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.

In present work studied the new boundary value problem for semi linear (Power-type nonlinearities) system equations of mixed hyperbolic -elliptic Keldysh type in the multivariate dimension with the changing time direction. Considered problem and equation belongs to the modern level partial differential equations. Applying methods of functional analysis, topological methods, “ε” -regularizing. and continuation by the parameter at the same time with aid of a prior estimates, under assumptions conditions on coefficients of equations of system, the existence and uniqueness of generalized and regular solutions of a boundary problem are established in a weighted Sobolev’s space. In this work one of main idea consists of show that the new boundary value problem which investigated in case of linear system equations can be well-posed when added nonlinear terms to this linear system equations, moreover in this case constructed new weithged spaces, the identity between of strong and weak solutions is established.

A four-dimensional delay differential equations (DDEs) model of malaria with standard incidence rate is proposed. By utilizing the limiting system of the model and Lyapunov direct method, the global stability of equilibria of the model is obtained with respect to the basic reproduction number ${R}_{0}$. Specifically, it shows that the disease-free equilibrium ${E}^{0}$ is globally asymptotically stable (GAS) for ${R}_{0}<1$, and globally attractive (GA) for ${R}_{0}=1$, while the endemic equilibrium $E^{\ast}$ is GAS and ${E}^{0}$ is unstable for ${R}_{0}>1$. Especially, to obtain the global stability of the equilibrium $E^{\ast}$ for $R_{0}>1$, the weak persistence of the model is proved by some analysis techniques.

In this paper, we consider a modification of the well-known logistic family using a family of fuzzy numbers. The dynamics of this modified logistic map is studied by computing its topological entropy with a given accuracy. This computation allows us to characterize when the dynamics of the modified family is chaotic. Besides, some attractors that appear in bifurcation diagrams are explained. Finally, we will show that the dynamics induced by the logistic family on the fuzzy numbers need not be complicated at all.

In the current paper, the best linear unbiased estimators (BLUEs) of location and scale parameters from location-scale family will be respectively proposed in cases when one parameter is known and when both are unknown under moving extremes ranked set sampling (MERSS). Explicit mathematical expressions of these estimators and their variances are derived. Their relative efficiencies with respect to the minimum variance unbiased estimators (MVUEs) under simple random sampling (SRS) are compared for the cases of some usual distributions. The numerical results show that the BLUEs under MERSS are significantly more efficient than the MVUEs under SRS.

This paper deals with the existence, uniqueness and continuous dependence of mild solutions for a class of conformable fractional differential equations with nonlocal initial conditions. The results are obtained by means of the classical fixed point theorems combined with the theory of cosine family of linear operators.

For a revised model of Caldentey and Stacchetti (Econometrica, 2010) in continuous-time insider trading with a random deadline which allows market makers to observe some information on a risky asset, a closed form of its market equilibrium consisting of optimal insider trading intensity and market liquidity is obtained by maximum principle method. It shows that in the equilibrium, (i) as time goes by, the optimal insider trading intensity is exponentially increasing even up to infinity while both the market liquidity and the residual information are exponentially decreasing even down to zero; (ii) the more accurate information observed by market makers, the stronger optimal insider trading intensity is such that the total expect profit of the insider is decreasing even go to zero while both the market liquidity and the residual information are decreasing; (iii) the longer the mean of random time, the weaker the optimal insider trading intensity is while the more both the residual information and the expected profit are, but there is a threshold of trading time, half of the mean of the random time, such that if and only if after it the market liquidity is increasing with the mean of random time increasing.

In this paper, solutions of the Camassa-Holm equation near the soliton $Q$ is decomposed by pseudo-conformal transformation as follows: $\lambda^{1/2}(t)u(t,\lambda(t)y+x(t))=Q(y)+\varepsilon(t,y)$, and the estimation formula with respect to $\varepsilon(t,y)$ is obtained: $|\varepsilon(t,y)|\leq Ca_3Te^{-\theta|y|}+|\lambda^{1/2}(t)\varepsilon_0|$. For the CH equation, we prove that the solution of the Cauchy problem and the soliton $Q$ is sufficiently close as $y\rightarrow\infty$, and the approximation degree of the solution and $Q$ is the same as that of initial data and $Q$, besides the energy distribution of $\varepsilon$ is consistent with the distribution of the soliton $Q$ in $H^2$.

A bio-safe dengue control strategy is to use Wolbachia, which can induce incomplete cytoplasmic incompatibility (CI) and reduce the mating competitiveness of infected males. In this work, we formulate a delay differential equation model, including both the larval and adult stages of wild mosquitoes, to assess the impacts of CI intensity ξ and mating competitiveness θ of infected males on the suppression efficiency. Our analysis identifies a CI intensity threshold ξ^* below which a successful suppression is impossible. When ξ≥ξ^*, the wild population will be eliminated ultimately if the releasing level exceeds the release amount threshold R^* uniformly. The dependence of R^* on ξ and θ, and the impact of temperature on suppression are further exhibited through numerical examples. Our analyses indicate that a slight reduction of ξ is more devastating than significantly decrease of θ in the suppression efficiency. To suppress more than 95% wild mosquitoes during the peak season of dengue in Guangzhou, the optimal starting date for releasing is sensitive to ξ but almost independent of θ. One percent reduction of ξ from 1 requires at least one week earlier in the optimal releasing starting date from 7 weeks ahead of the peak season of dengue.

A path-factor is a spanning subgraph $F$ of $G$ such that every component of $F$ is a path with at least two vertices. Let $k\geq2$ be an integer. A $P_{\geq k}$-factor of $G$ means a path factor in which each component is a path with at least $k$ vertices. A graph $G$ is a $P_{\geq k}$-factor covered graph if for any $e\in E(G)$, $G$ has a $P_{\geq k}$-factor including $e$. Let $\beta$ be a real number with $\frac{1}{3}\leq\beta\leq1$ and $k$ be a positive integer. We verify that (\romannumeral1) a $k$-connected graph $G$ of order $n$ with $n\geq5k+2$ has a $P_{\geq3}$-factor if $|N_G(I)|>\beta(n-3k-1)+k$ for every independent set $I$ of $G$ with $|I|=\lfloor\beta(2k+1)\rfloor$; (\romannumeral2) a $(k+1)$-connected graph $G$ of order $n$ with $n\geq5k+2$ is a $P_{\geq3}$-factor covered graph if $|N_G(I)|>\beta(n-3k-1)+k+1$ for every independent set $I$ of $G$ with $|I|=\lfloor\beta(2k+1)\rfloor$.

We consider the following quasilinear Schrödinger equation involving $p$-Laplacian \begin{align*} -\Delta_p u +V(x)|u|^{p-2}u-\Delta_p(|u|^{2\eta})|u|^{2\eta-2}u=\lambda\frac{|u|^{q-2}u}{|x|^{\mu}}+\frac{|u|^{2\eta p^*(\nu)-2}u}{|x|^\nu}\quad\text{in}\ \mathbb{R}^N, \end{align*} where $ N> p>1,\ \eta\ge \frac{p}{2(p-1)}$, $p< q<2\eta p^*(\mu)$, $p^*(s)=\frac{p(N-s)}{N-p}$, and $\lambda, \mu, \nu$ are parameters with $\lambda>0$, $\mu, \nu \in [0,p)$. Via the Mountain Pass Theorem and the Concentration Compactness Principle, we establish the existence of nontrivial ground state solutions for the above problem.

In this paper, we study the global behavior of solutions to the spherically symmetric(it means that the problem is 2+2 framework)coupled Nonlinear-Einstein-Klein-Gordon (NLEKG) system in the presence of a negative cosmological constant. We prove the well posedness of the NLEKG system in the Schwarzschild-AdS spacetimes and that the Schwarzschild-AdS spacetimes (the trivial black hole solutions of the EKG system for which ϕ = 0 identically) are asymptotically stable. Small perturbations of Schwarzschild-AdS initial data again lead to regular black holes, with the metric on the black hole exterior approaching a SchwarzschildAdS spacetime. Bootstrap argument on the black hole exterior, with the redshift effect and weighted Hardy inequalities playing the fundamental role in the analysis. Both integrated decay and pointwise decay estimates are obtained.

In this paper, the bilinear formalism, bilinear Bäcklund transformations and Lax pair of the (2+1)-dimensional KdV equation are constructed by the Bell polynomials approach. The N-soliton solution is derived directly from the bilinear form. Especially, based on the two-soliton solution, the lump solution is given out analytically by taking special parameters and using Taylor expansion formula. With the help of the multidimensional Riemann theta function, multiperiodic (quasiperiodic) wave solutions for the (2+1)-dimensional KdV equation are obtained by employing the Hirota bilinear method. Moreover, the asymptotic properties of the one- and two-periodic wave solution, which reveal the relations with the single and two-soliton solution, are presented in detail.

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.

In this paper, the approximate analytical oscillatory solutions to the generalized KolmogorovPetrovsky-Piskunov equation (gKPPE for short) are discussed by employing the theory of dynamical system and hypothesis undetermined method. According to the corresponding dynamical system of the bounded traveling wave solutions to the gKPPE, the number and qualitative properties of these bounded solutions are received. Furthermore, pulses (bell-shaped) and waves fronts (kink-shaped) of the gKPPE are given. In particular, two types of approximate analytical oscillatory solutions are constructed. Besides, the error estimations between the approximate analytical oscillatory solutions and the exact solutions of the gKPPE are obtained by the homogeneity principle. Finally, the approximate analytical oscillatory solutions are compared with the numerical solutions, which shows the two types of solutions are similar.

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 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.

A vertex-colored path $P$ is rainbow if its internal vertices have distinct colors; whereas $P$ is monochromatic if its internal vertices are colored the same. For a vertex-colored connected graph $G$, the rainbow vertex-connection number ${\rm rvc} (G)$ is the minimum number of colors used such that there is a rainbow path joining any two vertices of $G$; whereas the monochromatic vertex-connection number ${\rm mvc} (G)$ is the maximum number of colors used such that any two vertices of $G$ are connected by a monochromatic path. These two opposite concepts are the vertex-versions of rainbow connection number ${\rm rc} (G)$ and monochromatic connection number ${\rm mc} (G)$ respectively. The study on ${\rm rc} (G)$ and ${\rm mc} (G)$ of random graphs drew much attention, and there are few results on the rainbow and monochromatic vertex-connection numbers. In this paper, we consider these two vertex-connection numbers of random graphs and establish sharp threshold functions for them, respectively.

This paper concerns the controllability of autonomous and nonautonomous nonlinear discrete systems, in which linear parts might admit certain degeneracy. By introducing Fredholm operators and coincidence degree theory, sufficient conditions for nonlinear discrete systems to be controllable are presented. In addition, applications are given to illustrate main results.

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 consider the statistical inferences in a partially linear model when the model error follows an autoregressive process. A two-step procedure is proposed for estimating the unknown parameters by taking into account of the special structure in error. Since the asymptotic matrix of the estimator for the parametric part has a complex structure, an empirical likelihood function is also developed. We derive the asymptotic properties of the related statistics under mild conditions. Some simulations, as well as a real data example, are conducted to illustrate the finite sample performance.

This paper aims to prove the asymptotic behavior of the solution for the thermo-elastic von Karman system where the thermal conduction is given by Gurtin-Pipkins law. Existence and uniqueness of the solution are proved within the semigroup framework and stability is achieved thanks to a suitable Lyapunov functional. Therefore, the stability result clarified that the solutions energy functional decays exponentially at infinite time.

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.

Joint analysis of multiple phenotypes can have better interpretation of complex diseases and increase statistical power to detect more significant single nucleotide polymorphisms (SNPs) compare to traditional single phenotype analysis in genome-wide association analysis. Principle component analysis (PCA), as a popular dimension reduction method, has been broadly used in the analysis of multiple phenotypes. Since PCA transforms the original phenotypes into principal components (PCs), it is natural to think that by analyzing these PCs, we can combine information across phenotypes. Existing PCA-based methods can be divided into two categories, either selecting one particular PC manually or combining information from all PCs. In this paper, we propose an adaptive principle component test (APCT) which selects and combines the PCs adaptively by using Cauchy combination method. Our proposed method can be seen as a generalization of traditional PCA based method since it contains two existing methods as special situation. Extensive simulation shows that our method is robust and can generate powerful result in various situations. The real data analysis of stock mice data also demonstrate that our proposed APCT can identify significant SNPs that are missed by traditional methods.

In this paper we study the asymptotic behavior of the maximal position of a supercritical multiple catalytic branching random walk $(X_n)$ on $\mathbb Z$. If $M_n$ is its maximal position at time $n$, we prove that there is a constant $\alpha>0$ such that $M_n/n$ converges to $\alpha$ almost surely on the set of infinite number of visits to the set of catalysts. We also derive the asymptotic law of the centered process $M_n-\alpha n$ as $n\to \infty$. Our results are similar to those in [13]. However, our results are proved under the assumption of finite $L\log L$ moment instead of finite second moment. We also study the limit of $(X_n)$ as a measure-valued Markov process. For any function $f$ with compact support, we prove a strong law of large numbers for the process $X_n(f)$.

Let $(Z_{n})$ be a supercritical bisexual branching process in a random environment $\xi$. We study the almost sure (a.s.) convergence rate of the submartingale $\overline{W}_{n} =Z_{n}/I_{n}$ to its limit $\overline{W}$, where $(I_n)$ is an usually used norming sequence. We prove that under a moment condition of order $p \in (1,2),\overline{W}-\overline{W}_{n}=o(e^{-na})$ a.s. for some $a>0$ that we find explicitly; assuming the logarithmic moment condition holds, we have $\overline{W}-\overline{W}_{n}=o(n^{-\alpha})$ a.s.. In order to obtain these results, we provide the $L^{p}-$ convergence of $(\overline{W}_{n})$; similar conclusions hold for a bisexual branching process in a varying environment.

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.

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.

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.

Estimation of treatment effects is one of the crucial mainstays in economics and sociology studies. The problem will become more serious and complicated if the treatment variable is endogenous for the presence of unobserved confounding. The estimation and conclusion are likely to be biased and misleading if the endogeny of treatment variable is ignored. In this article, we propose the pseudo maximum likelihood method to estimate treatment effects in nonlinear models. The proposed method allows the unobserved confounding and random error terms to exist in an arbitrary relationship (such as, add or multiply), and the unobserved confounding have different influence directions on treatment variables and outcome variables. The proposed estimator is consistent and asymptotically normally distributed. Simulation studies show that the proposed estimator performs better than the special regression estimator, and the proposed method is stable for various distribution of error terms. Finally, the proposed method is applied to the real data that studies the influence of individuals have health insurance on an individual’s decision to visit a doctor.

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.

The alternating direction method of multipliers (ADMM) is one of the most successful and powerful methods for separable minimization optimization. Based on the idea of symmetric ADMM in two-block optimization, we add an updating formula for the Lagrange multiplier without restricting its position for multiblock one. Then, combining with the Bregman distance, in this work, a Bregman-style partially symmetric ADMM is presented for nonconvex multi-block optimization with linear constraints, and the Lagrange multiplier is updated twice with different relaxation factors in the iteration scheme. Under the suitable conditions, the global convergence, strong convergence and convergence rate of the presented method are analyzed and obtained. Finally, some preliminary numerical results are reported to support the correctness of the theoretical assertions, and these show that the presented method is numerically effective.

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.

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.

We consider non-autonomous ordinary differential equations in two cases. One is the one-dimensional case that admits a condition of hyperbolicity, and the other one is the higher-dimensional case that admits an exponential dichotomy. For differential equations of this kind, we give a rigorous treatment of the Harmonic Balance Method to look for almost periodic solutions.

In this paper, we analyze the existence, multiplicity and nonexistence of nontrivial radial convex solutions to the following system coupled by singular Monge-Ampère equations $$ \left \{ \begin{array}{l} \text{det}\ D^2u_1=\lambda h_1(|x|)f_1(-u_2), \qquad \text{in} \ \ \Omega,\\ \text{det}\ D^2u_2=\lambda h_2(|x|)f_2(-u_1), \qquad \text{in} \ \ \Omega,\\ u_1=u_2=0, \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \text{on} \ \ \partial \Omega \end{array} \right. $$ for a certain range of $\lambda >0$, $h_i$ are weight functions, $f_i$ are continuous functions with possible singularity at $0$ and satisfy a combined $N$-superlinear growth at $\infty$, where $i\in \{1,2\}$, $\Omega$ is the unit ball in $\mathbb{R}^N$. We establish the existence of a nontrivial radial convex solution for small $\lambda$, multiplicity results of nontrivial radial convex solutions for certain ranges of $\lambda$, and nonexistence results of nontrivial radial solutions for the case $\lambda\gg 1$. The asymptotic behavior of nontrivial radial convex solutions is also considered.

Let $\lambda K_{m,n}$ be a complete bipartite multigraph with two partite sets having $m$ and $n$ vertices, respectively. A $K_{p,q}$-factorization of $\lambda K_{m,n}$ is a set of $K_{p,q}$-factors of $\lambda K_{m,n}$ which partition the set of edges of $\lambda K_{m,n}$. When $\lambda =1$, Martin, in [Complete bipartite factorizations by complete bipartite graphs, Discrete Math., 167/168 (1997), 461-480], gave simple necessary conditions for such a factorization to exist, and conjectured those conditions are always sufficient. In this paper, we will study the $K_{p,q}$-factorization of $\lambda K_{m,n}$ for $p=1$, to show that the necessary conditions for such a factorization are always sufficient whenever related parameters are sufficiently large.

This paper studies a defined contribution (DC) pension fund investment problem with return of premiums clauses in a stochastic interest rate and stochastic volatility environment. In practice, most of pension plans were subject to the return of premiums clauses to protect the rights of pension members who died before retirement. In the mathematical modeling, we assume that a part of pension members could withdraw their premiums if they died before retirement and surviving members could equally share the difference between accumulated contributions and returned premiums. We suppose that the financial market consists of a risk-free asset, a stock, and a zero-coupon bond. The interest rate is driven by a stochastic affine interest rate model and the stock price follows the Heston’s stochastic volatility model with stochastic interest rates. Different fund managers have different risk preferences, and the hyperbolic absolute risk aversion (HARA) utility function is a general one including a power utility, an exponential utility, and a logarithm utility as special cases. We are concerned with an optimal portfolio to maximize the expected utility of terminal wealth by choosing the HARA utility function in the analysis. By using the principle of dynamic programming and Legendre transform-dual theory, we obtain explicit solutions of optimal strategies. Some special cases are also derived in detail. Finally, a numerical simulation is provided to illustrate our results.

The perfect matching polytope of a graph $G$ is the convex hull of the incidence vectors of all perfect matchings of $G$. A graph $G$ is PM-compact if the 1-skeleton graph of the prefect matching polytope of $G$ is complete. Equivalently, a matchable graph $G$ is PM-compact if and only if for each even cycle $C$ of $G$, $G-V(C)$ has at most one perfect matching. This paper considers the class of graphs from which deleting any two adjacent vertices or nonadjacent vertices, respectively, the resulting graph has a unique perfect matching. The PM-compact graphs in this class of graphs are presented.