known structures of Catalan numbers like dyck paths, binary trees, permutations, partitions and etc. We also discuss some algebraic interpretations and additional aspects of Catalan numbers. It would be interesting for readers to considering results on divisibility of CBC and Catalan numbers. It would be interesting to explore similar techniques for Structures of Motzkin numbers, Delannoy numbers, Narayana Numbers and etc.

Let $G$ be a finite non-trivial group. The intersection graph $\Gamma(G)$, is a graph whose vertices are all proper non-trivial subgroups of $G$, and there is an edge between two distinct vertices $H $ and $K$ if and only if $H\cap K\neq 1$. In this paper, we determine the clique number of the intersection graph of the cyclic groups of orders having at most three primes in their decomposition.

Let $G$ be a finite group. There are several ways to associate a graph to $G$ (see [7] and the references therein). The one we will consider in this paper, is denoted by $\Gamma(G)$ and is called the intersection graph of $G$. The intersection graph $\Gamma(G)$ of a nontrivial group $G$ is an undirected graph without loops and multiple edges defined as follows: the vertex set is the set of all proper non-trivial subgroups of $G$, and there is an edge between two distinct vertices $H $ and $K$ if and only if $H\cap K\neq 1$ where 1 denotes the trivial subgroup of $G$. The graph $\Gamma(G)$ has been extensively studied ( see, for example, [1, 8, 11, 12]). Currently, the present authors in [4], have determined all groups $G$ such that the clique number of $\Gamma(G)$ is less than 5, and also they have given a criterion for solvability of finite groups $G$, by the clique number of $\Gamma(G)$. More precisely, it has been proved that if $G$ is a finite group such that the clique number of $\Gamma(G)$ is less than 13, then $G$ is solvable. Note that 13 is the clique number of $\Gamma(A_5)$, where $A_5$ is the alternating group on 5 letters. Other researches in this topic are intersection graphs of a semigroup, a module, and ideals of a ring, were investigated in [5], [2] and [6, 10], respectively.

We start this section with the following definition:Definition 2.1. The subset $X$ of vertices of a finite graph $\Gamma$ is called a {\it clique}, if the induced subgraph by $X$ is a complete graph. The maximum size of a clique, among all cliques of $\Gamma$, is called the clique number of $\Gamma$ and we denote it by $\omega(\Gamma)$. If $\Gamma$ is empty (without vertex), then we define $\omega(\Gamma)=0$ and $\omega(\Gamma)=1$ if $\Gamma$ is null (with a non-empty vertex set with no edges). A clique $X$ in $\Gamma$ is called {\it maximal} if there is no clique $Y$ in $\Gamma$ such that $X\subsetneq Y$. Note that the maximum size, among all maximal cliques in $\Gamma(G)$, is $\omega(\Gamma(G))$. To prove the main theorems, we need the following result that its proof can be found in the most valid book of group theory. Proposition 2.2. If $G=\langle g\rangle$ is a cyclic group of ordsr $n$, then for any divisor $s$ of $n$, there is a unique subgroup $H=\langle g^{\frac{n}{s}}\rangle$ of $G$ of order $s$. The following result is a consequence of the above proposition. Corollary 2.3. Let $G$ be a finite cyclic group. Then, the intersection of two proper subgroups of $G$ is non-trivial if and only if their orders are not relatively prime. For a natural number $n$, we denote by $C_n$ the cyclic group of order $n$, $\pi(n)$ the set of prime divisors of $n$ and $d(n)$ the number of all divisors of $n$. Note that if $p$ is a prime number and $n$ is a multiple of $p$, then the number of divisors of $n$ with multiple $p$ is $d(\frac{n}{p})$. If $V(\Gamma(G))$ is the set of vertices of $\Gamma(G)$, then by Proposition 2.2, we have $|V(\Gamma(C_n))|=d(n)-2$. In this paper, we obtain $\omega(\Gamma(C_n))$, where $|\pi(n)|=3$.

Now, we state and prove our main results. First, we find the clique number of a cyclic group of a prime power order. Proposition 3.1. If $p$ is a prime and $m$ is a positive integer, then $\omega(\Gamma(C_{p^m}))=m-1$.Proof. Since $|V(\Gamma(C_n))|=d(n)-2$ and $d(p^m)=m+1$, we get the conclusion. In the sequel, assume that $p_1, p_2$ and $p_3$ are distinct primes. Also assume that $n_1, n_2$ and $n_3$ are positive integers such that $n_1\geq n_2\geq n_3$. In the following results, we obtain the clique number of the intersection graph of group $C_{p_1^{n_1}p_2^{n_2}}$. We recall that $d(p_1^{n_1}p_2^{n_2})=(n_1+1)(n_2+1)$. Proposition 3.2. We have$\omega(\Gamma(C_{p_1^{n_1}p_2^{n_2}}))=d(p_1^{n_1}p_2^{n_2})-2-d(p_2^{n_2})=n_1n_2+n_1-1$. Proof. Suppose that $G=C_{p_1^{n_1}p_2^{n_2}}$. Then, we define the subsets of $V(\Gamma(G))$ as follows:♦ For $1\leq i\leq 2$, $V_{p_i}(\Gamma(G))$ is the set of all   proper subgroups $H$ of $G$ such that $\pi(|H|)=\{p_i\}$.
♦ $V_{p_1p_2}(\Gamma(G))$ is the set of all proper subgroups $H$ of $G$ such that $\pi(|H|)=\{p_1, p_2\}$. It is clear that $\{V_{p_1}(\Gamma(G)), V_{p_2}(\Gamma(G)), V_{p_1p_2}(\Gamma(G)) \}$ forms a partition for $V(\Gamma(G))$. By Proposition 2.2, we have $|V_{p_i} (\Gamma(G))|=d(p_i^{n_i})-1=n_i$ and $|V_{p_1p_2}(\Gamma(G))|=d(\frac{n}{p_1p_2})-1=n_1n_2-1$. By Corollary 2.3, in $\Gamma(G)$, each element of the clique $V_{p_1}(\Gamma(G))$ does not join to any element of the clique $V_{p_2}(G)$ and moreover all elements of $V_{p_i}(\Gamma(G))$ for $i=1, 2$ join to all elements of the clique $V_{p_1p_2}(G)$. Therefore $V_{p_1}(\Gamma(G))\cup V_{p_1p_2}(\Gamma(G))$ and $V_{p_2}(\Gamma(G))\cup V_{p_1p_2}(\Gamma(G))$ are the only maximal cliques of $\Gamma(G)$ and since $n_1\geq n_2$, we have the result.Now we state the last main result. Theorem 3.3. Let $G= C_n$ where $n=p_1^{n_1}p_2^{n_2}p_3^{n_3}$. Then\\(1) If $n_1\geq n_2n_3$, then $\omega(\Gamma(G))=d(\frac{n}{p_1})-1=n_1+n_1n_2+n_1n_3+n_1n_2n_3-1$.\\(2) If $n_1\leq n_2n_3$, then $\omega(\Gamma(G))=d(\frac{p_1^{n_1}p_2^{n_2}}{p_1p_2})+d(\frac{p_1^{n_1}p_3^{n_3}}{p_1p_3})+d(\frac{p_2^{n_2}p_3^{n_3}}{p_2p_3})+d(\frac{n}{p_1p_2p_3})-1$\\$$\hspace{5cm}=n_1n_2+n_1n_3+n_2n_3+n_1n_2n_3-1.~~~~~~~~~~~~~~~~~~~~~~~\hspace{4cm}$$Proof. Similar to the proof of Proposition 3.2, we define the subsets of $V(\Gamma(G))$ as follows:\\♦ For $1\leq i\leq 3$, $V_{p_i}(\Gamma(G))$ is the set of all subgroups $H$ of $G$ such that $\pi(|H|)=\{p_i\}$. Therefore, $|V_{p_i}(\Gamma(G))|=d(p_i^{n_i})-1=n_i$.♦ $V_{p_ip_j}(\Gamma(G))$ is the set of all subgroups $H$ of $G$ such that $\pi(|H|)=\{p_i, p_j\}$ for $1\leq i<j\leq 3$. Therefore, $|V_{p_ip_j}(\Gamma(G))|=d(\frac{p_i^{n_i}p_j^{n_j}}{p_ip_j})=n_in_j$where $i\neq j$.♦ $V_{p_1p_2p_3}(\Gamma(G))$ is the set of all proper subgroups $H$ of $G$ such that $\pi(|H|)=\{p_1, p_2, p_3\}$. So, we have$|V_{p_1p_2p_3}(\Gamma(G))|=d(\frac{n}{p_1p_2p_3})-1=n_1n_2n_3-1$.By Proposition 2.2, the above sets forms a partition for $V(\Gamma(G))$. By Corollary 2.3, in $\Gamma(G)$, each element of the clique $V_{p_i}(\Gamma(G))$ does not join to any element of the clique $V_{p_j}(G)\cup V_{p_jp_k}(G)$ for all distinct $i, j, k$ and moreover all elements of $V_{p_i}(\Gamma(G))$ for $i=1, 2, 3$, join to all elements of the clique $V_{p_1p_2p_3}(G)\cup V_{p_ip_j}(G)$, where $1\leq i\neq j\leq 3$. Since $|G|$ has three prime divisors, the intersection of every two subgroups of $G$ of orders with two distinct prime divisors, is non-trivial (by Corollary 2.3). It follows that $V_{p_1p_2}(\Gamma(G))\cup V_{p_1p_3}(\Gamma(G))\cup V_{p_2p_3}(\Gamma(G))$ is a cloque in $\Gamma(G)$. Now, we define $W_i$ as follows for $1\leq i\leq 4$: $$W_1=V_{p_1}(\Gamma(G))\cup V_{p_1p_2}(\Gamma(G))\cup V_{p_1p_3}(\Gamma(G))\cup V_{p_1p_2p_3}(\Gamma(G)), \hspace{0.5cm}|W_1|=n_1+n_1n_2+n_1n_3+n_1n_2n_3-1$$ $$W_2=V_{p_2}(\Gamma(G))\cup V_{p_1p_2}(\Gamma(G))\cup V_{p_2p_3}(\Gamma(G))\cup V_{p_1p_2p_3}(\Gamma(G)), \hspace{0.5cm}|W_2|=n_2+n_1n_2+n_2n_3+n_1n_2n_3-1$$ $$W_3=V_{p_3}(\Gamma(G))\cup V_{p_1p_3}(\Gamma(G))\cup V_{p_2p_3}(\Gamma(G))\cup V_{p_1p_2p_3}(\Gamma(G)), \hspace{0.5cm}|W_3|=n_3+n_1n_3+n_2n_3+n_1n_2n_3-1$$ $$W_4= V_{p_1p_2}(\Gamma(G))\cup V_{p_1p_3}(\Gamma(G))\cup V_{p_2p_3}(\Gamma(G))\cup V_{p_1p_2p_3}(\Gamma(G)), \hspace{0.5cm}|W_4|=n_1n_2+n_1n_3+n_2n_3+n_1n_2n_3-1.$$ It is easy to see that $W_1, W_2, W_3$ and $W_4$ are the only maximal cliques in $\Gamma(G)$. Therefore, $$\omega(\Gamma(G))=max\{|W_1|, |W_2|, |W_3|, |W_4|\}.$$Since $n_1\geq n_2\geq n_3$, we have $|W_1|\geq |W_2|\geq |W_3|$. Thus$$\omega(\Gamma(G))=max\{|W_1|, |W_4|\}=max\{n_1+n_1n_2+n_1n_3+n_1n_2n_3-1, n_1n_2+n_1n_3+n_2n_3+n_1n_2n_3-1\},$$this completes the proof.

Let's denote $\mathcal{S}^{\ast}(f_c)$ as a family of analytic functions $f(z)=z+a_2z^2+a_3z^3+\cdots$ in the open unit disk $\mathbb{D}$ that satisfy the following relation for $c\in (0,1)$:$$\frac{zf'(z)}{f(z)}\prec f_c(z)=\frac{1}{\sqrt{1-cz}}, \quad z\in\mathbb{D}.$$First, we introduce the analytic functions $f_c(z)$ and examine their starlike and positivity properties of the real part. Then, we obtain their images in the open unit disk $\mathbb{D}$, which are Cassini ovals. Cassini ovals, due to their properties, have applications in solving various problems in fields such as geometry, physics, and mathematics. These curves are used in studying the motion of waves and electromagnetic waves in interstellar spaces, as well as in the design of engineering structures such as telescopes. In this article, with the help of integrals, we investigate the structure of mappings in this family and some properties including maximum and minimum moduli, bounds of the real part of these functions. Moreover, we obtain the relationships between the defined geometric ranks with this family, including the order of starlikeness and order of strong starlikeness.

Let $\mathcal{A}$ be a set of analytic functions of the form $f(z)=z+a2z^2+a3z^3+\cdots$ in the open unit disc $\mathbb{D}:=\left\{z\in\mathbb{C}\colon |z|<1\right\}$. A function $f\in\mathcal{A}$ is called univalent if it is one-to-one. In [5], two classes of starlike and convex functions with order $0\le \beta<1$ are defined as follows:\begin{equation}\label{starlike-convex}\mathcal{S}^{\ast}(\beta):=\left\{f\in\mathcal{A}\colon \Re\left(\frac{zf'(z)}{f(z)}\right)>\beta\right\},\quad \mathcal{K}(\beta):=\left\{f\in\mathcal{A}\colon zf'(z)\in\mathcal{S}^{\ast}(\beta)\right\}.\end{equation}Similarly, in [2], the class of functions called strongly starlike with order $0<\alpha\le 1$ is defined as:\[\mathcal{SS}^{\ast}(\alpha)=\left\{f\in\mathcal{A}\colon \left|\mathrm{Arg}\frac{zf'(z)}{f(z)}\right|<\frac{\alpha \pi}{2}\right\}.\]If $f$ and $g$ are two analytic functions in $\mathbb{D}$, we say that $f$ is subordinate to $g$ \cite{Dur}, denoted by $f\prec g$, if and only if there exists an analytic function $w$ with $w(0)=0$ such that for all $z\in\mathbb{D}$:\[\left|w(z)\right|<1, \quad f(z)=g(w(z)).\]If $g$ is univalent, we have:\[f(z)\prec g(z) \Longleftrightarrow f(0)=g(0),\quad f(\mathbb{D})\subset g(\mathbb{D}).\]Given $c\in(0,1)$, analytic functions $f_c$ are defined as follows:(1.2)$$f_c(z):=\frac{1}{\sqrt{1-cz}}=1+\frac{c}{2}z+\frac{3c^2}{8}z^2+\cdots$$in the principal branch of the complex logarithm, where $\log 1=0$. These functions are univalent in $\mathbb{D}$ and map the open unit disc $\mathbb{D}$ into the interior of the Cassinian ovals given by the Cartesian equation:\begin{equation}\label{Cassinian-Ovals}(x^2+y^2)^2-\frac{2}{1-c^2}(x^2-y^2)+\frac{1}{1-c^2}=0,\end{equation}or the polar equation:\begin{equation}\label{Cassinian-Ovals1}r^4-\frac{2r^2 }{1-c^2} \cos(2\theta)=\frac{1}{c^2-1}.\end{equation}

In this section, we will first derive the structure of functions in the class $\mathcal{S}^{\ast}(f_c)$, and then using the stated theorems, we will determine the order of starlikeness and strongly starlikeness of functions in the class $\mathcal{S}^{\ast}(f_c)$. Theorem 2.1. A function $f$ belongs to the class $\mathcal{S}^{\ast}(f_c)$ if and only if there exists a function $p \prec f_c$ such that\begin{equation}\label{thm-1-0}f(z)=z\exp\left(\int_{0}^{z}\frac{p(t)-1}{t}dt\right), \quad z\in\mathbb{D}.\end{equation} If we set $p(z)=f_c(z)$ in theorem (2.1), then we get(2.2) $$F_c(z):=z\exp\left(\int_{0}^{z}\frac{f_c(t)-1}{t}dt\right)=\frac{4z}{(1+\sqrt{1-cz})^2}, \quad z\in\mathbb{D}.$$This function $F_c(z)$ is an extreme function for the class $\mathcal{S}^{\ast}(f_c)$. Figure 2 illustrates the image of the open unit disk $\mathbb{D}$ under the mapping $F_c(z)$ for $c=3/4$. Theorem 2.2. Let $f_c$ be the given function described in (1.2). Then $f_c$ is convex and satisfies the following conditions:\begin{equation}\label{max-min0}\max_{|z|=r<1}\left|f_c(z)\right|=f_c(r),\quad \min_{|z|=r<1}\left|f_c(z)\right|=f_c(-r).\end{equation} In the following theorem, we obtain bounds for the real part and strongly starlike mappings of the functions $f_c$. Theorem 2.3. Suppose $c\in(0,1)$. Then we have the following:(1) \[f_c(\mathbb{D})\subset \left\{w\in\mathbb{C}\colon \frac{1}{\sqrt{1+c}}<\Re(w)<\frac{1}{\sqrt{1-c}}\right\},\](2)\[f_c(\mathbb{D})\subset \left\{w\in\mathbb{C}\colon \left|\mathrm{Arg}(w)\right|<\frac12 \arccos\sqrt{1-c^2}\right\}.\] Theorem 2.4. If $f\in \mathcal{S}^{\ast}(fc)$ and $|z|=r<1$, then the following hold:(1) \[\frac{zf'(z)}{f(z)}\prec \frac{zF'_c(z)}{F_c(z)},\quad \frac{f(z)}{z}\prec\frac{F_{c}(z)}{z},\](2) \[F'_c(-r)\le \left|f'(z)\right|\le F'_c(r),\](3) \[-F_c(-r)\le |f(z)|\le F_c(r),\](4) \[\left|\arg{(f(z)/z)}\right|\le \max{|z|=r}\arg\left(\frac{1}{(1+\sqrt{1-cz})^2}\right),\](5) Either $f$ is a rotation of $F_c$ or\[\left\{w\in \mathbb{C} \colon\ |w|\leq-F_c(-1)=\frac{4}{(1+\sqrt{1+c})^2}\right\}\subsetf(\mathbb{D}),\]where in all cases, the function $F_c$ is defined as per equation (2.2).\end{thm}In the following theorem, we determine the subordination order and strong subordination order for the class of functions $\mathcal{S}^{\ast}(f_c)$. Theorem 2.5. The class of functions $\mathcal{S}^{\ast}(f_c)$ has the following properties:(1) For $0\le \beta\le \frac{1}{\sqrt{1+c}}$, we have\[\mathcal{S}^{\ast}(f_c)\subset \mathcal{S}^{\ast}(\beta).\](2) For $\frac{1}{\pi}\arccos\sqrt{1-c^2}\le \alpha\le 1$, we have\[\mathcal{S}^{\ast}(f_c)\subset \mathcal{SS}^{\ast}(\alpha).\]

The class $\mathcal{S}^{\ast}(f_c)$ consists of functions that can be represented in a specific form involving the function $f_c$, which is a special function related to the starlikeness property. The function $F_c(z)$, derived from $f_c(z)$, is an extreme function for the class $\mathcal{S}^{\ast}(f_c)$ and has specific properties, including convexity and bounds on its maximum and minimum modulus on the unit circle. The presented theorems provide bounds for the real part of the functions $f_c$ and establish relationships related to subordination and strong subordination order for the class of functions $\mathcal{S}^{\ast}(f_c)$. Overall, the obtained theorems and their proofs contribute to understanding the structural properties, order of starlikeness and strongly starlikeness, as well as subordination order within the class of functions $\mathcal{S}^{\ast}(f_c)$, for different values of the parameter $c$.

The above abstract has been extracted by the translator from the original article (J. Klaška, Real-world applications of number theory, South Bohemia Mathematical Letters, 25 no. 1 (2017) 39–47.) The present paper is concerned with practical applications of the number theory and is intended for all readers interested in applied mathematics. Using examples we show how human creativity can change the results of the pure mathematics into a practical usable form. Some historical notes are also included.

German mathematician Johann Carl Friedrich Gauss (30 April 1777-23 February 1855), regarded as one of the greatest mathematicians of all time, claimed: "Mathematics is the queen of the sciences and number theory is the queen of mathematics." However, for many years number theory had only few practical applications. It is well known that the great English number theorist Godfrey Harold Hardy (7 February 1877-1 December 1947) believed that number theory had no practical applications. See his essay "A Mathematician's Apology" [16]. Over the 20th and 21st centuries, this situation has changed significantly. Contrary to Hardy's opinion, many practical and interesting applications of number theory have been discovered. The present paper brings some remarkable examples of number theory applications in the real world. The paper can be regarded as a loose continuation of the author's preceding work [19] and [20]. Let $n$ be a positive integer, $\geq 2$. Then, the equation (1.1)$$a_1x_1+\ldots+a_nx_n=m$$is said to be a linear Diophantine equation if all unknowns $x_1,\ldots,x_n$ and all coefficients $a_1,\ldots,a_n,m$ are integers. For general methods for solving (1.1), see for example [5], [25], and [24, pp. 27-31]. In the following we give some interesting examples of using Diophantine equations in the natural sciences.

As the first example we show some applications of a linear Diophantine equation to problems in chemistry. In particular, we will deal with the balancing of chemical equations. See [6]. Consider a chemical equation written in the form (2.1)$$x_1A_{a_1}B_{b_1}C_{c_1}\ldots +x_2A_{a_2}B_{b_2}C_{c_3}\cdots +\cdots\rightarrow x'_1A'_{a'_1}B'_{b'_1}C'_{c'_1}\cdots +x'_2A'_{a'_2}B'_{b'_2}C'_{c'_3}\cdots +\cdots$$where $A, B, C,\ldots$ are the elements occurring in the reaction, $a_1, b_1, c_1,\ldots,a'_1, b'_1, c'_1,\ldots$ are positive integers or $0$, and $x_1, x_2,\ldots,x'_1, x'_2,\ldots$ are the unknown coeffcients of the reactants and products. Then, we hav (2.2)\begin{array}{rcl}x_1a_1+x_2a_2+\cdots&=&x'_1a'_1+x'_2a'_2+\cdots\\x_1b_1+x_2b_2+\cdots&=&x'_1b'_1+x'_2b'_2+\cdots\\x_1c_1+x_2c_2+\cdots&=&x'_1c'_1+x'_2c'_2+\cdots\\&\cdots&\end{array} Clearly, each equation of (2.2) expresses the law of conservation of the number of atoms for any particular element $A,B,C,\ldots$. Finding all integer solutions $[x_1,x_2,\ldots,x'_1,x'_2,\ldots]$ of (2.2) is a nice elementary problem of Diophantine analysis. In the second example we show how linear Diophantine equations can be used to determine the molecular formula [6]. Assume that a substance with a molecular weight of $m$ contains elements $A, B, C,\ldots$ with atomic weights $a, b, c,\ldots$ and that $x,y,z,\ldots$ represent the numbers of atoms of $A, B, C,\ldots$ in a molecule. Then, we have (2.3)$$ax+by+cz+\ldots=m.$$Let $\alpha,\beta,\gamma,\ldots$ denote the integers nearest the values $a, b, c,\ldots$ and $\mu$ denote theinteger nearest m. Then, (2.3) can be replaced by the linear Diophantine equation (2.4)$$\alpha x+\beta y+\gamma z+\cdots=\mu.$$ If we require that the values $x, y, z,\ldots$ in (2.4) should be reasonably small, we can solve (2.4) under a condition (2.5)$$-\frac{1}{2}<(a-\alpha)x+(b-\beta)y+(c-\gamma)z+\cdots<\frac{1}{2}.$$If more solutions of (2.4) are obtained, the true values may be found by substituting into (2.3) and finding which of them satisfies (2.3) with minimum deviation from $m$. In the third example we focus on an interesting problem in virology. Recall, that virus particles consist of protein subunits ordered geometrically according to strict symmetry rules. These rules highly depend on the chemical properties of the protein. As the last example we establish the number $p(n)$ which is the number of partitions of $n$ and it is known under the name of \emph{partitio numerorum}. Two interesting connections between the problem \emph{partitio numerorum} and physics will now be mentioned. First recall that the Hardy-Ramanujan formula (2.6)$$p(n)\sim \frac{1}{4n\sqrt{3}}\cdot\exp(\pi\sqrt{\frac{2n}{3}})\ \text{for}\ n\rightarrow\infty$$has been used, with great success, in quantum physics. The formula (2.6) was discovered in 1917 by G. H. Hardy and the brilliant Indian mathematician Srinivasa Ramanujan (22 December 1887-26 April 1920). The second very important application of Hardy-Ramanujan formula can be found in the problems of statistical mechanics.

Finally, some further significant applications of the number theory will be shortly mentioned. Above all, it is well known that the theory of Fibonacci numbers has many applications in physics, chemistry, biology, economy, and architecture. Listing 163 chronological references to papers published from 1611 to 2011, paper [19] can serve as an introduction to this field. Further fields of number theory with important applications include the theory of sequences over finite fields [20]. This theory found an application in the testing of Einstein's general relativity or in testing the global warming of oceans. Furthermore, using methods of elementary number theory, practical problems have been solved concerning to the splicing of telephone cables [21]. Many further interesting applications can be found in the book Number Theory and the Periodicity of Matter [3]. Lastly, new attractive applications of the number theory include cryptography, coding theory, and random number generation. With the rise of computers, these fields develop very rapidly with their importance continuously increasing.