In order to measure the uncertainty of financial asset returns in the stock market, this paper presents a new model, called SV-dtC model, a stochastic volatility (SV) model assuming that the stock return has a doubly truncated Cauchy distribution, which takes into account the high peak and fat tail of the empirical distribution simultaneously. Under the Bayesian framework, a prior and posterior analysis for the parameters is made and Markov Chain Monte Carlo (MCMC) is used for computing the posterior estimates of the model parameters and forecasting in the empirical application of Shanghai Stock Exchange Composite Index (SSE-CI) with respect to the proposed SV-dtC model and two classic SV-N (SV model with Normal distribution) and SV-T (SV model with Student-t distribution) models. The empirical analysis shows that the proposed SV-dtC model has better performance by model checking, including independence test (Projection correlation test), Kolmogorov-Smirnov test(K-S test) and Q-Q plot. Additionally, deviance information criterion (DIC) also shows that the proposed model has a significant improvement in model fit over the others.

We investigate the global structures of the non-selfsimilar solutions for $n$-dimensional ($n$-D) non-homogeneous Burgers equation, in which the initial data has two different constant states, which are separated by a $({n-1})$-dimensional sphere. We first obtain the expressions of $n$-D shock waves and rarefaction waves emitting from the initial discontinuity. Then, by estimating the new kind of interactions of the related elementary waves, we obtain the global structures of the non-selfsimilar solutions, in which ingenious techniques are proposed to construct the $n$-D shock waves. The asymptotic behaviors with geometric structures are also proved.

In this paper, we are concerned with the the Schrödinger-Newton system with $L^2$-constraint. Precisely, we prove that there cannot exist multi-peak normalized solutions concentrating at $k$ different critical points of $V(x)$ under certain assumptions on asymptotic behavior of $V(x)$ and its first derivatives near these points. Especially, the critical points of $V(x)$ in this paper must be degenerate.
The main tools are a local Pohozaev type of identity and the blow-up analysis. Our results also show that the asymptotic behavior of concentrated points to Schrödinger-Newton problem is quite different from the classical Schrödinger equations, which is mainly caused by the nonlocal term.

In this paper we mainly deal with the global well-posedness and large-time behavior of the 2D tropical climate model with small initial data. We first establish the global well-posedness of solution in the Besov space, then we obtain the optimal decay rates of solutions by virtue of the frequency decomposition method. Specifically, for the low frequency part, we use the Fourier splitting method of Schonbek and the spectrum analysis method, and for the high frequency part, we use the global energy estimate and the behavior of exponentially decay operator.

A coloring of graph $G$ is an injective coloring if its restriction to the neighborhood of any vertex is injective, which means that any two vertices get different colors if they have a common neighbor. The injective chromatic number $\chi_i(G)$ of $G$ is the least integer $k$ such that $G$ has an injective $k$-coloring. In this paper, we prove that (1) if $G$ is a planar graph with girth $g\geq 6$ and maximum degree $\Delta \geq 7$, then $\chi_i(G)\leq \Delta +2$; (2) if $G$ is a planar graph with $\Delta \geq24$ and without 3,4,7-cycles, then $\chi_i(G)\leq \Delta +2$.

An independent set in a graph $G$ is a set of pairwise non-adjacent vertices. The independence polynomial of $G$ is the polynomial $\sum\limits_{A} x^{|A|}$, where the sum is over all independent sets $A$ of $G$. In 1987, Alavi, Malde, Schwenk and Erdős conjectured that the independence polynomial of any tree or forest is unimodal. Although this unimodality conjecture has attracted many researchers' attention, it is still open. Recently, Basit and Galvin even asked a much stronger question whether the independence polynomial of every tree is ordered log-concave. Note that if a polynomial has only negative real zeros then it is ordered log-concave and unimodal. In this paper, we observe real-rootedness of independence polynomials of rooted products of graphs. We find some trees whose rooted product preserves real-rootedness of independence polynomials. In consequence, starting from any graph whose independence polynomial has only real zeros, we can obtain an infinite family of graphs whose independence polynomials have only real zeros. In particular, applying it to trees or forests, we obtain that their independence polynomials are unimodal and ordered log-concave.

In this paper the authors discuss a numerical simulation problem of three-dimensional compressible contamination treatment from nuclear waste. The mathematical model, a nonlinear convection-diffusion system of four PDEs, determines four major physical unknowns: the pressure, the concentrations of brine and radionuclide, and the temperature. The pressure is solved by a conservative mixed finite volume element method, and the computational accuracy is improved for Darcy velocity. Other unknowns are computed by a composite scheme of upwind approximation and mixed finite volume element. Numerical dispersion and nonphysical oscillation are eliminated, and the convection-dominated diffusion problems are solved well with high order computational accuracy. The mixed finite volume element is conservative locally, and get the objective functions and their adjoint vector functions simultaneously. The conservation nature is an important character in numerical simulation of underground fluid. Fractional step difference is introduced to solve the concentrations of radionuclide factors, and the computational work is shortened significantly by decomposing a three-dimensional problem into three successive one-dimensional problems. By the theory and technique of a priori estimates of differential equations, we derive an optimal order estimates in $L^2$ norm. Finally, numerical examples show the effectiveness and practicability for some actual problems.

The spatial and spatiotemporal autoregressive conditional heteroscedasticity (STARCH) models receive increasing attention. In this paper, we introduce a spatiotemporal autoregressive (STAR) model with STARCH errors, which can capture the spatiotemporal dependence in mean and variance simultaneously. The Bayesian estimation and model selection are considered for our model. By Monte Carlo simulations, it is shown that the Bayesian estimator performs better than the corresponding maximum-likelihood estimator, and the Bayesian model selection can select out the true model in most times. Finally, two empirical examples are given to illustrate the superiority of our models in fitting those data.

In this paper we deal with the initial-boundary value problem for the coupled Keller-Segel-Stokes system with rotational flux, which is corresponding to the case that the chemical is produced instead of consumed, $$ \left\{ \begin{array}{l} n_t+u\cdot\nabla n=\Delta n-\nabla\cdot(nS(x,n,c)\nabla c),\quad x\in \Omega, t>0, \\ c_t+u\cdot\nabla c=\Delta c-c+n,\quad x\in \Omega, \ \ t>0, \\ u_t+\nabla P=\Delta u+n\nabla \phi,\quad x\in \Omega, \ \ t>0, \\ \nabla\cdot u=0,\quad x\in \Omega, t>0 \end{array} \right. (KSS) $$ subject to the boundary conditions $(\nabla n-nS(x,n,c)\nabla c)\cdot\nu=\nabla c\cdot\nu=0$ and $u=0$, and suitably regular initial data $(n_0 (x),c_0 (x),u_0 (x))$, where $\Omega\subset \mathbb{R}^3$ is a bounded domain with smooth boundary $\partial\Omega$. Here $S$ is a chemotactic sensitivity satisfying $|S(x,n,c)|\leq C_S(1+n)^{-\alpha}$ with some $C_S> 0$ and $\alpha> 0$. The greatest contribution of this paper is to consider the large time behavior of solutions for the system (KSS), which is still open even in the 2D case. We can prove that the corresponding solution of the system (KSS) decays to $(\frac{1}{|\Omega|}\int_{\Omega}n_0,\frac{1}{|\Omega|}\int_{\Omega}n_0,0)$ exponentially, if the coefficient of chemotactic sensitivity is appropriately small. As a precondition to consider the asymptotic behavior, we also show the global existence and boundedness of the corresponding initial-boundary problem KSS with a simplified method. We find a new phenomenon that the suitably small coefficient $C_S$ of chemotactic sensitivity could benefit the global existence and boundedness of solutions to the model KSS.

The existence and stability of stationary solutions for a reaction-diffusion-ODE system are investigated in this paper. We first show that there exist both continuous and discontinuous stationary solutions. Then a good understanding of the stability of discontinuous stationary solutions is gained under an appropriate condition. In addition, we demonstrate the influences of the diffusion coefficient on stationary solutions. The results we obtained are based on the super-/sub-solution method and the generalized mountain pass theorem. Finally, some numerical simulations are given to illustrate the theoretical results.

Let $\mathbf{G}=\{G_i: i\in[n]\}$ be a collection of not necessarily distinct $n$-vertex graphs with the same vertex set $V$, where $\mathbf{G}$ can be seen as an edge-colored (multi)graph and each $G_i$ is the set of edges with color $i$. A graph $F$ on $V$ is called rainbow if any two edges of $F$ come from different $G_i$s'. We say that $\mathbf{G}$ is rainbow pancyclic if there is a rainbow cycle $C_{\ell}$ of length $\ell$ in $\mathbf{G}$ for each integer $\ell\in [3,n]$. In 2020, Joos and Kim proved a rainbow version of Dirac's theorem: If $\delta(G_i)\geq\frac{n}{2}$ for each $i\in[n]$, then there is a rainbow Hamiltonian cycle in $\mathbf{G}$. In this paper, under the same condition, we show that $\mathbf{G}$ is rainbow pancyclic except that $n$ is even and $\mathbf{G}$ consists of $n$ copies of $K_{\frac{n}{2},\frac{n}{2}}$. This result supports the famous meta-conjecture posed by Bondy.

The classical Archimedean approximation of $\pi$ uses the semiperimeter or area of regular polygons inscribed in or circumscribed about a unit circle in $\mathbb{R}^2 $ and it is well-known that by using linear combinations of these basic estimates, modern extrapolation techniques can greatly speed up the approximation process. % reduce the associated approximation errors. Similarly, when $n$ vertices are randomly selected on the circle, the semiperimeter and area of the corresponding random inscribed and circumscribing polygons are known to converge to $\pi$ almost surely as $ n \to \infty $, and by further applying extrapolation processes, faster convergence rates can also be achieved through similar linear combinations of the semiperimeter and area of these random polygons. In this paper, we further develop nonlinear extrapolation methods for approximating $\pi$ through certain nonlinear functions of the semiperimeter and area of such polygons. We focus on two types of extrapolation estimates of the forms $ \mathcal{X}_n = \mathcal{S}_n^{\alpha} \mathcal{A}_n^{\beta} $ and $ \mathcal{Y}_n (p) = \left( \alpha \mathcal{S}_n^p + \beta \mathcal{A}_n^p \right)^{1/p} $ where $ \alpha + \beta = 1 $, $ p \neq 0 $, and $ \mathcal{S}_n $ and $ \mathcal{A}_n $ respectively represents the semiperimeter and area of a random $n$-gon inscribed in the unit circle in $ \mathbb{R}^2 $, and $ \mathcal{X}_n $ may be viewed as the limit of $ \mathcal{Y}_n (p) $ when $ p \to 0 $. By deriving probabilistic asymptotic expansions with carefully controlled error estimates for $ \mathcal{X}_n $ and $ \mathcal{Y}_n (p) $, we show that the choice $ \alpha = 4/3 $, $ \beta = -1/3 $ minimizes the approximation error in both cases, and their distributions are also asymptotically normal.

We are concerned with singularity formation of strong solutions to the two-dimensional (2D) full compressible magnetohydrodynamic equations with zero resistivity in a bounded domain. By energy method and critical Sobolev inequalities of logarithmic type, we show that the strong solution exists globally if the temporal integral of the maximum norm of the deformation tensor is bounded. Our result is the same as Ponce's criterion for 3D incompressible Euler equations. In particular, it is independent of the magnetic field and temperature. Additionally, the initial vacuum states are allowed.

This paper considers a linear regression model involving both quantitative and qualitative factors and an $m$-dimensional response variable y. The main purpose of this paper is to investigate $D$-optimal designs when the levels of the qualitative factors interact with the levels of the quantitative factors. Under a general covariance structure of the response vector y, here we establish that the determinant of the information matrix of a product design can be separated into two parts corresponding to the two marginal designs. Moreover, it is also proved that $D$-optimal designs do not depend on the covariance structure if we assume hierarchically ordered system of regression models.

This paper mainly studies the contact extension of conservative or dissipative systems, including some old and new results for wholeness. Then extension of contact system is corresponding to the symplectification of contact Hamiltonian system. This is a reciprocal process and the relation between symplectic system and contact system has been discussed. We have an interesting discovery that by adding a pure variable $p$, the slope of the tangent of the orbit, every differential system can be regarded as an independent subsystem of contact Hamiltonian system defined on the projection space of contact phase space.

This paper aims to extend a space-time spectral method to address the multi-term time-fractional subdiffusion equations with Caputo derivative. In this method, the Jacobi polynomials are adopted as the basis functions for temporal discretization and the Lagrangian polynomials are used for spatial discretization. An efficient spectral approximation of the weak solution is established. The main work is the demonstration of the well-posedness for the weak problem and the derivation of a posteriori error estimates for the spectral Galerkin approximation. Extensive numerical experiments are presented to perform the validity of a posteriori error estimators, which support our theoretical results.

Let $G$ be a simple graph and $G^{\sigma}$ be the oriented graph with $G$ as its underlying graph and orientation $\sigma$. The rank of the adjacency matrix of $G$ is called the rank of $G$ and is denoted by $r(G)$. The rank of the skew-adjacency matrix of $G^{\sigma}$ is called the skew-rank of $G^{\sigma}$ and is denoted by $sr(G^{\sigma})$. Let $V(G)$ be the vertex set and $E(G)$ be the edge set of $G$. The cyclomatic number of $G$, denoted by $c(G)$, is equal to $|E(G)|-|V(G)|+\omega(G)$, where $\omega(G)$ is the number of the components of $G$. It is proved for any oriented graph $G^{\sigma}$ that $-2c(G)\leqslant sr(G^{\sigma})-r(G)\leqslant2c(G)$. In this paper, we prove that there is no oriented graph $G^{\sigma}$ with $sr(G^{\sigma})-r(G)=2c(G)-1$, and in addition, there are infinitely many oriented graphs $G^{\sigma}$ with connected underlying graphs such that $c(G)=k$ and $sr(G^{\sigma})-r(G)=2c(G)-\ell$ for every integers $k, \ell$ satisfying $0\leqslant\ell\leqslant4k$ and $\ell\neq1$.

We consider a Neumann problem driven by a ($p(z),q(z)$)-Laplacian (anisotropic problem) plus a parametric potential term with $\lambda>0$ being the parameter. The reaction is superlinear but need not satisfy the Ambrosetti-Rabinowitz condition. We prove a bifurcation-type theorem describing the changes in the set of positive solutions as the parameter $\lambda$ moves on $\mathring{\mathbb{R}}_+=(0,+\infty)$.

In this paper, we investigate the following $p$-Kirchhoff equation \begin{eqnarray*} \left\{\begin{array}{ll} \big(a+b\int_{\mathbb{R}^{N}}(|\nabla u|^{p}+|u|^{p})dx\big)\big(-\Delta_{p}u+|u|^{p-2}u\big)=|u|^{s-2}u+\mu u,~x\in \mathbb{R}^{N},\\ \int_{\mathbb{R}^{N}}|u|^{2}dx=\rho, \end{array} \right. \end{eqnarray*} where $a> 0, \,b \geq 0 , \,\rho>0$ are constants, $p^{\ast}=\frac{Np}{N-p}$ is the critical Sobolev exponent, $\mu$ is a Lagrange multiplier, $-\Delta_{p}u=-{\rm div}(|\nabla u|^{p-2}\nabla u), \ 2<p<N<2p, \ \mu\in\mathbb{R}$, and $ s\in(2\frac{N+2}{N}p-2,~p^{\ast})$. We demonstrate that the $p$-Kirchhoff equation has a normalized solution using the mountain pass lemma and some analysis techniques.

This paper considers the random coefficient autoregressive model with time-functional variance noises, hereafter the RCA-TFV model. We first establish the consistency and asymptotic normality of the conditional least squares estimator for the constant coefficient. The semiparametric least squares estimator for the variance of the random coefficient and the nonparametric estimator for the variance function are constructed, and their asymptotic results are reported. A simulation study is presented along with an analysis of real data to assess the performance of our method in finite samples.

For an integer $r\geq 2$ and bipartite graphs $H_i$, where $1\leq i\leq r$, the bipartite Ramsey number $br(H_1,H_2,\cdots,H_r)$ is the minimum integer $N$ such that any $r$-edge coloring of the complete bipartite graph $K_{N, N}$ contains a monochromatic subgraph isomorphic to $H_i$ in color $i$ for some $1\leq i\leq r$. We show that if $r\geq 3, \alpha_1,\alpha_2>0, \alpha_{j+2}\geq [(j+2)!-1]\sum\limits^{j+1}_{i=1}\alpha_i$ for $j=1,2,\cdots,r-2$, then $br(C_{2\lfloor \alpha_1 n\rfloor},C_{2\lfloor \alpha_2 n\rfloor},\cdots,C_{2\lfloor \alpha_r n\rfloor})=\big(\sum\limits^r_{j=1} \alpha_j+o(1)\big)n.$

This article analyzes the Pareto optimal allocations, agreeable trades and agreeable bets under the maxmin Choquet expected utility (MCEU) model. We provide several useful characterizations for Pareto optimal allocations for risk averse agents. We derive the formulation descriptions for non-existence agreeable trades or agreeable bets for risk neutral agents. We build some relationships between ex-ante stage and interim stage on agreeable trades or bets when new information arrives.

The classical Ambarzumyan's theorem states that if the Neumann eigenvalues of the Sturm-Liouville operator $-\frac{d^2}{dx^2}+q$ with an integrable real-valued potential $q$ on $[0,\pi]$ are $\{n^2:n\geq 0\}$, then $q=0$ for almost all $x\in [0,\pi]$. In this work, the classical Ambarzumyan's theorem is extended to the Dirac operator on equilateral tree graphs. We prove that if the spectrum of the Dirac operator on graphs coincides with the unperturbed case, then the potential is identically zero.

Considering that HBV belongs to the DNA virus family and is hepatotropic, we model the HBV DNA-containing capsids as a compartment. In this paper, a delayed HBV infection model is established, where the general incidence function and two infection routes including cell-virus infection and cell-cell infection are introduced. According to some preliminaries, including well-posedness, basic reproduction number and existence of two equilibria, we obtain the threshold dynamics for the model. We illustrate numerical simulations to verify the above theoretical results, and furthermore explore the impacts of intracellular delay and cell-cell infection on the global dynamics of the model.

In this paper, the authors discuss a three-dimensional problem of the semiconductor device type involved its mathematical description, numerical simulation and theoretical analysis. Two important factors, heat and magnetic influences are involved. The mathematical model is formulated by four nonlinear partial differential equations (PDEs), determining four major physical variables. The influences of magnetic fields are supposed to be weak, and the strength is parallel to the $z$-axis. The elliptic equation is treated by a block-centered method, and the law of conservation is preserved. The computational accuracy is improved one order. Other equations are convection-dominated, thus are approximated by upwind block-centered differences. Upwind difference can eliminate numerical dispersion and nonphysical oscillation. The diffusion is approximated by the block-centered difference, while the convection term is treated by upwind approximation. Furthermore, the unknowns and adjoint functions are computed at the same time. These characters play important roles in numerical computations of conductor device problems. Using the theories of priori analysis such as energy estimates, the principle of duality and mathematical inductions, an optimal estimates result is obtained. Then a composite numerical method is shown for solving this problem.

This paper concerns a global optimality principle for fully coupled mean-field control systems. Both the first-order and the second-order variational equations are fully coupled mean-field linear FBSDEs. A new linear relation is introduced, with which we successfully decouple the fully coupled first-order variational equations. We give a new second-order expansion of $Y^\varepsilon$ that can work well in mean-field framework. Based on this result, the stochastic maximum principle is proved. The comparison with the stochastic maximum principle for controlled mean-field stochastic differential equations is supplied.

The strong chromatic index of a graph is the minimum number of colors needed in a proper edge coloring so that no edge is adjacent to two edges of the same color. An outerplane graph with independent crossings is a graph embedded in the plane in such a way that all vertices are on the outer face and two pairs of crossing edges share no common end vertex. It is proved that every outerplane graph with independent crossings and maximum degree $\Delta$ has strong chromatic index at most $4\Delta-6$ if $\Delta\geq 4$, and at most 8 if $\Delta\leq 3$. Both bounds are sharp.

The law of the iterated logarithm (LIL) for the performance measures of a two-station queueing network with arrivals modulated by independent queues is developed by a strong approximation method. For convenience, two arrival processes modulated by queues comprise the external system, all others are belong to the internal system. It is well known that the exogenous arrival has a great influence on the asymptotic variability of performance measures in queues. For the considered queueing network in heavy traffic, we get all the LILs for the queue length, workload, busy time, idle time and departure processes, and present them by some simple functions of the primitive data. The LILs tell us some interesting insights, such as, the LILs of busy and idle times are zero and they reflect a small variability around their fluid approximations, the LIL of departure has nothing to do with the arrival process, both of the two phenomena well explain the service station's situation of being busy all the time. The external system shows us a distinguishing effect on the performance measures: an underloaded (overloaded, critically loaded) external system affects the internal system through its arrival (departure, arrival and departure together). In addition, we also get the strong approximation of the network as an auxiliary result.

The Gerdjikov-Ivanov (GI) hierarchy is derived via recursion operator, in this article, we mainly investigate the third-order flow GI equation. In the framework of the Riemann-Hilbert method, the soliton matrices of the third-order flow GI equation with simple zeros and elementary high-order zeros of Riemann-Hilbert problem are constructed through the standard dressing process. Taking advantage of this result, some properties and asymptotic analysis of single soliton solution and two soliton solution are discussed, and the simple elastic interaction of two soliton are proved. Compared with soliton solution of the classical second-order flow, we find that the higher-order dispersion term affects the propagation velocity, propagation direction and amplitude of the soliton. Finally, by means of a certain limit technique, the high-order soliton solution matrix for the third-order flow GI equation is derived.

Motivated by a discrete-time process intended to measure the speed of the spread of contagion in a graph, the burning number $b(G)$ of a graph $G$, is defined as the smallest integer $k$ for which there are vertices $x_1,\cdots, x_k$ such that for every vertex $u$ of $G$, there exists $i\in \{1,\cdots, k\}$ with $d_G(u, x_i)\le k-i$, and $d_G(x_i, x_j) \ge j-i$ for any $1\le i<j \le k$. The graph burning problem has been shown to be NP-complete even for some acyclic graphs with maximum degree three. In this paper, we determine the burning numbers of all short barbells and long barbells, respectively.

In this paper, we consider a system which has $k$ statistically independent and identically distributed strength components and each component is constructed by a pair of statistically dependent elements with doubly type-II censored scheme. These elements $(X_1, Y_1 ),$ $(X_2, Y_2 ),$ $\cdots$, $(X_k, Y_k)$ follow a bivariate Kumaraswamy distribution and each element is exposed to a common random stress $T$ which follows a Kumaraswamy distribution. The system is regarded as operating only if at least $s$ out of $k$ ($1\leq s\leq k$) strength variables exceed the random stress. The multicomponent reliability of the system is given by $R_{s,k}$=$P$(at least $s$ of the $(Z_1, \cdots, Z_k)$ exceed $T)$ where $Z_i=\min(X_i, Y_i ), \ i=1,\cdots, k.$ The Bayes estimates of $R_{s,k}$ have been developed by using the Markov Chain Monte Carlo methods due to the lack of explicit forms. The uniformly minimum variance unbiased and exact Bayes estimates of $R_{s,k}$ are obtained analytically when the common second shape parameter is known. The asymptotic confidence interval and the highest probability density credible interval are constructed for $R_{s,k}$. The reliability estimators are compared by using the estimated risks through Monte Carlo simulations.

In this paper, we prove a local Hamilton type gradient estimate for positive solution of the nonlinear parabolic equation $$ u_{t}(x,t)=\Delta u(x,t) +au(x,t)\ln u(x,t)+ bu^{\alpha}(x,t), $$ on $\mathbf{M}\times (-\infty, \infty)$ with $\alpha\in\mathbf{R}$, where $a$ and $b$ are constants. As application, the Harnack inequalities are derived.

Aiming at the problem that the asset's fluctuation influences the borrower's repayment ability, a loan with a new and flexible repayment method is designed, which depends on the asset value of the borrower. The repayment method can reduce the loan default probability, but it causes the uncertainty of the pay off time. Because the repayment term is related to the regular repayment amount in this method, a boundary for the regular repayment amount is set up in order to avoid too long repayment term. This will balance the benefit of borrowers and lenders and improve the applicability of this method. By establishing a mathematical model of the residual value of the loan, this model can be transformed into an initial-boundary problem of a partial differential equation. The analytic solution and the expected time to pay off the loan are obtained. Finally, numerical analysis are shown.

Let $G$ be a finite group generated by $S$ and $C(G,S)$ the Cayley digraphs of $G$ with connection set $S$. In this paper, we give some sufficient conditions for the existence of hamiltonian circuit in $C(G,S)$, where $G=Z_m\rtimes H$ is a semiproduct of $Z_m$ by a subgroup $H$ of $G$. In particular, if $m$ is a prime, then the Cayley digraph of $G$ has a hamiltonian circuit unless $G=Z_m\times H$. In addition, we introduce a new digraph operation, called $\varphi$-semiproduct of $\Gamma_1$ by $\Gamma_2$ and denoted by $\Gamma_1 \rtimes_\varphi\Gamma_2$, in terms of mapping $\varphi:V(\Gamma_2)\longrightarrow\{1,-1\}$. Furthermore we prove that $C(Z_m, \{a\}) \rtimes_{\varphi}C(H,S)$ is also a Cayley digraph if $\varphi$ is a homomorphism from $H$ to $\{1,-1\}\le Z_m^*$, which produces some classes of Cayley digraphs that have hamiltonian circuits.